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Signals and Communication Technology
Aparna Vyas
Soohwan Yu
Joonki Paik
Multiscale
Transforms with
Application
to Image
Processing
Signals and Communication Technology
More information about this series at http://www.springer.com/series/4748
Aparna Vyas Soohwan Yu
Joonki Paik
•
Multiscale Transforms
with Application to Image
Processing
123
Aparna Vyas
Image Processing and Intelligent Systems
Laboratory, Graduate School of Advanced
Imaging Science, Multimedia and Film
Chung-Ang University
Seoul
South Korea
Joonki Paik
Image Processing and Intelligent Systems
Laboratory, Graduate School of Advanced
Imaging Science, Multimedia and Film
Chung-Ang University
Seoul
South Korea
Soohwan Yu
Image Processing and Intelligent Systems
Laboratory, Graduate School of Advanced
Imaging Science, Multimedia and Film
Chung-Ang University
Seoul
South Korea
ISSN 1860-4862
ISSN 1860-4870 (electronic)
Signals and Communication Technology
ISBN 978-981-10-7271-0
ISBN 978-981-10-7272-7 (eBook)
https://doi.org/10.1007/978-981-10-7272-7
Library of Congress Control Number: 2017959155
© Springer Nature Singapore Pte Ltd. 2018
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
Printed on acid-free paper
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The registered company is Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
Preface
Digital image processing is a popular, rapidly growing area of electrical and
computer engineering. Digital image processing has enabled various intelligent
applications such as face recognition, signature recognition, iris recognition,
forensics, automobile detection, and military vision applications. Its growth is
leveraged by technological innovations in the fields of computer processing, digital
imaging, and mass storage devices. Traditional analog imaging applications are
now switching to digital systems to utilize their usability and affordability.
Important examples include photography, medicine, video production, remote
sensing, and security monitoring. These sources produce a huge volume of digital
image data every day. Theoretically, image processing can be considered as the
processing of a two-dimensional image using a digital computer. The outcome of
image processing could be an image, a set of features, or characteristics related to
the image. Most image processing methods treat an image as a two-dimensional
signal and implement standard signal processing techniques.
Many image processing techniques were of only academic interest because
of their computational complexity. However, recent advances in processing and
memory technology made image processing a vital and cost-effective technology in
a host of applications. Multi-scale image transformations, such as Fourier transform, wavelet transform, complex wavelet transform, quaternion wavelet transform,
ridgelet transform, contourlet transform, curvelet transform, and shearlet transform,
play an extremely crucial role in image compression, image denoising, image
restoration, image enhancement, and super-resolution. Fourier transform is a
powerful tool that has been available to signal and image analysis for many years.
However, the problem with using Fourier transform is frequency analysis cannot
offer high frequency and time resolution at the same time. To overcome this
problem, windowed Fourier transform or short-time Fourier transform was introduced. Although the short-time Fourier transform has the ability to provide time
information, a complete multiresolution analysis is not possible. Wavelet is a
solution to the multiresolution problem. A wavelet has the important property of not
having a fixed-width sampling window. The wavelet transform can be classified
into (i) continuous wavelet transform and (ii) discrete wavelet transform. The
v
vi
Preface
discrete wavelet transform (DWT) algorithms have a firm position in processing of
images in many areas of research and industry.
The main focus of classical wavelets includes compression and efficient representation. Important features which play a role in analysis of functions in two
variables are dilation, translation, spatial and frequency localization, and singularity
orientation. Singularities of functions in more than one variable vary in dimensionality. Important singularities in one dimension are simply points. In two
dimensions, zero- and one-dimensional singularities are important. A smooth singularity in two dimensions may be a one-dimensional smooth manifold. Smooth
singularities in two-dimensional images often occur as boundaries of physical
objects. Efficient representation in two dimensions is a hard problem. To overcome
this problem, new multi-scale transformations such as ridgelet transform, contourlet
transform, curvelet transform, and shearlet transform were introduced. Recently,
these multi-scale transforms have become increasingly important in image
processing.
In this book, we will provide a complete introduction of multi-scale image
transformations followed by their applications to various image processing algorithms including image denoising, image restoration, image enhancement, and
super-resolution. The book is mainly divided into three parts. The readers will learn
about the basic introduction of image processing in the first part in Chaps. 1 and 2.
The second part starts with Fourier transform followed by wavelet transform and
new multi-scale constructions. The third part deals with applications of the
multi-scale transform in image processing.
The chapters of the present book consist of both tutorial and advanced theory.
Therefore, the book is intended to be a reference for graduate students and
researchers to obtain state-of-the-art knowledge on multi-scale image processing
applications. The technique of solving problems in the transform domain is common in applied mathematics as used in research and industry, but we do not devote
as much time to it as we should in the undergraduate curriculum. Also, the book is
intended to be used as a reference manual for scientists who are engaged in image
processing research, developers of image processing hardware and software systems, and practicing engineers and scientists who use image processing as a tool in
their applications.
Appendices summarize mostly used mathematical background in the book.
Seoul, South Korea
Aparna Vyas
Soohwan Yu
Joonki Paik
Contents
Part I
Introduction to Image Processing
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2 Fourier Analysis and Fourier Transform . . . . . . . . . . . . . .
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Periodic Functions . . . . . . . . . . . . . . . . . . . . .
2.2.2 Frequency and Amplitude . . . . . . . . . . . . . . . .
2.2.3 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Fourier Series of Periodic Functions . . . . . . . .
2.2.5 Complex Form of Fourier Series . . . . . . . . . . .
2.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 2D-Fourier Transform . . . . . . . . . . . . . . . . . . .
2.3.2 Properties of Fourier Transform . . . . . . . . . . .
2.4 Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . .
2.4.1 1D-Discrete Fourier Transform . . . . . . . . . . . .
2.4.2 Inverse 1D-Discrete Fourier Transform . . . . . .
2.4.3 2D-Discrete Fourier Transform and 2D-Inverse
Discrete Fourier Transform . . . . . . . . . . . . . . .
2.4.4 Properties of 2D-Discrete Fourier Transform . .
2.5 Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . .
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1 Fundamentals of Digital Image Processing
1.1 Image Acquisition of Digital Camera .
1.1.1 Introduction . . . . . . . . . . . . .
1.2 Sampling . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
Part II
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Multiscale Transform
vii
viii
Contents
2.6
The Discrete Cosine Transform . . . . . . . . . . . . . . . .
2.6.1 1D-Discrete Cosine Transform . . . . . . . . . .
2.6.2 2D-Discrete Cosine Transform . . . . . . . . . .
2.7 Heisenberg Uncertainty Principle . . . . . . . . . . . . . . .
2.8 Windowed Fourier Transform or Short-Time Fourier
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1 1D and 2D Short-Time Fourier Transform . .
2.8.2 Drawback of Short-Time Fourier Transform
2.9 Other Spectral Transforms . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Wavelets and Wavelet Transform . . . . . . . . . . . . . . . . . . . . .
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Multiresolution Analysis . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 The Wavelet Series Expansions . . . . . . . . . . . . .
3.4.2 Discrete Wavelet Transform . . . . . . . . . . . . . . .
3.4.3 Motivation: From MRA to Discrete Wavelet
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 The Quadrature Mirror Filter Conditions . . . . . .
3.5 The Fast Wavelet Transform . . . . . . . . . . . . . . . . . . . . .
3.6 Why Use Wavelet Transforms . . . . . . . . . . . . . . . . . . . .
3.7 Two-Dimensional Wavelets . . . . . . . . . . . . . . . . . . . . . .
3.8 2D-discrete Wavelet Transform . . . . . . . . . . . . . . . . . . .
3.9 Continuous Wavelet Transform . . . . . . . . . . . . . . . . . . .
3.9.1 1D Continuous Wavelet Transform . . . . . . . . . .
3.9.2 2D Continuous Wavelet Transform . . . . . . . . . .
3.10 Undecimated Wavelet Transform or Stationary Wavelet
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Biorthogonal Wavelet Transform . . . . . . . . . . . . . . . . . .
3.11.1 Linear Independence and Biorthogonality . . . . .
3.11.2 Dual MRA . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11.3 Discrete Transform for Biorthogonal Wavelets . .
3.12 Scarcity of Wavelet Transform . . . . . . . . . . . . . . . . . . .
3.13 Complex Wavelet Transform . . . . . . . . . . . . . . . . . . . . .
3.14 Dual-Tree Complex Wavelet Transform . . . . . . . . . . . . .
3.15 Quaternion Wavelet and Quaternion Wavelet Transform .
3.15.1 2D Hilbert Transform . . . . . . . . . . . . . . . . . . . .
3.15.2 Quaternion Algebra . . . . . . . . . . . . . . . . . . . . .
3.15.3 Quaternion Multiresolution Analysis . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Contents
ix
4 New Multiscale Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Ridgelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The Continuous Ridgelet Transform . . . . . . . . . . . .
4.2.2 Discrete Ridgelet Transform . . . . . . . . . . . . . . . . . .
4.2.3 The Orthonormal Finite Ridgelet Transform . . . . . . .
4.2.4 The Fast Slant Stack Ridgelet Transform . . . . . . . . .
4.2.5 Local Ridgelet Transform . . . . . . . . . . . . . . . . . . . .
4.2.6 Sparse Representation by Ridgelets . . . . . . . . . . . . .
4.3 Curvelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 The First Generation Curvelet Transform . . . . . . . . .
4.3.2 Sparse Representation by First Generation Curvelets
4.3.3 The Second-Generation Curvelet Transform . . . . . . .
4.3.4 Sparse Representation by Second Generation
Curvelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Contourlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Contourlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Multiscale Decomposition . . . . . . . . . . . . . . . . . . . .
4.5.2 Directional Decomposition . . . . . . . . . . . . . . . . . . .
4.5.3 The Discrete Contourlet Transform . . . . . . . . . . . . .
4.6 Shearlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Shearlet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7.1 Continuous Shearlet Transform . . . . . . . . . . . . . . . .
4.7.2 Discrete Shearlet Transform . . . . . . . . . . . . . . . . . .
4.7.3 Cone-Adapted Continuous Shearlet Transform . . . . .
4.7.4 Cone-Adapted Discrete Shearlet Transform . . . . . . .
4.7.5 Compactly Supported Shearlets . . . . . . . . . . . . . . . .
4.7.6 Sparse Representation by Shearlets . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Part III
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Application of Multiscale Transforms to Image Processing
5 Image Restoration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Model of Image Degradation and Restoration Process .
5.2 Image Quality Assessments Metrics . . . . . . . . . . . . . .
5.3 Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Noise Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Additive Noise Model . . . . . . . . . . . . . . . . .
5.4.2 Multiplicative Noise Model . . . . . . . . . . . . . .
5.5 Types of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Amplifier (Gaussian) Noise . . . . . . . . . . . . . .
5.5.2 Rayleigh Noise . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Uniform Noise . . . . . . . . . . . . . . . . . . . . . . .
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Contents
5.5.4 Impulsive (Salt and Pepper) Noise . . . . . . . . . . .
5.5.5 Exponential Noise . . . . . . . . . . . . . . . . . . . . . . .
5.5.6 Speckle Noise . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Image Deblurring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Gaussian Blur . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Motion Blur . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6.3 Rectangular Blur . . . . . . . . . . . . . . . . . . . . . . . .
5.6.4 Defocus Blur . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Superresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Classification of Image Restoration Algorithms . . . . . . . .
5.8.1 Spatial Filtering . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.2 Frequency Domain Filtering . . . . . . . . . . . . . . . .
5.8.3 Direct Inverse Filtering . . . . . . . . . . . . . . . . . . . .
5.8.4 Constraint Least-Square Filter . . . . . . . . . . . . . . .
5.8.5 IBD (Iterative Blind Deconvolution) . . . . . . . . . .
5.8.6 NAS-RIF (Nonnegative and Support Constraints
Recursive Inverse Filtering) . . . . . . . . . . . . . . . .
5.8.7 Superresolution Restoration Algorithm Based on
Gradient Adaptive Interpolation . . . . . . . . . . . . .
5.8.8 Deconvolution Using a Sparse Prior . . . . . . . . . .
5.8.9 Block-Matching . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.10 LPA-ICI Algorithm . . . . . . . . . . . . . . . . . . . . . .
5.8.11 Deconvolution Using Regularized Filter (DRF) . .
5.8.12 Lucy-Richardson Algorithm . . . . . . . . . . . . . . . .
5.8.13 Neural Network Approach . . . . . . . . . . . . . . . . .
5.9 Application of Multiscale Transform in Image Restoration
5.9.1 Image Restoration Using Wavelet Transform . . . .
5.9.2 Image Restoration Using Complex Wavelet
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.3 Image Restoration Using Quaternion Wavelet
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9.4 Image Restoration Using Ridgelet Transform . . . .
5.9.5 Image Restoration Using Curvelet Transform . . . .
5.9.6 Image Restoration Using Contourlet Transform . .
5.9.7 Image Restoration Using Shearlet Transform . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Image Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Spatial Domain Image Enhancement Techniques . .
6.2.1 Gray Level Transformation . . . . . . . . . . . .
6.2.2 Piecewise-Linear Transformation Functions
6.2.3 Histogram Processing . . . . . . . . . . . . . . . .
6.2.4 Spatial Filtering . . . . . . . . . . . . . . . . . . . .
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Contents
6.3
Frequency Domain Image Enhancement Techniques . . . . .
6.3.1 Smoothing Filters . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Sharpening Filters . . . . . . . . . . . . . . . . . . . . . . .
6.3.3 Homomorphic Filtering . . . . . . . . . . . . . . . . . . .
6.4 Colour Image Enhancement . . . . . . . . . . . . . . . . . . . . . . .
6.5 Application of Multiscale Transforms in Image
Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1 Image Enhancement Using Fourier Transform . . .
6.5.2 Image Enhancement Using Wavelet Transform . .
6.5.3 Image Enhancement Using Complex Wavelet
Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.4 Image Enhancement Using Curvelet Transform . .
6.5.5 Image Enhancement Using Contourlet Transform .
6.5.6 Image Enhancement Using Shearlet Transform . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
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219
222
223
225
228
Appendix A: Real and Complex Number System . . . . . . . . . . . . . . . . . . . 233
Appendix B: Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Appendix C: Linear Transformation, Matrices . . . . . . . . . . . . . . . . . . . . 239
Appendix D: Inner Product Space and Orthonormal Basis . . . . . . . . . . . 241
Appendix E: Functions and Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 245
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
About the Authors
Aparna Vyas was born in Allahabad, India, in 1983. She received her B.Sc. in
Science and M.Sc. in Mathematics from the University of Allahabad, Allahabad,
India, in 2004 and 2006, respectively. She received her Ph.D. degree in
Mathematics from the University of Allahabad, Allahabad, India, in 2010. She was
Assistant Professor, Department of Mathematics, School of Basic Sciences,
SHIATS, Allahabad, India, since 2006–2013. In 2014, she joined Manav Rachna
University, Faridabad, India, as Assistant Professor. She was a postdoctoral fellow
at Soongsil University from August 2016 to September 2016. Currently, she is a
postdoctoral fellow in Chung-Ang University under the BK21 Plus Project. She has
more than 10 years of teaching and research experience. She is also a life member
of the Indian Mathematical Society and Ramanujan Mathematical Society. Her
research interests include wavelet analysis and image processing.
Soohwan Yu was born in Incheon, Korea, in 1988. He received his B.S. in
Information and Communication Engineering from Suwon University, Korea, in
2013. He received his M.S. in Image Engineering from Chung-Ang University,
Korea, in 2016, where he is currently pursuing his Ph.D. in Image Engineering. His
research interests include image enhancement, super-resolution, and image
restoration.
Joonki Paik was born in Seoul, Korea, in 1960. He received his B.S. in Control
and Instrumentation Engineering from Seoul National University in 1984. He
received his M.S. and Ph.D. degrees in Electrical Engineering and in Computer
Science from Northwestern University in 1987 and in 1990, respectively. From
1990 to 1993, he worked at Samsung Electronics, where he designed image stabilization chipsets for consumer camcorders. Since 1993, he has been a faculty
member at Chung-Ang University, Seoul, South Korea. Currently, he is Professor
in the Graduate School of Advanced Imaging Science, Multimedia and Film. From
1999 to 2002, he was Visiting Professor in the Department of Electrical and
Computer Engineering at the University of Tennessee, Knoxville. Dr. Paik was the
recipient of the Chester-Sall Award from the IEEE Consumer Electronics Society,
the Academic Award from the Institute of Electronic Engineers of Korea, and the
xiii
xiv
About the Authors
Best Research Professor Award from Chung-Ang University. He served the
Consumer Electronics Society of IEEE as a member of the editorial board. Since
2005, he has been the head of the National Research Laboratory in the field of
image processing and intelligent systems. In 2008, he worked as a full-time technical consultant for the System LSI division at Samsung Electronics, where he
developed various computational photographic techniques including an extended
depth-of-field (EDoF) system. From 2005 to 2007, he served as Dean of the
Graduate School of Advanced Imaging Science, Multimedia and Film. From 2005
to 2007, he was Director of the Seoul Future Contents Convergence (SFCC) Cluster
established by the Seoul Research and Business Development (R&BD) Program.
Dr. Paik is currently serving as a member of the Presidential Advisory Board for
Scientific/Technical Policy for the Korean government and as a technical consultant
for the Korean Supreme Prosecutors Office for computational forensics.
Part I
Introduction to Image Processing
Chapter 1
Fundamentals of Digital Image Processing
1.1 Image Acquisition of Digital Camera
1.1.1 Introduction
The concept of the digital image was first introduced in the transportation of the
digital image using submarine cable system in the early twenty century [3]. In addition, the advance in the computational hardware and processing unit lead to the
development of modern digital image processing techniques. Specifically, the digital
image processing started in the application field of remote sensing. In 1964, the Jet
Propulsion Laboratory applied the digital image processing technique to improve
the visual quality of the transmitted digital image by Ranger 7 [1, 3]. In the medical
imaging, the image processing techniques were applied to develop the computerized tomography for medical imaging devices in early 1970s, which generates a
two-dimensional image and three-dimensional volume of the inside of the object by
passing the X-ray [3]. In addition to the remote sensing and medical imaging, the
digital image processing techniques have been widely used in various application
fields such as consumer electronics, defense, robot vision, surveillance systems, and
artificial intelligence systems.
In the modern image acquisition system, the image signal processing (ISP) chain
plays an important role to obtain the high-quality digital image as shown in Fig. 1.1.
The light pass the lens and color filter array (CFA). Since the imaging sensor without the CFA absorbs the light in all spectrum bands, we cannot obtain the color
information. To generate the color image, the digital camera uses the common CFA
called Bayer pattern, which consists of two green (G), one red (R), and one blue
(B) filter because the human visual system is more sensitive to the light in the green
wavelength [2]. The advanced CFA replaces the one green filter with white filter to
increase the amount of light.
In an imaging sensor such as charge coupled device (CCD) or complementary
metal oxide semiconductor (CMOS), the photon reacts to each semiconductor and
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_1
3
4
1 Fundamentals of Digital Image Processing
Fig. 1.1 The block diagram of the image signal processing chain
converts the electrical charges to the electric analog signal. The analog front end
(AFE) module of ISP chain performs the sampling and quantization processes to
convert the analog signal to the digital signal. Sequentially, the AFE module also
controls the gain of the acquired signal to increase the signal-to-noise ratio (SNR).
In low-illumination condition, since the amount of the photons is decreased to react
an imaging sensor, the digital image having low contrast is acquired with low SNR
[4, 5, 7, 8]. In addition, the recent imaging devices increases the spatial resolution
of a digital image by drastically reducing the physical size of each pixel. However,
the reduced pixel size results in the chrominance noise called the cross-talk because
of the interference of the photons among the pixels and it also reduces the SNR in
each color channel.
The digital back end (DBE) module performs the digital image processing techniques to improve the quality of an input image. First, the DBE module performs
the demosaicing to separate the color information from the raw image data by using
the interpolation algorithms [6]. In addition, the image enhancement techniques are
performed to improve the dynamic range of an image. The auto white balance (AWB)
performs the color constancy, which makes the digital image be acquired under the
neutral light condition by correcting the chromaticity. Finally, the noise reduction
should be performed to remove the amplified noise in the image enhancement process.
Additionally, since the demosaicing and noise reduction techniques may decrease
the quality of the image by the blurring and jagging artifacts, the image restoration
techniques called super-resolution can be performed to obtain the high-resolution
image.
1.2 Sampling
The sampling and quantization are major processes performed in the AFE module of
ISP chain to convert a continuous image signal to a series of discrete signals. In this
section, we briefly describe the mathematical background of the sampling theorem.
1.2 Sampling
5
Fig. 1.2 The sampling operation of a continuous function using an impulse train.: a 1D continuous
function, b impulse train with period T , and c the sampled function by the multiplication of a and b
Let x(t) be a one-dimensional (1D) continuous function, the sampling operation
can be regarded as the multiplication of x(t) and impulse train p(t) with period T ,
for k = · · · , −2, −1, 0, 1, 2, . . ., as
xs (t) = x(t) p(t) = x(t)
∞
δ(t − kT ).
(1.2.1)
k=−∞
Figure 1.2 shows the sampling operation of an 1D continuous function using an
impulse train. As shown in Fig. 1.2, a continuous function is sampled with interval
T and the amplitude of an impulse train varies with that of the continuous function
x(t). The black dots represent the sampled values of x(t) at the location kT .
Since the impulse train is a periodic function with period T , it can be expressed
as the Fourier series expansion as
p(t) =
∞
n=−∞
an e j
2πk
T
t
,
(1.2.2)
6
1 Fundamentals of Digital Image Processing
where
an =
1
T
−T /2
T /2
1
T
1
= .
T
=
T /2
−T /2
p(t)e− j
δ(t)e− j
2πk
T t
2πk
T t
dt
dt
(1.2.3)
The Fourier transform of the impulse train is defined by using (1.2.2) as
∞
p(t)e− j2πut dt
∞
∞
1 j 2πk t − j2πut
=
e T
dt
e
T k=−∞
−∞
∞ ∞
1 k
=
e− j2π(u− T )t dt
T k=−∞ −∞
∞
1 k
=
δ u−
,
T k=−∞
T
P(u) =
−∞
(1.2.4)
where P(u) is the Fourier transform of p(t). The Fourier transform of impulse
train is an impulse train with period 1/T . In addition, since the multiplication of
Fourier transformed functions in the spatial domain is the convolution in the frequency domain, the Fourier transform of the sampled function in (1.2.1) can be
expressed as
X s (u) = X (u) ∗ P(u)
∞
=
X (τ )P(u − τ )dτ
−∞
∞
1 k
.
=
X u−
T k=−∞
T
(1.2.5)
where X s (u) represents the Fourier transform of the sampled function xs (t), and ∗
the convolution operator. The Fourier transform of the sampled function xs (t) is also
a sequence of the Fourier transform of x(t) at the location k/T with period 1/T .
Figure 1.3 shows that how a continuous function is sampled in the different
sampling rate 1/T in the frequency domain. Figure 1.3a is the Fourier transform of a continuous function x(t), which is filtered using the band-limit filter in
−u max ≤ u ≤ u max . Figure 1.3b shows the Fourier transform of the sampled function
with the higher sampling rate. Since the sampled signal is completely separated, X (u)
can be reconstructed from X s (u) using the band-limit filter which used in Fig. 1.3a.
1.2 Sampling
7
Fig. 1.3 Comparative results using different sampling rate.: a the Fourier transform of band-limited
continuous function, b the Fourier transform of the sampled function using T1 < 2u max , and c the
Fourier transform of the sampled function using T1 > 2u max
On the other hand, if the band-limited signal is sampled at lower sampling rate, the
Fourier transform of the sampled function is overlapped as shown in Fig. 1.3c. It
implies that the sampling operation is performed at the sampling rate higher than
twice the maximum frequency u max to completely reconstruct X (u). This is called
as Nyquist-Shannon sampling theorem.
1
> 2u max .
T
(1.2.6)
In the two-dimensional (2D) case, the sampling can be performed using an 2D
impulse train. Given the 2D continuous function x(t, q) and impulse train p(t, q),
the 2D sampled function xs (t, q) can be written as
xs (t, q) = x(t, q) p(t, q) = x(t, q)
∞
∞
δ(t − kT )δ(q − l Q).
(1.2.7)
l=−∞ k=−∞
Since the 2D impulse train is a periodic function, it can be expressed the Fourier
series expansion as 1D impulse train in (1.2.2). The Fourier series expansion of 2D
impulse train is defined as
8
1 Fundamentals of Digital Image Processing
∞
∞
p(t) =
bn e j
2πk
T t
2πl
Q
ej
q
,
(1.2.8)
l=−∞ k=−∞
where
1
bn =
TQ
Q/2
−T /2
T /2
−Q/2
Q/2
−T /2
−Q/2
1
TQ
1
=
.
TQ
=
T /2
p(t, q)e− j
δ(t, q)e− j
2πk
T t
2πk
T
e− j
2πl
Q
q
t − j 2πl
Q q
e
dtdq
dtdq
(1.2.9)
The Fourier transform of p(t, q) is defined as
∞
∞
p(t, q)e− j2πut e− j2πvq dtdq
∞ ∞
∞
∞
1 j 2πk t j 2πl
q
=
e T e Q
e− j2πut e− j2πvq dtdq
T
Q
−∞ −∞
l=−∞ k=−∞
P(u, v) =
−∞
−∞
∞ ∞ ∞
∞
1 k
− j2π v− Ql q
=
e− j2π(u− T )t e
dtdq
T Q l=−∞ k=−∞ −∞ −∞
∞
∞
1 l
k
=
δ v−
,
δ u−
T Q l=−∞ k=−∞
T
Q
(1.2.10)
The Fourier transform of 2D impulse train is a periodic impulse train with period
1/T and 1/Q in u and v directions. Let X s (u, v) be the Fourier transform of 2D
sampled function xs (t, q), X s (u, v) can be estimated using the convolution theorem
in the frequency domain as
X s (u, v) = X (u, v) ∗ P(u, v)
∞ ∞
=
X (τu , τv )P(u − τu , v − τv )dτu dτv
−∞ −∞
=
1
TQ
=
1
TQ
=
1
TQ
k
l
δ u − τu −
δ v − τv −
dτu dτv
T
Q
−∞ −∞
l=−∞ k=−∞
∞
∞ ∞ ∞
k
l
X (τu , τv )δ u − τu −
δ v − τv −
dτu dτv
T
Q
l=−∞ k=−∞ −∞ −∞
∞
∞
k
l
X u − ,v −
.
T
Q
∞
∞
X (τu , τv )
∞
∞
l=−∞ k=−∞
(1.2.11)
1.2 Sampling
9
Fig. 1.4 The spectrum of the sampled function with periods 1/T and 1/Q in the frequency domain
In the same manner as the sampling in the 1D case, the Fourier transform of
the sampling in the spatial domain results in the multiply copied version of the
frequency spectrum X (u, v) at the location k/T and l/Q. Figure 1.4 shows that the
periodic frequency spectrum of the sampled function with periods 1/T and 1/Q in
the frequency domain.
In order to reconstruct the original 2D continuous signal, the Nyquist-Shannon
sampling rate should be satisfied as
1
< 2u max ,
T
(1.2.12)
1
< 2vmax .
Q
(1.2.13)
and
where u max and vmax respectively represent the maximum frequency of sampled
spectrum in the u and v directions. Figure 1.5 shows that the spectrum of the sampled
function using the sampling rate lower than the twice of the maximum frequency
along the u and v directions, respectively.
10
1 Fundamentals of Digital Image Processing
Fig. 1.5 The Fourier transform of the sampled function with the sampling rate than the NyquistShannon sampling rate: a the spectrum of sampled function using T1 < 2u max , and b the spectrum
of sampled function using Q1 < 2vmax
References
1. Andrews, H.C., Tescher, A.G., Kruger, R.P.: Image processing by digital computer. IEEE Spectr.
9(7), 20–32 (1972)
2. Bayer, B.E.: Color imaging array. U.S. Patent No. 3,971,065 (1976)
3. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Prentice Hall, New Jersey
(2006)
4. Ko, S., Yu, S., Kang, W., Park, C., Lee, S., Paik, J.: Artifact-free low-light video enhancement
using temporal similarity and guide map. IEEE Trans. Ind. Electron. 64(8), 6392–6401 (2017)
References
11
5. Ko, S, Yu, S., Kang, W., Park, S., Moon, B., Paik, J.: Variational framework for low-light image
enhancement using optimal transmission map and combined l1 and l2-minimization. Signal
Process.: Image Commun. 58, 99–110 (2017)
6. Malvar, H.S., He, L., Cutler, R.: High quality linear interpolation for demosaicing of bayerpatterned color images. Proc IEEE Int. Conf. Acoust. Speech Signal Process. 34(11), 2274–2282
(2004)
7. Park, S., Yu, S., Moon, B., Ko, S., Paik, J.: Low-light image enhancement using variational
optimization-based retinex model. IEEE Trans. Consum. Electron. 63(2), 178–184 (2017)
8. Park, S., Moon, B., Ko, S., Yu, S., Paik, J.: Low-light image restoration using bright channel
prior-based variational retinex model. EURASIP J. Image Video Process. 2017(1), 1–11 (2017)
Part II
Multiscale Transform
Chapter 2
Fourier Analysis and Fourier Transform
2.1 Overview
The origins of Fourier analysis in science can be found in Ptolemy’s decomposing
celestial orbits into cycles and epicycles and Pythagoras’ decomposing music into
consonances. Its modern history began with the eighteenth century work of Bernoulli,
Euler, and Gauss on what later came to be known as Fourier series. J. Fourier in
1822 [Theorie analytique de la Chaleur] was the first to claim that arbitrary periodic
functions could be expanded in a trigonometric (later called a Fourier) series, a
claim that was eventually shown to be incorrect, although not too far from the truth.
It is an amusing historical sidelight that this work won a prize from the French
Academy, in spite of serious concerns expressed by the judges (Laplace, Lagrange,
and Legendre) regarding Fourier’s lack of rigor. Fourier was apparently a better
engineer than mathematician. Dirichlet later made rigorous the basic results for
Fourier series and gave precise conditions under which they applied. The rigorous
theoretical development of general Fourier transforms did not follow until about one
hundred years later with the development of the Lebesgue integral.
Fourier analysis is a prototype of beautiful mathematics with many-faceted applications not only in mathematics, but also in science and engineering. Since the work
on heat flow of Jean Baptiste Joseph Fourier (March 21, 1768 to May 16, 1830) in the
treatise entitled Theorie Analytique de la Chaleur, Fourier series and Fourier transforms have gone from triumph to triumph, permeating mathematics such as partial
differential equations, harmonic analysis, representation theory, number theory and
geometry. Their societal impact can best be seen from spectroscopy to the effect that
atoms, molecules and hence matters can be identified by means of the frequency
spectrum of the light that they emit. Equipped with the fast Fourier transform in
computations and fulled by recent technological innovations in digital signals and
images, Fourier analysis has stood out as one of the very top achievements of mankind
comparable with the Calculus of Sir Isaac Newton.
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_2
15
16
2 Fourier Analysis and Fourier Transform
The Fourier transform is of fundamental importance to image processing. It allows
us to perform tasks which would be impossible to perform any other way; its efficiency allows us to perform other tasks more quickly. The Fourier Transform provides, among other things, a powerful alternative to linear spatial filtering; it is more
efficient to use the Fourier transform than a spatial filter for a large filter. The Fourier
Transform also allows us to isolate and process particular image frequencies, and so
perform low-pass and high-pass filtering with a great degree of precision.
2.2 Fourier Series
The concept of frequency and the decomposition of waveforms into elementary
“harmonic” functions first arose in the context of music and sound.
2.2.1 Periodic Functions
A periodic function is a function that repeats its value in regular intervals or periods.
A function f is said to be periodic with period T (T = 0) if f (x + T ) = f (x) for
all values of x in the domain. The most important examples are the trigonometric
functions (i.e. sine or cosine), which repeat values over the intervals of 2π.
The sine function f (x) = sin(x) has the value 0 at the origin and performs exactly
one full cycle between the origin and the point x = 2π. Hence f (x) = sin(x) is a
periodic function with period 2π, i.e.
sin(x) = sin(x + 2π) = sin(x + 4π) = · · · = sin(x + 2nπ),
(2.2.1)
for all n ∈ Z. The same is true for cosine function except its value is 1 at the origin
i.e. cos(0) = 1, see Fig. 2.1.
2.2.2 Frequency and Amplitude
The number of oscillations of sin(x) over the distance T = 2π is one and thus the
value of the angular frequency ω = 2π/T = 1. If f (x) = sin(3x), we obtain a
compressed sine wave that oscillates three times faster than the original function
sin(x). The function sin(3x) performs five full cycles over a distance of 2π and thus
has the angular frequency ω = 3 and a period T = 2π/3, see Fig. 2.2.
In general, the period T relates the angular frequency ω as
T =
2π
ω
(2.2.2)
2.2 Fourier Series
Fig. 2.1 Sine and Cosine Graph
Fig. 2.2 sin(x) and sin(3x) Graph
17
18
2 Fourier Analysis and Fourier Transform
for ω > 0. The relationship between the angular frequency ω and the common
frequency f is given by
f =
ω
1
=
T
2π
or
ω = 2π f,
(2.2.3)
where f is measured in cycles per length or time unit. A sine or cosine function
oscillates between the peak values +1 and −1 and its amplitude is 1. Multiplying
by a constant a ∈ R changes the peak values of the function to +a and −a and its
amplitude to a. In general, the expression
a · sin(ωx)
and
a · cos(ωx)
(2.2.4)
denotes a sine or cosine function with amplitude a and angular velocity ω, evaluated
at position (or point in time) x.
2.2.3 Phase
Phase is the position of a point in time (an instant) on a waveform cycle. A complete
cycle is defined as the interval required for the waveform to return to its arbitrary
initial value. In sinusoidal functions or in waves “phase” has two different, but closely
related, meanings. One is the initial angle of a sinusoidal function at its origin and is
sometimes called phase offset or phase difference. Another usage is the fraction of
the wave cycle that has elapsed relative to the origin.
Shifting a sine function along the x-axis by distance ϕ,
sin(x) → sin(x − ϕ),
(2.2.5)
changes the phase of the sine wave and ϕ denotes the phase angle of the resulting
function, see Fig. 2.3. Thus, we have
sin(nx) = cos(ωx − π/2).
(2.2.6)
i.e. cosine and sine functions are orthogonal in a sense and we can use this fact to
create new sinusoidal function with arbitrary frequency, phase and amplitude. In
particular, adding a cosine and a sine function with the identical frequency ω and
arbitrary amplitude A and B, respectively create another sinusoids
A · cos(ωx) + B · sin(ωx) = C · cos(ωx − ϕ).
where c =
√
A2 + B 2 and ϕ = tan −1 (B/A).
(2.2.7)
2.2 Fourier Series
19
Fig. 2.3 sin(x), sin(x − π/4) and sin(x − π) Graph
2.2.4 Fourier Series of Periodic Functions
As we seen earlier, sinusoidal function of arbitrary frequency, amplitude and phase
can be described as the sum of suitably weighted cosine and sine functions. Is it
possible to write non-sinusoidal functions to sum of cosine and sine functions? It
was Fourier [Jean Baptiste Joseph de Fourier (1768–1830)] who first extended this
idea to arbitrary functions and showed that (almost) any periodic function f (x) with
a fundamental frequency ω0 can be described as a infinite sum of harmonic sinusoids
i.e.
∞
A0 +
f (x) =
[An cos(ω0 nx) + Bn sin(ω0 nx)].
(2.2.8)
2
n=1
This is called Fourier series and A0 , An and Bn are called Fourier coefficients of the
function f (x), where
1 π
A0 =
f (x)d x,
(2.2.9)
π −π
An =
1
π
Bn =
1
π
π
−π
π
−π
f (x)cos(ω0 nx)d x,
(2.2.10)
f (x)sin(ω0 nx)d x.
(2.2.11)
20
2 Fourier Analysis and Fourier Transform
A Fourier series is an expression of a periodic function f (x) in terms of an infinite
sum of sines and cosines. Fourier series make use of the orthogonality relationships of
sine and cosine functions, since these functions form a complete orthogonal system
over [−π, π] or any interval of length 2π. The computation and study of Fourier
series is known as Harmonic analysis and is extremely useful as a way to break up an
arbitrary periodic function into a set of simple terms that can be plugged in, solved
individually and then recombined to obtain the solution of the original problem of
an approximation to it to whatever accuracy is desired or practical.
More general form of Fourier series is
∞
f (x) =
A0 +
[An cos(nx) + Bn sin(nx)],
2
n=1
where
A0 =
1
π
π
1
An =
π
1
Bn =
π
−π
π
−π
π
(2.2.12)
f (x)d x,
(2.2.13)
f (x)cos(nx)d x,
(2.2.14)
f (x)sin(nx)d x.
(2.2.15)
−π
The miracle of Fourier series is that as long as f (x) is continuous (or even piecewisecontinuous, with some caveats discussed in the Stewart text), such a decomposition
is always possible.
2.2.5 Complex Form of Fourier Series
The Fourier series representation for a periodic function f, can be expressed more
simply using complex exponentials. Moreover, because of the unique properties of
the exponential function, Fourier series are often easier to manipulate in complex
form. The transition from the real form to the complex form of a Fourier series is
made using the following identities, called Euler identities,
eiθ = cos(θ) + isin(θ) and
e−iθ = cos(θ) − isin(θ).
(2.2.16)
By adding these identities, and then dividing by 2, or subtracting them, and then
dividing by 2i, we have
cos(θ) =
eiθ + e−iθ
2
and
sin(θ) =
eiθ − e−iθ
.
2i
(2.2.17)
2.2 Fourier Series
21
The complex Fourier series is obtained from (2.2.12) by writing cos(nx) and sin(nx)
in their complex exponential form and rearranging as follows:
inx
inx
∞ e + e−inx
e − e−inx
A0 An
+
+ Bn
f (x) =
2
2
2i
n=1
=
∞ −1 A−m + i B−m imx
A0 An − i Bn inx
+
e +
e
2
2
2
m=−∞
n=1
where we substituted m = n in the last term on the last line. Equation clearly suggests
the much simpler complex form of the Fourier series
f (x) =
∞
Cn einx ,
(2.2.18)
f (x)e−inx d x.
(2.2.19)
n=−∞
with the coefficients given by
Cn =
1
2π
π
−π
Note that the Fourier coefficients Cn are complex valued. It is seen that for a realvalued function f (x), the following holds for the complex coefficients Cn
C−n = Cn ,
(2.2.20)
where Cn denotes the complex conjugate of Cn .
2.3 Fourier Transform
In the previous section we have seen how to expand a periodic function as a trigonometric series. This can be thought of as a decomposition of a periodic function in
terms of elementary modes, each of which has a definite frequency allowed by the
periodicity. This concept can be generalized to functions periodic on any interval.
If the function has period L, then the frequencies must be integer multiples of
the fundamental frequency k = 2π/L. The Fourier series of functions of arbitrary
periodicity is
∞
f (x) =
A0 +
[An cos(2πnx/L) + Bn sin(2πnx/L)],
2
n=1
(2.3.1)
22
2 Fourier Analysis and Fourier Transform
where
1
A0 =
L
1
An =
L
Bn =
1
L
L/2
−L/2
L/2
−L/2
L/2
f (x)d x,
(2.3.2)
f (x)cos(2πnx/L)d x,
(2.3.3)
f (x)sin(2πnx/L)d x,
(2.3.4)
−L/2
or in the exponential notation,
f (x) =
∞
Cn ei2πnx/L ,
(2.3.5)
f (x)e−i2πnx/L d x.
(2.3.6)
n=−∞
where
1
Cn =
L
L/2
−L/2
Fourier series was a powerful one and forms the backbone of the Fourier transform.
The Fourier transform can be viewed as an extension of the above Fourier series to
non-periodic functions and allows us to deal with non-periodic functions. A nonperiodic function can be thought of as a periodic function in the limit L → ∞.
Clearly, the larger L is, the less frequently the function repeats, until in the limit
L → ∞ the function does not repeat at all. In the limit L → ∞ the allowed
frequencies become a continuum and the Fourier sum goes over to a Fourier integral.
Consider a function f (x) defined on the real line. If f (x) were periodic with
period L, then f (x) can be expand by Eq. (2.3.5) as Fourier series converging to it
almost everywhere within each period [−L/2, L/2]. Even if f (x) is not periodic,
we can still define a function
f (x) =
∞
Cn ei2πnx/L ,
(2.3.7)
n=−∞
with the same Cn as above. Consider the limit in which L become very large.
Define
kn =
2nπ
,
L
2.3 Fourier Transform
23
then
f L (x) =
∞
Cn eikn x .
(2.3.8)
n=−∞
It is clear that for very large L the sum contains a very large number of waves with
wave-vector kn and that each successive wave differs from the last by a tiny change
in wave-vector (or wavelength),
k = kn+1 − kn =
2π
.
L
In the limit L → ∞ the allowed k becomes a continuous variable, the discrete
coefficients, Cn , become a continuous function of k, denoted by C(k) and the summation can be replaced by an integral and we have
f (x) =
1
2π
C(k) =
∞
C(k)eikx d x,
(2.3.9)
−∞
∞
−∞
f (x)e−ikx d x.
(2.3.10)
The functions f and C are called a Fourier transform pair, C is called the Fourier
transform of f and f is called the inverse Fourier transform of C.
This prompts us to define the 1D-Fourier transform of the function f (x) as
fˆ(k) =
∞
−∞
f (x)e−ikx d x,
(2.3.11)
provided that the integral exists. Not every function f (x) has a Fourier transform.
A sufficient condition is that it be square-integrable; that is, so that the following
integral converges:
|| f ||2 =
∞
−∞
| f (x)|2 d x.
(2.3.12)
If in addition of being square-integrable, the function is continuous, then one also
has the inversion formula
∞
1
ˇ
(2.3.13)
fˆ(k)eikx d x.
f (x) = ( f ) =
2π −∞
24
2 Fourier Analysis and Fourier Transform
2.3.1 2D-Fourier Transform
Two-dimensional (2D) Fourier transform of the function f (x, y) is defined as
fˆ(k, l) =
∞
−∞
∞
f (x, y)e−i(kx+ly) d xd y,
−∞
(2.3.14)
provided that the integral exists and a the two-dimensional (2D) inverse Fourier
transform is defined by
ˇf ) = 1
f (x, y) = ( 2π
∞
−∞
∞
−∞
fˆ(k, l)ei(kx+ly) dkdl.
(2.3.15)
2.3.2 Properties of Fourier Transform
Let fˆ(k) and ĝ(k) are Fourier transform of functions f (t) and g(t), respectively.
Then we have the following:
1. Linearity
a
f + bg(k) = a f (k) + b
g (k),
here a, b are constants, i.e. if we add two functions then the Fourier transform of
the resulting function is simply the sum of the individual Fourier transforms and if
we multiply a function by any constant then we must multiply the Fourier transform
by the same constant.
2. Shifting There are two basic shift properties of the Fourier transform:
i. Time Shifting
f (k)e±ikt0 .
( f
(t ± t0 ))(k) = ii. Frequency Shifting
±ik0 x )(k) = ( f (t)e
f (k ± k0 ).
Here t0 and k0 are constants. i.e. Translating a function in one domain corresponds
to a multiplication by a complex exponential function in the other domain.
3. Scaling
1 k
f (ax)(k) = f ( ),
a a
2.3 Fourier Transform
25
here a is constant. When a signal is expanded in time, it is compressed in frequency,
and vice versa i.e. we cannot be simultaneously short in time and short in frequency.
4. Differentiation
i. Time differentiation property
f (t)(k) = ik f (k).
Differentiating a function is said to amplify the higher frequency components because
of the additional multiplying factor k.
ii. Frequency differentiation property
d
f (k)
t
f (t)(k) = i
.
dk
5. Conjugate Symmetry The Fourier transform is conjugate symmetric for time
functions that are real-valued,
f (−k) = f (k).
From this it follows that the real part and the magnitude of the Fourier transform of
real valued time functions are even functions of frequency and that the imaginary
part and phase are odd functions of frequency. By property of conjugate symmetry,
in displaying or specifying the Fourier transform of a real-valued time function it is
necessary to display the transform only for positive values of k.
6. Reversal
f
(−x)(k) = f (−k),
for
x, k ∈ R.
7. Duality This property relates to the fact that the analysis equation and synthesis
1
equation look almost identical except for a factor of 2π
and the difference of a minus
sign in the exponential in the integral.
f (t) =
1
2π
∞
−∞
fˆ(k)eikt dk ⇐⇒
fˆ(k) =
∞
−∞
f (t)e−ikt dt
8. Convolution The convolution theorem states that convolution in time domain corresponds to multiplication in frequency domain and vice versa:
( f
∗ g)(t)(k) = f (k)
g (k).
9. Parseval’s Relation
∞
−∞
| f (t)|2 dt =
1
2π
∞
−∞
| fˆ(k)|2 dk.
26
2 Fourier Analysis and Fourier Transform
2.4 Discrete Fourier Transform
We assume that vectors in C N , i.e., sequences of N complex numbers, are indexed
from 0 to N − 1 instead of {1, 2, 3, . . . , N }. we regard x as a function defined on the
finite set
(2.4.1)
Z N = {0, 1, 2, . . . , N − 1},
and we identify x with column vector
⎡
⎤
x0
x1
.
.
.
⎢
⎢
⎢
x =⎢
⎢
⎢
⎣
⎥
⎥
⎥
⎥.
⎥
⎥
⎦
x N −1
This allows us to write the product of N × N matrix A by x as Ax. To save space,
we usually writ such a x horizontally instead of vertically x = (x0 , x1 , . . . , x N −1 ).
In order to be consistent with the notation for functions used later in the infinite
dimensional context, we write l 2 (Z N ) in place of C N . So, formally,
l 2 (Z N ) = {x = (x0 , x1 , . . . , x N −1 ) :
x j ∈ C, 0 ≤ j ≤ N − 1}.
(2.4.2)
With the usual component-wise addition and scalar multiplication, l 2 (Z N ) is an N dimensional vector space over C. One basis for l 2 (Z N ) is an standard or Euclidean
basis E = {e0 , e1 , . . . , e N −1 }, where
1, if n = j
e j (n) =
0, otherwise.
(2.4.3)
In this notation, the complex inner product on l 2 (Z N ) is
x, y =
N −1
xk y k ,
(2.4.4)
k=0
with the associated norm
||x|| =
N −1
k=0
called the l 2 -norm.
1/2
|xk |
2
,
(2.4.5)
2.4 Discrete Fourier Transform
27
2.4.1 1D-Discrete Fourier Transform
Definition 2.1 Define E 0 , E 1 , . . . , E N −1 ∈ l 2 (Z N ) by
1
E m (n) = √ e2πimn/N ,
N
for
0 ≤ m, n ≤ N − 1.
(2.4.6)
Clearly, the set {E 0 , E 1 , . . . , E N −1 } is an orthonormal basis for l 2 (Z N ). We have
x=
N −1
x, E m E m ,
(2.4.7)
x, E m y, E m ,
(2.4.8)
m=0
x, y =
N −1
m=0
||x|| =
2
N −1
| x, E m |2 .
(2.4.9)
m=0
By definition of inner product
x, E m =
N −1
n=0
N −1
1
1 xn √ e2πimn/N = √
xn e−2πimn/N .
N
N m=0
(2.4.10)
Definition 2.2 Suppose x = (x0 , x1 , . . . , x N −1 ) ∈ l 2 (Z N ). For m = 0, 1, 2, . . . ,
N − 1, define
N −1
xn e−2πimn/N .
(2.4.11)
xm =
n=0
Then x = (
x0 , x1 , . . . , x N −1 ) ∈ l 2 (Z N ). The map : l 2 (Z N ) → l 2 (Z N ), which takes
x to x , is called the 1D-discrete Fourier transform (DFT).
It can easily see that xm , m ∈ Z is periodic with period N :
xm+N =
N
−1
m=0
xn e−2πi(m+N )n/N =
N
−1
m=0
xn e−2πimn/N e−2πi N n/N =
N
−1
xn e−2πimn/N = xm ,
m=0
since e−2πi N n/N = e−2πin = 1, for every n ∈ Z. Comparing the Eqs. (2.4.10) and
(2.4.11), we have
√
(2.4.12)
xm = N x, E m .
28
2 Fourier Analysis and Fourier Transform
Remark 2.1 Equation (2.4.12) actually defines the DFT coefficients xk for any index
k and resulting xk are periodic with period N in the index. We will thus sometimes
refers to the xk on the other ranges of the index, for example −N /2 < k ≤ N /2
when N is even. Actually, even if N is odd, the range −N /2 < k ≤ N /2 works
because k is required to be an integer.
Equation (2.4.12) leads to the following reformulation of formulae (2.4.7), (2.4.8)
x1 , . . . , x N −1 ) and y = (
y0 , y1 , . . . , y N −1 ) ∈ l 2 (Z N ). Then
and (2.4.9). Let x = (
x0 , (i) Fourier Inversion Formula:
N −1
1 xm e−2πimn/N ,
xn =
N m=0
for
n = 0, 1, 2, . . . , N − 1.
(2.4.13)
(ii) Parseval’s Relation:
x, y =
N −1
1 1
xm , xm ym =
ym .
N m=0
N
(2.4.14)
N −1
1 1
xm ||2 .
|
xm |2 = ||
N m=0
N
(2.4.15)
(ii) Plancherel Theorem:
||x||2 =
The DFT can be represented by matrix, since Eq. (2.4.11) shows that the map
taking x to x is a linear transformation. Define
w N = e−2πi/N .
Then we have
e−2πimn/N = w mn
N
and
xm =
e2πimn/N = w −mn
N ,
and
N −1
xn w mn
N .
(2.4.16)
n=0
Definition 2.3 Let W N be the matrix [wmn ]0≤m,n≤N −1 , such that wmn = w mn
N . Hence
⎡
1 1
1
⎢ 1 wN
w 2N
⎢
⎢
2
⎢
w 4N
WN = ⎢ 1 wN
⎢·
·
·
⎢
⎣·
·
·
−1)
2(N −1)
w
1 w (N
N
N
..
··
·
·
·
·
1
w NN −1
−1)
· w 2(N
N
·
·
·
·
−1)(N −1)
· w (N
N
⎤
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎦
2.4 Discrete Fourier Transform
29
Regarding x, x ∈ l 2 (Z N ) as column vectors (as Eq. 2.4.11), the m th component
of W N x is
N −1
wmn xn =
n=0
N −1
xn w mn
xm
N =
0 ≤ m ≤ N − 1.
n=0
Hence
x = W N x.
(2.4.17)
Now, we only compute a simple example in order to demonstrate the definitions.
We could use Eq. (2.4.12) for this, but it is easier to use Eq. (2.4.17). The values of
matrices W2 and W4 are as follows:
⎡
⎤
1 1 1 1
⎢ 1 −i −1 i ⎥
1 1
⎥
W2 =
and W4 = ⎢
⎣ 1 −1 1 −1 ⎦ .
1 −1
1 i −1 −i
Example 2.1 Let x = (1, 0, −3, 4) ∈ l 2 (Z4 ). Find x.
Solution
⎡
1
⎢1
x = W4 x = ⎢
⎣1
1
1
−i
−1
i
1
−1
1
−1
⎤
⎤ ⎡
⎤⎡
2
1
1
⎥
⎢
⎥ ⎢
i ⎥
⎥ ⎢ 0 ⎥ = ⎢ 4 + 4i ⎥ .
−1 ⎦ ⎣ −3 ⎦ ⎣ −6 ⎦
4 − 4i
4
−i
The matrix W N has a lot of structure. This structure can even be exploited to
develop an algorithm called the fast Fourier transform that provides a very efficient
method for computing DFT’s without actually doing the full matrix multiplication.
Definition 2.4 (Convolution) For x, y ∈ l 2 (Z N ), the convolution x ∗ y ∈ l 2 (Z N ) is
the vector with components
(x ∗ y)(m) =
N −1
x(m − n)y(n),
(2.4.18)
m=0
for all m.
Suppose x, y ∈ l 2 (Z N ). Then for each m,
xm ym .
(x
∗ y)m = (2.4.19)
30
2 Fourier Analysis and Fourier Transform
2.4.2 Inverse 1D-Discrete Fourier Transform
To interpret the Fourier inversion formula (2.4.13), we make the following definition:
Definition 2.5 Let m = 0, 1, 2, . . . , N − 1. Define Fm ∈ l 2 (Z N ) by
Fm (n) =
1 2πimn/N
e
,
N
for
0 ≤ n ≤ N − 1.
(2.4.20)
Then F = {F0 , F1 , . . . , FN −1 } is called the Fourier basis for l 2 (Z N ).
Form Eq. (2.4.6) we have
1
Fm = √ E m .
N
(2.4.21)
Since E m is orthonormal basis for l 2 (Z N ), F is an orthogonal basis for l 2 (Z N ). With
this notation, Eq. (2.4.13) becomes
x=
N −1
xm Fm .
(2.4.22)
m=0
The Fourier inversion formula (2.4.13) shows that the linear transformation
: l 2 (Z N ) → l 2 (Z N ) is a one-one map. Therefore is invertible. Hence, Eq. (2.4.13)
gives us a formula for the inverse of discrete Fourier transform and it is denoted byˇ.
Definition 2.6 Let y = (y0 , y1 , . . . , y N −1 ) ∈ l 2 (Z N ). Define
y̌n =
N −1
1 yn e2πimn/N
N m=0
For
n = 0, 1, 2, . . . , N − 1.
(2.4.23)
Then y̌ = ( y̌0 , y̌1 , . . . , y̌ N −1 ) ∈ l 2 (Z N ). The mapˇ : l 2 (Z N ) → l 2 (Z N ), which takes
y to y̌, is called the 1D-inverse discrete Fourier transform (IDFT).
We can easily see that y̌m , m ∈ Z is also periodic function with period N . Fourier
inversion formula states that for x ∈ l 2 (Z N ),
(
xˇn ) = xn or (x̌n ) = xn , for n = 0, 1, 2, . . . , N − 1.
Since the DFT is an invertible linear transformation, the matrix W N is invertible and
x . Substituting x = y and equivalently x = y̌ in equations,
we must have x = W N−1
we have
(2.4.24)
y̌ = W N−1 y.
2.4 Discrete Fourier Transform
31
In the notation of formula (2.4.16), formula (2.4.21) becomes
y̌ =
N −1
yn
n=0
N −1
1 −mn 1 mn
wN =
yn w N .
N
N
n=0
w mn
This shows that the (n, m) entry of W N−1 is NN , which is 1/N times of the complex
conjugate of the (n, m) entry of W N . If we denote by W N the matrix whose entries
are the complex conjugates of the entries of W N , we have
W N−1 =
1
WN .
N
⎡
We have
W2−1 =
1 1 1
2 1 −1
and W4−1
1
1⎢
1
= ⎢
4 ⎣1
1
1
i
−1
−i
1
−1
1
−1
⎤
1
−i ⎥
⎥.
−1 ⎦
i
Example 2.2 Let y = (2, 4 + 4i, −6, 4 − 4i) ∈ l 2 (Z4 ). Find y̌.
⎡
Solution
W4−1
1 1
1⎢
1 i
= ⎢
4 ⎣ 1 −1
1 −i
1
−1
1
−1
⎤⎡
⎤ ⎡
⎤
1
2
1
⎢
⎥ ⎢
⎥
−i ⎥
⎥ ⎢ 4 + 4i ⎥ = ⎢ 0 ⎥ .
−1 ⎦ ⎣ −6 ⎦ ⎣ −3 ⎦
i
4 − 4i
4
2.4.3 2D-Discrete Fourier Transform and 2D-Inverse
Discrete Fourier Transform
The definition of the two-dimensional (2D) discrete Fourier transform is very similar
to that for one dimension. The forward and inverse transforms for an M × N matrix,
where for notational convenience we assume that the m indices are from 0 to M − 1
and the n indices are from 0 to N − 1 are:
x(r,s) =
M−1
N −1
x(m.n) e−2πi( M + N ) ,
mr
ns
(2.4.25)
m=0 n=0
and
x̌(r,s) =
M−1 N −1
1 mr
ns
x(m,n) e2πi( M + N ) .
M N m=0 n=0
(2.4.26)
32
2 Fourier Analysis and Fourier Transform
2.4.4 Properties of 2D-Discrete Fourier Transform
All the properties of the one-dimensional DFT transfer into two dimensions. But
there are some further properties not previously mentioned, which are of particular
use for image processing.
1. Similarity First notice that the forward and inverse transforms are very similar,
with the exception of the scale factor 1/M N in the inverse transform, and the negative
sign in the exponent of the forward transform. This means that the same algorithm,
only very slightly adjusted, can be used for both the forward an inverse transforms.
2. The DFT as a spatial filter Note that the values
e±2πi( M + N )
mr
ns
(2.4.27)
are independent of the values x or x̌. This means that they can be calculated in
advance, and only then put into the formulas above. It also means that every value
of x(r,s) is obtained by multiplying every value of x̌(r,s) by a fixed value, and adding
up all the results. But this is precisely what a linear spatial filter does: it multiplies
all elements under a mask with fixed values, and adds them all up. Thus we can
consider the DFT as a linear spatial filter which is as big as the image. To deal with
the problem of edges, we assume that the image is tiled in all directions, so that the
mask always has image values to use.
3. Separability Notice that the discrete Fourier transform filter elements can be
expressed as products:
mr
ns
mr
ns
(2.4.28)
e2πi( M + N ) = e2πi( M ) e2πi( N ) .
The first product value
e2πi( M )
mr
(2.4.29)
depends only on m and r, and is independent of n and s. Conversely, the second
product value
ns
(2.4.30)
e2πi( N )
depends only on n and s, and is independent of m and r . This means that we can
break down our formulas above to simpler formulas that work on single rows or
columns:
M−1
xr e−2πimr/M ,
(2.4.31)
xm =
r =0
xr = (
xˇm )r =
M−1
1 xm e2πimr/M .
M m=0
(2.4.32)
2.4 Discrete Fourier Transform
33
If we replace m and r with n and s we obtain the corresponding formulas for the DFT
of matrix columns. These formulas define the one-dimensional DFT of a vector, or
simply the DFT.
The 2D DFT can be calculated by using this property of separability; to obtain
the 2D DFT of a matrix, we first calculate the DFT of all the rows, and then calculate
the DFT of all the columns of the result. Since a product is independent of the order,
we can equally well calculate a 2D DFT by calculating the DFT of all the columns
first, then calculating the DFT of all the rows of the result.
4. Linearity An important property of the DFT is its linearity. the DFT of a sum is
equal to the sum of the individual DFT’s, and the same goes for scalar multiplication:
x + y =
x +
y
and
= k
kx
x,
where k is a scalar, and x and y are matrices.
This property is of great use in dealing with image degradation such as noise
which can be modelled as a sum:
d = f + η,
where f is the original image, η is the noise, and d is the degraded image. Since
d = f +
η
we may be able to remove or reduce η by modifying the transform. As we shall see,
some noise appears on the DFT in a way which makes it particularly easy to remove.
5. Convolution Theorem This result provides one of the most powerful advantages
of using the DFT. Suppose we wish to convolve an image M with a spatial filter S.
Our method has been place S over each pixel of M in turn, calculate the product of
all corresponding gray values of M and elements of S, and add the results. The result
is called the digital convolution of M and S, is denoted by
M ∗ S.
This method of convolution can be very slow, especially if S is large.
The DC coefficient The value x(0,0) of the DFT is called the DC coefficient. If we
put r = s = 0 in the Eq. (3.2.25), we have
x(0,0) =
M−1
N −1
m=0 n=0
x(m,n) e
n0
−2πi( m0
M + N )
=
M−1
N −1
m=0 n=0
x(m,n) .
34
2 Fourier Analysis and Fourier Transform
That is, this term is equal to the sum of all terms in the original matrix.
6. Shifting For purposes of display, it is convenient to have the DC coefficient in
the centre of the matrix. This will happen if all elements x(m,n) in the matrix are
multiplied by (−1)m+n before the transform.
7. Conjugate Symmetry An analysis of the Discrete Fourier transform definition
leads to a symmetry property, i.e., if we make the substitutions m = −m and n = −n
in Eq. (2.4.25), then
x(m,n) = x(−m+ pM,−n+q N ) ,
for any integers p and q. This means that half of the transform is a mirror image of
the conjugate of the other half. We can think of the top and bottom halves, or the left
and right halves, being mirror images of the conjugates of each other.
8. Displaying Transforms Having obtained the Fourier transform x of an image x,
we would like to see what it looks like. As the elements x are complex numbers, we
can’t view them directly, but we can view their magnitude |
x |, since these will be
numbers of type double, generally with large range. The display of the magnitude of
a Discrete Fourier transform is called the spectrum of the transform.
2.5 Fast Fourier Transform
The fast Fourier transform (FFT) is a class of algorithms for computing the DFT
and IDFT efficiently and it is one of the most influential algorithms of the twentieth
century. Cooley and Tukey [6] in 1965 published the modern version of the FFT
algorithms, but most of the ideas appeared earlier. Given the matrix W N , it is not surprising that shortcuts can be found for computing the matrix-vector product W N x. All
modern software uses the FFT algorithm for computing discrete Fourier transforms
and the details are usually transparent to the user. There are many of points of view
on the FFT as a computer scientist would classify the FFT as a classic divide and
conquer algorithm, a mathematician might view the FFT as a natural consequence
of the structure of certain finite groups and other practitioners might simply view it
as a efficient technique for organizing the sums involved in Eq. (2.4.11).
From Eq. (2.4.17), we have x̂ = W N x, where W N is the matrix in Definition 2.3.
Clearly, direct computation of x̂ takes N 2 complex multiplications. More precisely,
we could also count the number of additions. Though, we get a good idea of the
speed of computation by just considering the number of complex multiplications
required because multiplication is much slower on a computer than addition. Here,
complex multiplication, we mean the multiplication of two complex numbers. This
would appear to require four real multiplications, but by a trick, it requires only three
real multiplications. In signal and image processing, the vectors under consideration
can be very large. Computation of the DFTs of these vectors in real time by direct
means may be beyond the capacity of ones computational hardware. So a fast algo-
2.5 Fast Fourier Transform
35
rithm is needed known as FFT. Our intention is merely to demonstrate a very brief
introduction to the idea behind one form of the FFT algorithm.
Now, We begin with the simplest version of the FFT, in which the length N of the
vector is assumed to be even.
Theorem 2.1 Let M ∈ N, with N = 2M and let x ∈ l 2 (Z N ). Define u, v ∈ l 2 (Z M )
by
u k = x2k
f or
k = 0, 1, 2, . . . , M − 1, or u k = x0 , x2 , x4 , . . . x N −4 , x N −2 ,
and
vk = x2k+1
f or
k = 0, 1, 2, . . . , M − 1 or vk = x1 , x3 , x5 , . . . x N −3 , x N −1 .
Let x̂ denote the DFT of x defined on N points, that is, x̂ = W N x. Let û and v̂ denote
the DFTs of u and v respectively,defined on M = N /2 points, that is, û = W M u and
v̂ = W M v. Then for m = 0, 1, 2, . . . , M − 1,
x̂(m) = û(m) + e−2πim/N v̂(m).
(2.5.1)
Also, for m = M, M + 1, M + 2, . . . , N − 1, let l = m − M. Note that the
corresponding values of l are l = 0, 1, 2, . . . , M − 1. Then
x̂(m) = x̂(l + M) = û(l) − e−2πil/N v̂(l).
(2.5.2)
Proof By definition, for m = 0, 1, 2, . . . , N − 1,
xm =
N −1
xn e−2πimn/N ,
n=0
The sum over n = 0, 1, 2, . . . , N − 1 can be broken up into the sum over the
even values n = 2k, k = 0, 1, 2, . . . , M − 1, plus the sum over the odd values
n = 2k + 1, k = 0, 1, 2, . . . , M − 1 :
xm =
M−1
x2k e−2πi2 km/N +
k=0
=
M−1
M−1
x2k+1 e−2πi(2k+1)m/N ,
k=0
u k e−2πikm/(N /2) + e−2πim/N
k=0
=
M−1
k=0
M−1
vk e−2πikm/(N /2) ,
k=0
u k e−2πikm/M + e−2πim/N
M−1
k=0
vk e−2πikm/M .
36
2 Fourier Analysis and Fourier Transform
In the case m = 0, 1, 2, . . . , M − 1, the last expression is û(m) + e−2πim/N v̂(m), so
we have Eq. (2.5.1). Now suppose m = M, M + 1, M + 2, . . . , N − 1. By writing
m = l + M as in the statement of the theorem and substituting this for m above, we
get
xm =
M−1
u k e−2πik(l+m)/M + e−2πi(l+m)/N
k=0
=
M−1
M−1
vk e−2πik(l+m)/M ,
k=0
u k e−2πikl/M − e−2πil/N
M−1
k=0
vk e−2πikl/M ,
k=0
since the exponentials e−2πikl/M are periodic with period M, and e−2πi M/N =
= −1 for N = 2M. Hence Eq. (2.5.2) proves.
e
−πi
Notice that the same values are used in Eqs. (2.5.1) and (2.5.2) and to apply
Eqs. (2.5.1) and (2.5.2), we first compute û and v̂. Each can be computed directly with
M 2 complex multiplications since each of these is a vector of length M = N /2.Then
compute the products e−2πim/N v̂(m) for m = 0, 1, 2, . . . , M − 1, this requires an
additional M multiplications. Rest is done using only additions and subtractions
of these quantities, which we do not count. Hence, the total number of complex
multiplications required to compute x̂ by Eqs. (2.5.1) and (2.5.2) is at most
2M 2 + M = 2
N
2
2
+
1
N
= (N 2 + N ).
2
2
For N large, this is essentially N 2 /2, whereas the number of complex multiplications
required to compute x̂ directly is N 2 . Thus, Theorem 2.1 already cuts the computation
time nearly in half.
If N is divisible by 4 instead of just 2, we can proceed further. Similarly, if
N is divisible by 8, we can carry this one step further, and so on. Since u and v
have even order, we can then apply the same method to reduce the time required to
compute them. A more general way to describe this is to define # N , for any positive
integer N , to be the least number of complex multiplications required to compute
the DFT of a vector of length N . If N = 2M, then Eqs. (2.5.1) and (2.5.2) reduce
the computation of x̂ to the computation of two DFTs of size M, plus M additional
complex multiplications. Hence
# N ≤ 2# M + M.
(2.5.3)
The most favorable case is when N is a power of 2. He we have the following:
Theorem 2.2 Let N = 2n for some n ∈ N. Then
2.5 Fast Fourier Transform
37
#N ≤
1
N log2 N .
2
Proof For n = 1, a vector of length 21 is of the form x = (a, b). Then x̂ = (a +
b, a − b). Notice that this computation does not require any complex multiplications,
so #2 = 0 < 1 = (2log2 2)/2. The result holds for n = 1. By induction, suppose it
holds for n = k − 1. Then for n = k, we have by Eq. (2.5.3) and
1
1
1
#2k ≤ 2#2k−1 + 2k−1 ≤ 2 2k−1 (k − 1) + 2k−1 = k2k−1 = k2k = N log2 N .
2
2
2
Hence result holds for n = k. Thus the result true for all n.
For a vector of size 262, 144 = 218 , the FFT reduces the number of complex
multiplications needed to compute the DFT from 6.87 × 1010 to 2, 359, 296, thus
making the computation more than 29, 000 times faster. Hence, if it takes 8 hours to
do this via DFT directly, then it would take about 1 second to do it via the FFT. As N
increases, this ratio becomes more extreme to the point that some computations that
can be done by the FFT in a reasonable length of time could not be done directly in
an entire lifetime. The FFT is usually implemented without explicit recursion. The
FFT is not limited to N that are the powers of 2. What if N is not even? If N is prime,
the method of the FFT does not apply. However, an efficient FFT algorithm can be
derived most easily when N is “highly Composite” i.e. factors completely into small
integers. In general, if N is composite, say N = pq, a generalization of Theorem 2.1
can be applied.
Theorem 2.3 Let p, q ∈ N, and N = pq. Let x ∈ l 2 (Z N ). Define w0 , w1 , . . . ,
w p−1 ∈ l 2 (Zq ) by
wl (k) = xkp+l
f or
k = 0, 1, 2, . . . , q − 1.
For b = 0, 1, 2, . . . , q − 1, define vb ∈ l 2 (Z p ) by
vb (l) = e−2πibl/N ŵl (b)
f or
l = 0, 1, 2, . . . , p − 1.
Then for a = 0, 1, 2, . . . , p − 1 and b = 0, 1, 2, . . . , q − 1,
x̂(aq + b) = v̂b (a).
(2.5.4)
Note that by the division algorithm, every m = 0, 1, 2, . . . , N − 1 is of the form
aq + b for some a ∈ {0, 1, 2, . . . , p − 1} and b ∈ {0, 1, 2, . . . , q − 1}, so Eq. (2.5.4)
gives the full DFT of x.
Proof We can write each n = 0, 1, . . . , N − 1 uniquely in the form kp + l for some
k ∈ {0, 1, 2, . . . , q − 1} and l ∈ {0, 1, 2, . . . , p − 1}. Hence
38
2 Fourier Analysis and Fourier Transform
xaq+b =
N −1
xn e−2πi(aq+b)n/N =
n=0
p−1 q−1
xkp+l e−2πi(aq+b)(kp+l)/( pq) .
l=0 k=0
Note that
e−2πi(aq+b)(kp+l)/( pq) = e−2πiak e−2πial/ p e−2πibk/q e−2πibl/( pq) .
Since e−2πiak = 1 and pq = N , using the definition of wl (k) we obtain
xaq+b =
p−1
e−2πial/ p e−2πibl/N
l=0
=
p−1
q−1
e−2πibk/q
k=0
e−2πial/ p e−2πibl/N ŵl (b)
l=0
=
p−1
l=0
e−2πial/ p vb (l) = vb (a).
This proof shows the basic principle behind the FFT. In computing xaq+b , the
same quantities vb (l), 0 ≤ l ≤ p − 1, arise for each value of a. The FFT algorithm
recognizes this and computes these values only once. We first compute the vectors
w
l , for l = 0, 1, ..., p − 1. Each of these is a vector of length q, so computing
each w
l requires #q complex multiplications. So this step requires a total of p#q
complex multiplications. The next step is to multiply each w
l (b) by e e−2πibl/N to
obtain the vectors vb (l). This requires a total of pq complex multiplications, one
for each of the q values of b and p values of l. Finally we compute the vectors v̂b
for b = 0, 1, ..., q − 1. Each vb is a vector of length p, so each of the q vectors v̂b
requires # p complex multiplications, for a total of q # p multiplications. Adding up,
we have an estimate for the number of multiplications required to compute a DFT
of size N = pq, namely
(2.5.5)
# pq ≤ p #q + q # p + pq.
This estimate can be used inductively to make various estimates on the time required
to compute the FFT. The advantage of using the FFT is greater the more composite
N is. There are many variations on the FFT algorithm, sometimes leading to slight
advantages over the basic one given here. But the main point is that the DFT of
a vector of length N = 2n can be computed with at most n2n−1 = (N /2)log2 N
complex multiplications as opposed to N 2 = 22n if done directly.
Since x̌n = N1 x̂ N −n and x ∗ y = (x̂ ŷ)ˇ, the FFT algorithm can be used to compute
the IDFT and convolutions quickly also. In computing IDFT, at most (N /2)log2 N
steps requires if N is a power of 2. We do not count division by N because integer
division is relatively fast. If x, y ∈ l 2 (Z N ), for N a power of 2, it takes at most
2.6 The Discrete Cosine Transform
39
N log2 N multiplications to compute x̂ and ŷ, and N multiplications to compute x̂ ŷ,
and at most (N /2)log2 N multiplications to take the IDFT of x̂ ŷ. Thus overall it takes
no more than N + (3N /2)log2 N multiplications to compute z ∗ w.
2.6 The Discrete Cosine Transform
The Fourier transform and the DFT are designed for processing complex valued
signals, and they always produce a complex-valued spectrum even in the case where
the original signal was strictly real-valued. The reason is that neither the real nor the
imaginary part of the Fourier spectrum alone is sufficient to represent (i.e., reconstruct) the signal completely. In other words, the corresponding cosine (for the real
part) or sine functions (for the imaginary part) alone do not constitute a complete set
of basis functions.
Further, we know that a real-valued signal has a symmetric Fourier spectrum, so
only one half of the spectral coefficients need to be computed without losing any
signal information.
There are several spectral transformations that have properties similar to the DFT
but do not work with complex function values. The discrete cosine transform (DCT)
is well known example that is particularly interesting in our context because it is frequently used for image and video compression. The DCT uses only cosine functions
of various wave numbers as basis functions and operators on real-valued signals and
spectral coefficients. Similarly, there is also a discrete sine transform (DST) based
on a system of sine functions.
2.6.1 1D-Discrete Cosine Transform
In the one-dimensional case, the discrete cosine transform (DCT) for a signal g(n)
of length N is defined as
G(m) =
N −1
2 m(2n + 1)
,
g(n) · cm cos π
N n=0
2N
(2.6.1)
for 0 ≤ m < N , and the inverse discrete cosine transform (IDCT) is
g(n) =
N −1
2 m(2n + 1)
,
G(m) · cm cos π
N m=0
2N
for 0 ≤ n < N , respectively, with
(2.6.2)
40
2 Fourier Analysis and Fourier Transform
cm =
√1 ,
2
if m = 0
1,
otherwise.
(2.6.3)
Note that the index variables (n, m) are used differently in the forward transform and
the inverse transform, so the two transform are, in contrast to the DFT, not symmetric.
One may ask why it is possible that the DCT can work without any sine functions,
while they are essential in the DFT. The trick is to divide all frequencies in half such
that they are spaced more densely and thus the frequency resolution in the spectrum
is doubled. Comparing the cosine parts of the DFT basis functions (Eq. 2.4.11) and
those of the DCT (Eq. 2.6.1), we have
DFT: CnN (m) = cos 2π mn
N
2m(n+0.5)
DCT: DnN (m) = cos π m(2n+1)
=
cos
π
,
2N
2N
one can only see that the period of the DCT basis functions (2N /m) is double the
period of DFT functions (M/m) and DCT functions are also phase-shifted by 0.5
units. of course, much more efficient (fast) algorithms exist. Moreover, the DCT can
also be computed in O(Mlog2 M) time using FFT. The DCT is often used for image
compression, in particular JPEG compression, where the size of the transformed sub
images is fixed at 8 × 8 and the processing can highly be optimized.
2.6.2 2D-Discrete Cosine Transform
The two-dimensional form of the DCT follows immediately from the one-dimensional
definition, resulting in 2D forward transform
N1 −1 N
2 −1
2 g(n 1 , n 2 )
N1 N2 n =0 n =0
1
2
m 1 (2n 1 + 1)
m 2 (2n 2 + 1)
· cm 2 cos π
,
· cm 1 cos π
2N1
2N2
G(m 1 , m 2 ) =
(2.6.4)
for 0 ≤ m 1 < N1 , 0 ≤ m 2 < N2 , and the 2D-inverse DCT is
N1 −1 N
2 −1
2 G(m 1 , m 2 )
N1 N2 n =0 n =0
1
2
m 1 (2n 1 + 1)
m 2 (2n 2 + 1)
· cm 2 cos π
,
· cm 1 cos π
2N1
2N2
g(n 1 , n 2 ) =
for 0 ≤ m 1 < N1 , 0 ≤ m 2 < N2 .
(2.6.5)
2.7 Heisenberg Uncertainty Principle
41
2.7 Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle was originally stated in physics, and claims
that it is impossible to know both the position and momentum of a particle simultaneously. However, it has an analog basis in signal processing. In terms of signals,
the Heisenberg uncertainty principle is given by the rule that it is impossible to know
both the frequency and time at which they occur. The time and frequency domains are
complimentary. If one is local, the other is global. Formally, the uncertainty principle
is expressed as
1
(2.7.1)
(t)2 (ω)2 ≥ .
4
In the case of an impulse signal, which assumes a constant value for a brief period
of time, the frequency spectrum is finite; whereas in the case of a step signal which
extends over infinite time, its frequency spectrum is a single vertical line. This fact
shows that we can always localize a signal in time or in frequency but not both
simultaneously. If a signal has a short duration, its band of frequency is wide and
vice-versa.
2.8 Windowed Fourier Transform or Short-Time Fourier
Transform
The short-time Fourier transform (STFT) is a modified version of Fourier transform.
The Fourier transform separates the input signal into a sum of sinusoids of different
frequencies and also identities their respective amplitudes. Thus, the Fourier transform gives the frequency-amplitude representation of an input signal. The Fourier
transform is not an effective tool to analyses non-stationary signals. In STFT, the
non-stationary signal is divided into small portions, which are assumed to be stationary. This is done using a window function of a chosen width, which is shifted and
multiplied with the signal to obtain small stationary signals.
2.8.1 1D and 2D Short-Time Fourier Transform
The short-time Fourier transform maps a signal into two-dimensional function of
time and frequency. The STFT of a one-dimensional signal f (t) is represented by
X (τ , ω) where
∞
X (τ , ω) =
f (t)g ∗ (t − τ )e−iωt dt,
(2.8.1)
−∞
42
2 Fourier Analysis and Fourier Transform
Fig. 2.4 The time-frequency tiling of STFT
Here, g ∗ denotes the conjugate of g, f (t) represents the input signal, gτ ,ω (t) =
g(t − τ )eiωt is a temporal window with finite support and X (τ , ω) is the time frequency atom. Also, the non-stationary signal f (t) is assumed to be approximately
stationary in the span of the temporal window gτ ,ω (t).
In the case of a 2D signal f (x, y), the space-frequency atom or STFT is given by
X (τ1 , τ2 , ω1 , ω2 ) =
∞
−∞
∞
−∞
f (x, y)g ∗ (x − τ1 , x − τ2 )e−i(ω1 x+ω2 y) d xd y, (2.8.2)
where τ1 , τ2 represents the spatial position of the two-dimensional window
gτ1 ,τ2 ,ω1 ,ω2 (x, y) and ω1 , ω2 represents the spatial frequency parameters. The performance of STFT for specific application depends on the choice of the window.
Different types of windows that can be used in STFT are Hamming, Hanning, Gaussian and Kaiser windows. The time-frequency tiling of STFT is given in Fig. 2.4.
2.8.2 Drawback of Short-Time Fourier Transform
The main drawback of STFT is that once a particular size time window is chosen, the
window remains the same for all frequencies. To analyze the signal effectively, a more
flexible approach is needed where the window size can vary in order to determine
more accurately either the time or frequency information of the signal. This problem
is known as resolution problem.
2.9 Other Spectral Transforms
Apparently, the Fourier transform is not the only way to represent a given signal in
frequency space; in fact, numerous similar transforms exist. Some of these, such as
the discrete cosine transform, also use sinusoidal basis functions, while others, such
2.9 Other Spectral Transforms
43
as the Hadamard transform (also known as the Walsh transform), build on binary
0/1-functions. All of these transforms are of global nature; i.e., the value of any
spectral coefficient is equally influenced by all signal values, independent of the
spatial position in the signal. Thus a peak in the spectrum could be caused by a
high-amplitude event of local extent as well as by a widespread, continuous wave
of low amplitude. Global transforms are therefore of limited use for the purpose of
detecting or analyzing local events because they are incapable of capturing the spatial
position and extent of events in a signal. A solution to this problem is to use a set of
local, spatially limited basis functions (wavelets) in place of the global, spatially fixed
basis functions. The corresponding wavelet transform, of which several versions have
been proposed, allows the simultaneous localization of repetitive signal components
in both signal space and frequency space.
References
1. Bachmann, G., Narici, L., Beckenstein, E.: Fourier and Wavelet Analysis. Springer, New York
(1999)
2. Bracewell, R.N.: The Fourier Transform and its Applications. Mcgraw-Hill International Editors (2000)
3. Brigham, E.O.: The Fast Fourier Transform: An Introduction to its Theory and Applications.
Prentice Hall, New Jersey (1973)
4. Broughton, S.A., Bryan, K.: Discrete Fourier Analysis and Wavelets: Applications to Signal
and Image Processing. Wiley, Inc., Hoboken, New Jersey (2009)
5. Chu, E.: Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms. CRC Press, Boca Raton (2008)
6. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex fourier series.
Math. Comput. 19, 297–301 (1965)
7. Folland, G.B.: Fourier Analysis and Its Applications. The Wadsworth and Brooks/Cole Mathematics Series (1992)
8. Frazier, M.W.: An Introduction to Wavelets through Linear Algebra. Springer, Berlin (1999)
9. Rao, K.R., Yip, P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic, New York (1990)
10. Rjasanow, S., Steinbach, O.: Fast Fourier Transform and its Applications, Prentice Hall, New
Jersey (1988)
11. Walker, J.S.: Fourier analysis and wavelet analysis. Notices AMS. 44(6), 658–670 (1997)
Chapter 3
Wavelets and Wavelet Transform
3.1 Overview
Wavelet transforms are the most powerful and the most widely used tool in the field of
image processing. Wavelet transform has received considerable attention in the field
of image processing due to its flexibility in representing non-stationary image signals
and its ability in adapting to human visual characteristics. Wavelet transform is an
efficient tool to represent an image. The wavelet transform allows multiresolution
analysis of an image. The aim of the transform is to extract relevant information
from an image. A wavelet transform divides a signal into a number of segments,
each corresponding to a different frequency band.
Fourier transform is a powerful tool that has been available to signal analysis
for many years. It gives information regarding the frequency content of a signal.
However, the problem with using Fourier transform is that frequency analysis cannot
offer both good frequency and time resolution at the same time. A Fourier transform
does not give information about the time at which a particular frequency has occurred
in the signal. Hence, a Fourier transform is not an effective tool to analyses a nonstationary signal. To overcome this problem, windowed Fourier transform, or shorttime Fourier transform, was introduced. Even though a short-time Fourier transform
has the ability to provide time information, but multiresolution is not possible with
short-time Fourier transforms. Wavelet is the answer to the multiresolution problem.
A wavelet has the important property of not having a fixed-width sampling window.
The wavelet transform can be broadly classified into (i) continuous wavelet transform,
and (ii) discrete wavelet transform. For long signals, continuous wavelet transform
can be time consuming since it needs to integrate over all times. To overcome the time
complexity, discrete wavelet transform was introduced. Discrete wavelet transforms
can be implemented through subband coding. The DWT is useful in image processing
because it can simultaneously localize signals in time and scale, whereas the DFT
or DCT can localize in the frequency domain.
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_3
45
46
3 Wavelets and Wavelet Transform
The first literature that relates to the wavelet transform is Haar wavelet. It was proposed by the mathematician Alfrd Haar in 1909. However, the concept of the wavelet
did not exist at that time. Until 1981, the concept was proposed by the geophysicist
Jean Morlet. Afterward, Morlet and the physicist Alex Grossman invented the term
wavelet in 1984. Before 1985, Haar wavelet was the only orthogonal wavelet people
know. A lot of researchers even thought that there was no orthogonal wavelet except
Haar wavelet. Fortunately, the mathematician Yves Meyer constructed the second
orthogonal wavelet called Meyer wavelet in 1985. As more and more scholars joined
in this field, the 1st international conference was held in France in 1987. In 1989,
Stephane Mallat and Meyer proposed the concept of multiresolution [21–25]. In the
same year, Ingrid Daubechies found a systematical method to construct the compact
support orthogonal wavelet. In 1989, Mallat proposed the fast wavelet transform.
With the appearance of this fast algorithm, the wavelet transform had numerous
applications in the signal and image processing field.
3.2 Wavelets
The notion of wavelets came into being because the Fourier analysis which depends
on oscillating building blocks is poorly suited to signals that change suddenly. A
wavelet is crudely a function which together with its dilates and their translates
determine all functions of our need.
Definition 3.1 A function ψ ∈ L 2 (R) is called a wavelet if it posses the following
properties:
1. It is square integrable, or equivalently, has finite energy:
∞
−∞
|ψ(x)|2 d x < ∞,
(3.2.1)
2. The function integrates to zero, or equivalently, its Fourier transform denoted as
ψ̂(ξ) is zero at the origin:
∞
−∞
|ψ(x)|d x = 0,
(3.2.2)
3. The Fourier transform of ψ(x) must satisfy the admissibility condition given by
Cψ =
∞
−∞
|ψ̂(ξ)|2
dξ < ∞,
|ξ|
(3.2.3)
where ψ̂(ξ) is the Fourier transform of ψ(x).
Equation (3.2.1) implies that most of the energy in ψ(x) is confined to a finite
interval, or ψ(x) has good space localisation. Ideally, the function is exactly zero
3.2 Wavelets
47
outside the finite interval and this implies that the function is compactly supported
function. Equation (3.2.2) suggests that the function is either oscillatory or has a wavy
appearance. Equation (3.2.3) is useful in formulating the inverse Fourier transform.
From Eq. (3.2.3), it is obvious that ψ(ξ) must have a sufficient decay in frequency.
This means that the Fourier transform of a wavelet is localised, that is, a wavelet
mostly contains frequencies from a certain frequency band. Since the Fourier transform is zero at the origin, and the spectrum decays at high frequencies, a wavelet
has a band pass characteristics. Thus a wavelet is a ‘small wave’ that exhibits good
time-frequency localisation.
We generate a doubly-indexed family of wavelets from ψ by dilating and translating,
x −b
,
(3.2.4)
ψa,b (x) = a −1/2 ψ
a
where a, b ∈ R, a = 0 (we use negative as well as positive a at this point). The
normalization has been chosen so that ||ψa,b || = ||ψ|| for all a, b ∈ R. We will
assume that ||ψ|| = 1. The discretization of the dilation parameter, we choose a =
−j
a0 , where j ∈ Z, and assume a0 > 1, and discretize b by taking only the integer
(positive and negative) multiples of one fixed b0 , where b0 (> 0) is appropriately
chosen so that the ψ(x − kb0 ) cover the whole line. For different values of j, the
j/2
j
j
−j
width of a0 ψ(a0 x) is a0 times of the width of ψ(x), so that the choice b = kb0 a0
will ensure that the discretized wavelets at level j cover the line in the same way
−j
−j
that the ψ(x − kb0 ) do. Thus we choose a = a0 , b = kb0 a0 , where j, k ∈ Z and
a0 > 1, b0 > 0, are fixed. Hence discrete form of wavelet is
−j
x − kb0 a0
j/2
j/2
j
(3.2.5)
ψ j,k (x) = a0 ψ
= a0 ψ(a0 x − kb0 ).
−j
a0
For a0 = 2, b0 = 1, we have
ψ j,k (x) = 2 j/2 ψ(2 j x − k).
(3.2.6)
Alternatively, A function ψ ∈ L 2 (R) is called an orthonormal wavelet if the system
{ψ j,k } j,k∈Z forms an orthonormal basis for L 2 (R). This is also known as dyadic
wavelet.
The Haar wavelet (1910) is the oldest wavelet which has limited application as it
is not continuous. To suit for approximating data with sharp discontinuities, wavelets
that could automatically adapt to different components of a signal were targeted for
investigation in early 1980s. Daubechies [9] brought a big breakthrough in 1988 and
her work immediately stimulated a rapid development in the theory and applications
of wavelet analysis. The Haar wavelet is not continuous, while the Shannon wavelet
as well as Meyer wavelets are smooth. Meyer wavelets can be so chosen that their
Fourier transforms are also smooth.
48
3 Wavelets and Wavelet Transform
Definition 3.2 An orthonormal wavelet ψ ∈ L 2 (R) is called compactly supported
wavelet if the support of the wavelet ψ is compact.
Definition 3.3 An orthonormal wavelet ψ ∈ L 2 (R) is said to be a band-limited
wavelet if its Fourier transform has compact support.
The Haar wavelet is compactly supported while the Shannon wavelet, the Journé
wavelet and Meyer wavelets are examples of band-limited wavelets. For more details
about Haar, Shannon and Meyers wavelets, one can read, Hernandez and Weiss [13]
and for Daubechies wavelets, Daubechies [8].
3.3 Multiresolution Analysis
The concept of Multiresolution analysis (MRA) was first introduced by Mallat [21,
22] and Meyer [25]. It is a general framework that makes constructing orthonormal
wavelets basis for the space L 2 (R). The MRA based compactly supported orthonormal wavelet systems were constructed by Daubechies [8]. Multiresolution analysis
(MRA) is a very well-known and unique mathematical theory that incorporates and
unifies various image processing techniques. The main purpose of this analysis is to
obtain different approximations of a function f (x) at different resolution. Furthermore, the MRA structure grants fast implementation of wavelet decomposition and
reconstruction which makes wavelets a very practical tool for image processing and
analysis. Multiresolution analysis is a family of closed subspaces of L 2 (R) satisfying
certain properties:
Definition 3.4 A pair ( V j j∈Z , ϕ) consisting of a family V j j∈Z of closed subspaces of L 2 (R) together with a function ϕ ∈ V0 is called a Multiresolution analysis
(MRA) if it satisfies the following conditions:
(a) V j ⊂ V j+1 , for all j ∈ Z,
(b) f ∈ V0 ⇐⇒ f (2 j ·) ∈ V j , for all j ∈ Z,
(c) ∩ j∈Z V j = {0},
(d) ∪ j∈Z V j = L 2 (R),
(e) {ϕ(· − k) : k ∈ Z}, denoted by ϕ0,k , is an orthonormal basis for V0 .
The function ϕ is called a scaling function for the given MRA. We note that an
orthonormal basis for V j , j ∈ Z is given by translates of normalized dilations
{2 j/2 ϕ(2 j · −k), k ∈ Z}, denoted by ϕ j,k , of ϕ. For each j ∈ Z, define the approximation operator P j on functions f (x) ∈ L 2 (R) by,
P j f (x) =
f, ϕ j,k ϕ j,k ,
(3.3.1)
and for each j ∈ Z, the detail operator Q j on functions f (x) ∈ L 2 (R) is defined
by,
3.3 Multiresolution Analysis
49
Q j f (x) = P j+1 f (x) − P j f (x).
(3.3.2)
A multiresolution analysis gives
(1) an orthogonal direct sum decomposition of L 2 (R), and
(2) a wavelet ψ called an MRA Wavelet.
Let W0 be the orthogonal complement of V0 in V1 ; that is, V1 = V0 ⊕ W0 . Then if
we dilate the elements of W0 by 2 j , we obtain a closed subspace W j of V j in V j+1 ,
such that
for each j ∈ Z.
(3.3.3)
V j+1 = V j ⊕ W j ,
Conditions (c) and (d) of the Definition 3.4 of an MRA provide
L 2 (R) = ⊕∞
j=−∞ W j .
(3.3.4)
Furthermore, the W j spaces inherit the scaling property (Definition 3.4(b)) from the
Vj :
(3.3.5)
f ∈ W0 ⇐⇒ f (2 j ·) ∈ W j .
Equation (3.3.5) ensures that if {ψ0,k ; k ∈ Z} is an orthonormal basis for W0 , then
{ψ j,k ; k ∈ Z} will likewise be an orthonormal basis for W j , for any j ∈ Z. Our main
aim is to finding ψ ∈ W0 such that the ψ(· − k) constitute an orthonormal basis for
W0 . To construct this ψ let us write some interesting properties of ϕ and W0 .
1. Since ϕ ∈ V0 ⊂ V1 and ϕ1,k is an orthonormal basis in V1 , we have
ϕ=
h(k)ϕ1,k ,
(3.3.6)
k∈Z
with h(k) = ϕ, ϕ1,k and
k∈Z
|h(k)|2 = 1, i.e. h(k) ∈ l 2 (Z). Hence,
ϕ(x) =
√
h(k)ϕ(2x − k),
2
(3.3.7)
k∈Z
is known as the scaling relation or the refinement equation. The sequence {h(k)}k∈Z is
called the scaling sequence or scaling filter associated with ϕ(x). By taking Fourier
transform both side of equation (3.3.7), we have
1
ϕ̂(ξ) = √
2
h(k)e−ikξ/2 ϕ̂(ξ/2),
(3.3.8)
k∈Z
where convergence in either sum holds in L 2 -sense. Equation (3.3.8) can be rewritten
as
(3.3.9)
ϕ̂(ξ) = m 0 (ξ/2)ϕ̂(ξ/2),
50
3 Wavelets and Wavelet Transform
where
1
m 0 (ξ) = √
2
h(k)e−ikξ .
(3.3.10)
k∈Z
Equality in Eq. (3.3.9) holds point-wise almost everywhere. Equation (3.3.10) shows
that m 0 is a 2π-periodic function in L 2 ([0, 2π]).
2. The orthonormality of the ϕ(x − k) leads to special properties for m 0 . We have
δk,0 =
ϕ(x)ϕ(x − k)d x =
|ϕ̂(ξ)| e
2 ikξ
dξ =
2π
0
|ϕ̂(ξ + 2πl)|2 eikξ dξ,
l∈Z
implying
|ϕ̂(ξ + 2πl)|2 = (2π)−1 a.e.
(3.3.11)
l∈Z
Substituting equation (3.3.9) in Eq. (3.3.11) and put η = ξ/2 leads to
|m 0 (η + πl)|2 |ϕ̂(η + πl)|2 = (2π)−1 ,
l∈Z
by splitting the sum into even and odd l, using the periodicity of m 0 and applying
(3.3.11) once more gives
|m 0 (ξ)|2 + |m 0 (ξ + π)|2 = 1 a.e.
(3.3.12)
3. Characterization of W0 : f ∈ W0 is equivalent to f ∈ V1 and f ⊥ V0 . Since,
f ∈ V1 , we have
f =
f (k)ϕ1,k ,
k∈Z
with f (k) = f, ϕ1,k . This implies
1
fˆ(ξ) = √
2
where
f (k)e−ikξ/2 ϕ̂(ξ/2) = m f (ξ/2)ϕ̂(ξ/2),
(3.3.13)
1
m f (ξ) = √
2
(3.3.14)
k∈Z
f (k)e−ikξ ;
k∈Z
clearly m f is a 2π-periodic function in L 2 ([0, 2π]), convergence in (3.3.14) holds
pointwise a.e.. The constraint f ⊥ V0 implies f ⊥ ϕ0,k for all k, i.e.,
3.3 Multiresolution Analysis
0=
fˆ(ξ)ϕ̂(ξ)e
51
ikξ
fˆ(ξ + 2πl)ϕ̂(ξ + 2πl) eikξ dξ,
dξ =
l
hence
fˆ(ξ + 2πl)ϕ̂(ξ + 2πl) = 0,
(3.3.15)
l
where the series in (3.3.15) converges absolutely in L 1 ([π, π]). Substituting (3.3.9)
and (3.3.13), regrouping the sums for odd and even l (which we allow to do, because
of the absolute convergence), using (3.3.11) leads to
m f (ξ)m 0 (ξ) + m f (ξ + π)m 0 (ξ + π) = 0 a.e..
(3.3.16)
Since m 0 (ξ) and m 0 (ξ + π) cannot vanish together on a set of nonzero measure
(because of (3.3.12)), this implies the existence of 2π-periodic function λ(ξ) so that
m f (ξ) = λ(ξ)m 0 (ξ + π) a.e.,
(3.3.17)
λ(ξ) + λ(ξ + π) = 0 a.e..
(3.3.18)
and
This last equation can be reorganize as
λ(ξ) = eiξ ν(2ξ),
(3.3.19)
where ν is 2π-periodic. Substituting equations (3.3.19) and (3.3.17) into Eq. (3.3.13)
gives
(3.3.20)
fˆ(ξ) = eiξ/2 m 0 (ξ/2 + π)ν(ξ)ϕ̂(ξ/2),
where ν is 2π-periodic.
4. The general form of Eq. (3.3.20) for the Fourier transform of f ∈ W0 suggests
that we take
(3.3.21)
ψ̂(ξ) = eiξ/2 m 0 (ξ/2 + π)ϕ̂(ξ/2),
as a candidate of our wavelet. Disregarding convergence questions, Eq. (3.3.20) can
be rewritten as
fˆ(ξ) =
ν(k)e−ikξ ψ̂(ξ),
k∈Z
or
f (x) =
ν(k)ψ(x − k),
k∈Z
52
3 Wavelets and Wavelet Transform
so that the ψ(· − k) are a good candidate for a basis of W0 . Next, we need to verify
that the ψ0,k are indeed an orthonormal basis for W0 . The properties of m 0 and ϕ̂
ensure that Eq. (3.3.21) defines indeed an L 2 -function i.e. ψ ∈ V1 and ψ ⊥ V0 , so
that ψ ∈ W0 . Orthonormality of the ψ0,k is easy to check:
ψ(x)ψ(x − k)d x =
|ψ̂(ξ)| e
2 ikξ
dξ =
2π
|ψ̂(ξ + 2πl)|2 dξ.
eikξ
0
l∈Z
Now
l∈Z
|ψ̂(ξ + 2πl)|2 =
l∈Z
|m 0 (ξ/2 + πl + π)|2 |ϕ̂(ξ/2 + πl)|2
= |m 0 (ξ/2 + π)|2
|ϕ̂(ξ/2 + 2πn)|2
n∈Z
+ |m 0 (ξ/2)|2
|ϕ̂(ξ/2 + π + 2πn)|2
n∈Z
= (2π)−1 [|m 0 (ξ/2 + π)|2 + |m 0 (ξ/2)|2 ]
= (2π)−1
a.e. (by Eq. (3.3.11))
a.e. (by Eq. (3.3.12)).
Hence ψ(x)ψ(x − k)d x = δk,0 . In order to check that the ψ0,k are indeed a basis
for all of W0 , it then suffices to check that any f ∈ W0 can be written as
f =
γ(n)ψ0,n ,
n∈Z
with
n∈Z
|γ(n)|2 < ∞, or
fˆ(ξ) = γ(ξ)ψ̂(ξ),
(3.3.22)
with γ is 2π-periodic and L 2 ([0, 2π]). Let us go back to Eq. (3.3.20), we have fˆ(ξ) =
2π
π
ν(ξ)ψ̂(ξ), with 0 |ν(ξ)|2 = 0 |λ(ξ)|2 . by Eq. (3.3.14)
2π
|m f (ξ)|2 dξ =
0
=
2π
0 π
|λ(ξ)|2 |m 0 (ξ + π)|2 dξ
|λ(ξ)|2 [|m 0 (ξ)|2
0
+ |m 0 (ξ + π)|2 ]dξ (use Eq. (3.3.18))
π
=
|λ(ξ)|2 (use Eq. (3.3.12)).
0
3.3 Multiresolution Analysis
53
2π
Hence 0 |ν(ξ)|2 dξ = 2π|| f ||2 < ∞, and f is of the form (3.3.22) with square
integrable 2π-periodic γ. We have thus proved the following theorem:
Theorem 3.1 Let {V j } j∈Z be a sequence of closed subspaces of L 2 (R) satisfying
conditions (a)-(e) of Definition (3.4). Then there exists an associated orthonormal
wavelet basis {ψ j.k ; j, k ∈ Z } for L 2 (R) such that
P j+1 f = P j f +
f, ψ j,k ψ j,k
f or all f ∈ L 2 (R),
(3.3.23)
k∈Z
where P j is the orthogonal projection onto V j . One possibility for the construction
of the wavelet ψ is
ψ̂(ξ) = eiξ/2 m 0 (ξ/2 + π)ϕ̂(ξ/2),
(with m 0 as defined by (3.3.10) and (3.3.6)), or equivalently
ψ(x) =
g(k)ϕ1,k (x) =
√
g(k)ϕ(2x − k),
2
k∈Z
(3.3.24)
k∈Z
where g(k) = (−1)k h(−k + 1), is called wavelet filter.
3.4 Wavelet Transform
The wavelet transform (WT) provides a time-frequency representation of the signal.
The wavelet transform was developed to overcome the shortcomings of the short-time
Fourier transform, which can be used to analyse non-stationary signals. The main
drawback of STFT is that it gives a constant resolution at all frequencies, while the
wavelet transform uses a multiresolution technique by which different frequencies
are analysed with different resolutions. The wavelet transform is generally termed
mathematical microscope in which big wavelets give an approximate image of the
signal, while the smaller wavelets zoom in on the small details. The basic idea of
the wavelet transform is to represent the signal to be analysed as a superposition
of wavelets. Historically, the continuous wavelet transform came first. It is completely different from the discrete wavelet transform. It is popular among physicists,
whereas the discrete wavelet transform is more common in numerical analysis, signal
processing and image processing.
3.4.1 The Wavelet Series Expansions
Likewise Fourier series expansion, a wavelet series is a representation of square
integrable real or complex valued function by a certain orthonormal series generated
54
3 Wavelets and Wavelet Transform
by a wavelet. A function f (x) ∈ L 2 (R) can be represented by a scaling function
expansion in subspace V J0 and some other number of wavelet expansions in subspace
W J0 , W J0 +1 , . . .. Thus
∞
c j0 (k)ϕ j0 ,k (x) +
f (x) =
d j (k)ψ j,k (x),
(3.4.1)
j= j0 k∈Z
k∈Z
where j0 is arbitrary starting scale and the c j0 (k) and d j (k), defined by,
c j0 (k) = f (x), ϕ j0 ,k (x) =
f (x)ϕ j0 ,k (x)d x,
(3.4.2)
f (x)ψ j,k (x)d x,
(3.4.3)
d j (k) = f (x), ψ j,k (x) =
are called approximation coefficient or scaling coefficient and detail coefficient or
wavelet coefficient, respectively.
3.4.2 Discrete Wavelet Transform
The wavelet series expansion maps a function of a continuous variable into a sequence
of coefficients. If the sequence being expanded is discrete, the resulting coefficients
are called the discrete wavelet transform (DWT). The discrete wavelet transform
coefficients of function f (n) are defined as
1
Wϕ ( j0 , k) = √
M
1
Wψ ( j, k) = √
M
M−1
f (n)ϕ j0 ,k (n),
(3.4.4)
n=0
M−1
f (n)ψ j,k (n),
j ≥ j0 ,
(3.4.5)
n=0
and Wϕ ( j0 , k) and Wψ ( j, k) are called approximation coefficient and detail coefficient, respectively. The DFT coefficients enable us to reconstruct the discrete signal
f (x) in l 2 (Z) by
1
f (n) = √
M
k
1
Wϕ ( j0 , k)ϕ j0 ,k (n) + √
M
∞
Wψ ( j, k)ψ j,k (n).
j= j0
(3.4.6)
k
Here f (n), ϕ j0 ,k (n) and ψ j,k (n) are discrete functions defined in [0, M − 1].
Generally, we take j0 = 0 and select M = 2 J , so that the summations in
3.4 Wavelet Transform
55
Eqs. (3.4.4)–(3.4.6) are performed over n = 0, 1, 2, . . . , M−1, j = 0, 1, 2, . . . , J −
1 and k = 0, 1, 2, . . . , 2 j − 1.
There is another way to define DWT which will be described in next subsection.
It is necessary to mention here that we are only providing the details of construction
of discrete wavelet transform from MRA. See Daubechies [7], Walnut [38] for more
details and proof of the theorems provided in this subsection.
3.4.3 Motivation: From MRA to Discrete Wavelet Transform
Let {V j } j∈Z be an MRA with scaling function ϕ(x). Then the refinement equation
of ϕ(x) is
√
ϕ(x) =
h(n)ϕ1,n (x) = 2
h(n)ϕ(2x − n),
n
and the corresponding wavelet ψ(x) is defined by
ψ(x) =
g(n)ϕ1,n (x) =
√
2
g(n)ϕ(2x − n),
n
where g(n) = (−1)n h(1 − n). For any j, k ∈ Z,
ϕ j,k (x) = 2 j/2 ϕ(2 j x − k)
√
h(n)ϕ(2(2 j x − k) − n)
= 2 j/2 2
(using Eq. (3.3.7))
n
= 2( j+1)/2
= 2( j+1)/2
n
n
h(n)ϕ(2 j+1 x − 2k − n)
h(n − 2k)ϕ(2 j+1 x − n)
h(n − 2k)ϕ j+1,n (x).
=
(3.4.7)
n
Similarly,
ψ j,k (x) =
g(n − 2k)ϕ j+1,n (x).
n
For k ∈ Z, define
c0 (k) = f, ϕ0,k ,
and for j ∈ N and k ∈ Z, define c j (k) and d j (k) by
(3.4.8)
56
3 Wavelets and Wavelet Transform
c j (k) = f, ϕ− j,k
and d j (k) = f, ψ− j,k .
Then, by using Eq. (3.4.7)
c j+1 (k) = f, φ− j−1,k
= f,
=
n
n
=
h(n − 2k)φ− j,n
h(n − 2k) f, φ− j,n
c j (n)h(n − 2k)
(3.4.9)
n
and by Eq. (3.4.8), we have
d j+1 (k) =
c j (n)g(n − 2k).
(3.4.10)
n
In order to see that the calculation of c j+1 (k) and d j+1 (k) is completely reversible,
recall that by Eq. (3.3.1)
P− j f (x) =
f, ϕ− j,n ϕ− j,n (x) =
n
c j (n)ϕ− j,n (x),
(3.4.11)
d j (n)ψ− j,n (x).
(3.4.12)
n
and
Q − j f (x) =
f, ψ− j,n ψ− j,n (x) =
n
n
Also, by Eq. (3.3.1), we have
P− j f (x) = P− j−1 f (x) + Q − j−1 f (x).
Using Eqs. (3.4.11) and (3.4.12), we have
c (n)ϕ− j,n (x) =
n j
c
(n)ϕ− j−1,n (x) +
n j+1
=
c
(n)
n j+1
+
=
n
k
d j+1 (n)
k
n
d j+1 (n)ψ− j−1,n (x)
h(k − 2n)ϕ− j,k (x)
k
g(k − 2n)ϕ− j,k (x)
c
(n)h(k − 2n) +
n j+1
n
d j+1 (n)g(k − 2n) ϕ− j,k (x).
By matching the coefficients, we conclude that
c j (k) =
c j+1 (n)h(k − 2n) +
n
d j+1 (n)g(k − 2n).
n
(3.4.13)
3.4 Wavelet Transform
57
We summarize these results in the following theorem:
Theorem 3.2 Let {V j } be an MRA with associated scaling function ϕ(x), scaling
filter h(k), wavelet function ψ(x) and wavelet filter g(k). Given a function f (x) in
L 2 (R), define for k ∈ Z,
c0 (k) = f, ϕ0,k ,
and for j ∈ N and k ∈ Z, define c j (k) and d j (k) by
c j (k) = f, ϕ− j,k
and d j (k) = f, ψ− j,k .
Then
c j+1 (k) =
c j (n)h(n − 2k)
d j+1 (k) =
n
c j (n)g(n − 2k)
n
and
c j (k) =
c j+1 (n)h(k − 2n) +
n
d j+1 (n)g(k − 2n).
n
The above theorem suggests that the key object in calculating f, ϕ and f, ψ is
the scaling filter h(k) and not the scaling function ϕ(x). It also suggests that as long as
Eq. (3.4.13) holds, Eqs. (3.4.9) and (3.4.10) define an exactly invertible transform for
signals. The question is: What conditions must the scaling filter h(k) satisfy in order
for the transform defined by Eqs. (3.4.9) and (3.4.10) to be invertible by Eq. (3.4.13)?
These properties will be referred as quadrature mirror filter (QMF) conditions and
will be used to define the discrete wavelet transform.
3.4.4 The Quadrature Mirror Filter Conditions
In this subsection, we will provide the QMF conditions, reformulate them in the
language of certain filtering operations on signals called the approximation and detail
operators, and finally give a very simple characterization of the QMF conditions that
will be used in the design of wavelet filter and scaling filter.
Theorem 3.3 Let {V j } be an MRA with scaling filter h(k) and the wavelet filter
g(k). Then
√
(a) n h(n) = 2,
(b) n g(n) = 0,
(c) k h(k)h(k − 2n) = k g(k)g(k − 2n) = δ(n),
(d) k g(k)h(k − 2n) = 0, f or all n ∈ Z, and
(e) k h(m − 2k)h(n − 2k) + k g(m − 2k)g(n − 2k) = δ(n − m).
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3 Wavelets and Wavelet Transform
Proof (a)
R
ϕ(x)d x =
h(n)2−1/2
h(n)21/2 ϕ(2x − n)d x =
R
n
ϕ(x)d x.
R
n
Since we know that R ϕ(x)d x = 0, we can cancel the nonzero factor
from both sides. it follows that
√
2.
h(n) =
R
ϕ(x)d x
(3.4.14)
n
(b) From Eq. (3.2.2),
0=
R
ψ(x)d x = 0, so that
R
ψ(x)d x =
g(n)2
R
1/2
ϕ(2x − n)d x =
g(n)2
n
−1/2
ϕ(x)d x.
R
n
Hence,
g(n) = 0.
(3.4.15)
n
This is also equivalent to the statement that
h(2n) =
n
h(2n + 1)
(3.4.16)
n
(c) Since {ϕ0,n (x)} and {ϕ1,n (x)} are the orthonormal systems on R,
R
ϕ(x)ϕ(x − n) =
R
h(k)ϕ1,k (x)
=
k
R
=
k
=
h(m)ϕ1,m (x − n)d x
m
k
k
h(k)ϕ1,k (x)
h(m)ϕ1,m+2n (x)d x
m
m
h(k)h(m − 2n)
R
ϕ1,k (x)ϕ1,m (x)d x
h(k)h(k − 2n).
Hence,
h(k)h(k − 2n) = δ(n).
(3.4.17)
k
Since {ψ0,n (x)}n∈Z is also an orthonormal systems on R, the same argument gives
g(k)g(k − 2n) = δ(n).
k
(3.4.18)
3.4 Wavelet Transform
59
(d) Since ψ0,n , ϕ0,m (x) = 0 for all m, n ∈ Z, we have
g(k)h(k − 2n) = 0,
(3.4.19)
k
for all n ∈ Z.
(e) Since for any signal c0 (n),
c0 (n) =
c1 (k)h(n − 2k) +
k
d1 (k)g(n − 2k),
k
where
c1 (k) =
c0 (m)h(m − 2k)
m
and
d1 (k) =
c0 (m)g(m − 2k),
m
it follows that
c0 (n) =
c0 (m)h(m − 2k)h(n − 2k) +
k
m
c0 (m)g(m − 2k)g(n − 2k),
m
k
=
c0 (m)
m
h(m − 2k)h(n − 2k) +
k
g(m − 2k)g(n − 2k) .
k
Hence we must have
h(m − 2k)h(n − 2k) +
k
g(m − 2k)g(n − 2k) = δ(n − m).
(3.4.20)
k
√
Remark 3.1 Condition (a) is referred to as a normalization condition. The value 2
arises from the fact that we have chosen to write the two-scale dilation equation as
ϕ(x) = n h(n)21/2 ϕ(2x − n). In some of the literature on wavelets and especially
on two-scale dilation equations, the equation is written ϕ(x) = n h(n)ϕ(2x − n).
This leads to the normalization, n h(n) = 2. The choice of normalization is just a
convention and has no real impact on any of the results that follow.
Remark 3.2 Conditions (c) and (d) are referred to as orthogonality conditions since
they are immediate consequences of the orthogonality of the scaling functions at a
given scale, the orthogonality of the wavelet functions at a given scale, and the fact
that the wavelet functions are orthogonal to all scaling functions at a given scale.
60
3 Wavelets and Wavelet Transform
Remark 3.3 Condition (e) is referred to as the perfect reconstruction (PR) condition
since it follows from the reconstruction formula for orthonormal wavelet bases.
Definition 3.5 Let c(n) be a signal.
(a) Given m ∈ Z, the shift operator τm is defined by
τm c(n) = c(n − m).
(3.4.21)
(b) the downsampling operator ↓ is defined by
(↑ c)(n) = c(2n),
(3.4.22)
i.e. (↓ c)(n) is formed by removing every odd term in c(n).
(c) the upsampling operator ↑ is defined by
n
c 2 , n is even
(↑ c)(n) =
0, i f n is odd
(3.4.23)
i.e. (↑ c)(n) is formed by inserting a zero between adjacent entries of c(n).
Definition 3.6 Given a signal c(n) and a filter h(k), define g(k) by g(k) =
(−1)k h(1 − k). Then the approximation operator H and detail operator G corresponding to h(k) are defined by
c(n)h(n − 2k),
(H c)(k) =
(Gc)(k) =
n
c(n)g(n − 2k).
(3.4.24)
n
The approximation adjoint H ∗ and detail adjoint G ∗ are defined by
(H ∗ c)(k) =
c(n)h(k − 2n),
(G ∗ c)(k) =
n
c(n)g(k − 2n).
(3.4.25)
n
Remark 3.4 The operators H and G can be thought of as convolution with the filters
h(n) = h(−n) and g(n) = g(−n) followed by downsampling. That is,
(H c)(n) =↓ (c ∗ h)(n),
(Gc)(n) =↓ (c ∗ g)(n).
(3.4.26)
Remark 3.5 H ∗ and G ∗ can be thought of as upsampling followed by convolution
with the filters h and g. That is,
(H ∗ c)(n) = (↑ c) ∗ h(n) and (G ∗ c)(n) = (↑ c) ∗ g(n).
(3.4.27)
Remark 3.6 The operators H ∗ and G ∗ are the formals adjoints of H and G. That is,
for all signals c(n) and d(n),
3.4 Wavelet Transform
H c, d =
61
(H c)(k)d(k) =
k
c(k)(H ∗ d)(k) = c, H ∗ d
(3.4.28)
c(k)(G ∗ d)(k) = c, G ∗ d
(3.4.29)
k
and
(Gc)(k)d(k) =
Gc, d =
k
k
Now, we can reformulate the conditions of Theorem 3.3(c)–(e) as follows:
Theorem 3.4 Given a scaling filter h(k) and the wavelet filter g(k) defined by g(k) =
(−1)k h(1 − k). Then
√
(a) n h(n) = 2,
(b) n g(n) = 0,
∗
(c) k h(k)h(k − 2n) =
=
k g(k)g(k − 2n) = δ(n) if and only if H H
∗
GG = I, where I is the identity operator on sequences,
(d) k g(k)h(k − 2n) = 0, f or all n ∈ Z, if and only if H G ∗ = G H ∗ =
0, and
(e) k h(m − 2k)h(n − 2k) + k g(m − 2k)g(n − 2k) = δ(n − m) if and only
if H ∗ H + G ∗ G = I.
Next, we will show that all of the conditions in Theorem 3.3 can be written as a
single condition (Theorem 3.4) on the auxiliary function m 0 (ξ) = √12 n h(n)e−inξ
plus the normalization condition m 0 (0) = 1. These two conditions will be referred
to as the quadrature mirror filter (QMF) conditions.
Theorem 3.5 Given a filter h(k), define g(k) by g(k) = (−1)k g(1 − k), m o (ξ) by
Eq. (3.3.10), m ψ (ξ) or m 1 (ξ) by Eq. (3.3.14), and the operators H , G, H ∗ and G ∗
by Eqs. (3.4.24) and (3.4.25). Then the following are equivalent:
(a) |m 0 (ξ)|2 + |m 0 (ξ + π)|2 = 1.
(b) n h(n)h(n − 2k) = δ(k).
(c) H ∗ H + G ∗ G = I.
(d) H H ∗ = GG ∗ = I.
Definition 3.7 Given a filter h(k) and m o (ξ). Then h(k) is a QMF provided that:
(a) m 0 (0) = 1 and
(b) |m 0 ( 2ξ )|2 + |m 0 ( 2ξ + π)|2 = 1 for all ξ ∈ R.
Theorem 3.6 Suppose that h(k) is a QMF and define g(k) by g(k) = (−1)k g(1 − k).
Then the following√
holds:
(a) n h(n) = 2,
(b) n g(n) = 0,
(c) n h(2n) = n h(2n + 1),
(c) k h(k)h(k − 2n) = k g(k)g(k − 2n) = δ(n),
(d) k g(k)h(k − 2n) = 0, f or all n ∈ Z, and
(e) k h(m − 2k)h(n − 2k) + k g(m − 2k)g(n − 2k) = δ(n − m).
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3 Wavelets and Wavelet Transform
Definition 3.8 Let h(k) be a QMF, define g(k) by g(k) = (−1)k g(1 − k) and let
H, G, H ∗ and G ∗ be given by Eqs. (3.4.24) and (3.4.25). Fix J ∈ N, the discrete
wavelet transform (DWT) of a signal c0 (n), is the collection of sequences
{d j (k) : 1 ≤ j ≤ J ; k ∈ Z} ∪ {c J (k) : k ∈ Z},
(3.4.30)
c J +1 (n) = (H c J )(n) and d J +1 (n) = (Gc J )(n).
(3.4.31)
where
The inverse discrete wavelet transform (IDWT) is defined by
c J (n) = (H ∗ c J +1 )(n) + (G ∗ d J +1 )(n).
(3.4.32)
If J = ∞, then the DWT of c0 is the collection of sequences
{d j (k) : j ∈ N; k ∈ Z}.
(3.4.33)
3.5 The Fast Wavelet Transform
The fast wavelet transform (FWT) is a mathematical algorithm designed to turn a
waveform or signal in the time domain into a sequence of coefficients based on
an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain
is replaced with the space domain. It has as theoretical foundation the device of a
finitely generated, orthogonal multiresolution analysis (MRA). Transform coding
is a widely used method of compressing image information. In a transform-based
compression system two-dimensional (2D) images are transformed from the spatial domain to the frequency domain. An effective transform will concentrate useful
information into a few of the low-frequency transform coefficients.
We first consider the refinement equation (3.3.4) of multiresolution analysis
ϕ(x) =
√
h ϕ (k)ϕ(2x − k).
2
(3.5.1)
k∈Z
By a scaling of x by 2 j , translation of x by n, and making m = 2n + k, we would
get
ϕ(2 j x − k) =
=
√
√
2
k∈Z
h ϕ (k)ϕ(2(2 j x − n) − k)
h ϕ (m − 2n)ϕ(2( j+1) x − m),
2
m∈Z
(3.5.2)
3.5 The Fast Wavelet Transform
63
and analogously,
ψ(2 j x − k) =
√
2
h ψ (m − 2n)ϕ(2( j+1) x − m),
(3.5.3)
m∈Z
where h ϕ (k) = (−1)k h ψ (1−k). A property that involves the convolution of a scaling
function and a wavelet coefficient can be derived by the following steps: We begin by
considering the definition of the discrete wavelet transform as shown in Eqs. (3.4.4)
and (3.4.5). By substituting equation (3.2.6) in Eq. (3.4.5), we get
1
Wψ ( j, k) = √
M
M−1
f (x)2 j/2 ψ(2 j x − k),
(3.5.4)
x=0
using Eq. (3.5.3), we have
1
Wψ ( j, k) = √
M
M−1
f (x)2
j/2
√
2
x=0
h ψ (m − 2n)ϕ(2
( j+1)
x − m) . (3.5.5)
m∈Z
Rearrange the summation part of the equation, we have
1
Wψ ( j, k) =
h ψ (m − 2n) √
M
m∈Z
M−1
f (x)2( j+1)/2 ϕ(2( j+1) x − m) , (3.5.6)
x=0
where the bracketed quantity is identical to Eq. (3.5.4) with j0 = j + 1. Therefore,
Wψ ( j, k) =
h ψ (m − 2n)Wϕ ( j + 1, m),
(3.5.7)
m∈Z
and similarly the DWT approximation coefficient at scale j + 1 can be expressed as
Wϕ ( j, k) =
h ϕ (m − 2n)Wϕ ( j + 1, m).
(3.5.8)
m∈Z
Equations (3.5.7) and (3.5.8) demonstrate that both the approximation coefficient
Wϕ ( j, k) and the detail coefficient Wψ ( j, k) can be obtained by convolving Wϕ ( j, k),
approximation coefficients at the scale j + 1, with the time-reversed scaling and
wavelet vectors, h ϕ (−n) and h ψ (−n) followed by the subsequent sub-sampling. The
Eqs. (3.5.7) and (3.5.8) can then expressed in the following convolution formats:
Wψ ( j, k) = h ψ (−n) ∗ Wϕ ( j + 1, n), for n = 2k, k ≥ 0,
(3.5.9)
Wϕ ( j, k) = h ϕ (−n) ∗ Wϕ ( j + 1, n), for n = 2k, k ≥ 0.
(3.5.10)
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3 Wavelets and Wavelet Transform
Fig. 3.1 An FWT analysis filter bank
Fig. 3.2 Two scale FWT analysis filter bank
For the commonly used discrete signal, say, a digital image, the original data can
be viewed as approximation coefficients with order J . That is, f [n] = W [J ; n] by
Eqs. (3.4.4) and (3.4.5), next level of approximation and detail can be obtained. This
algorithm is “fast” because one can find the coefficients level by level rather than
directly using Eqs. (3.4.4) and (3.4.5) to find the coefficients. For a sequence of length
M = 2 J , the number of mathematical operations involved is on the order of O(M).
That is, the number of multiplications and additions is linear with respect to the length
of the input sequence since the number of multiplications and additions involved in
the convolutions performed by the FWT analysis bank (see Fig. 3.1) is proportional
to the length of the sequences being convolved. Hence, the FWT compares favorably
with the FFT algorithm, which requires on the order of O(Mlog2 M) operations.
We simply note that the filter bank in Fig. 3.1 can be iterated to create multistage structures for computing DWT coefficients at two or more successive scales.
Figure 3.2 shows the two scale FWT analysis filter bank.
As one might expect, a fast inverse transform for the reconstruction of from the
results of the forward transform can be formulated, called the inverse fast wavelet
transform. It uses the scaling and wavelet vectors employed in the forward transform,
together with the level j − 1 approximation and detail coefficients, to generate the j
level approximation coefficients (see Fig. 3.3). As with the forward FWT, the inverse
filter bank can also be iterated as shown in Fig. 3.4.
3.6 Why Use Wavelet Transforms
65
Fig. 3.3 The inverse FWT synthesis filter bank
Fig. 3.4 Two scale inverse FWT synthesis filter bank
3.6 Why Use Wavelet Transforms
Wavelets and wavelet transforms (including the DWT) have many advantages over
rival multiscale analysis techniques, such as the fast Fourier transform (FFT). The
key advantages are listed below:
Structure Extraction wavelets can be used to analyze the structure (shape) of a
function, that is the coefficients d j,k tell you how much of the corresponding wavelet
ψ j,k makes up the function.
Localization If the function f (x) has a discontinuity at x ∗ then only the wavelets
ψ j,k (x) which overlap the discontinuity will be affected and the associated wavelet
coefficients, d j,k will show this.
Efficiency Wavelet transforms are usually much faster than other methods (or at
least as good as). For example the discrete wavelet transform is O(n) whereas the
FFT is O(nlogn).
Sparsity If the right wavelet function is chosen then the coefficients computed can
be very sparse. This means that the key information from the function can be summed
up concisely. This is obviously advantageous when it comes to storing details about
a function or sequence, as it will take up less space. When images are looked at this
is especially of use due to the fact the image files tend to be quite large.
66
3 Wavelets and Wavelet Transform
3.7 Two-Dimensional Wavelets
Two-dimensional (2D) wavelets are a natural extension from the single dimension
case. As a concept they can be applied to many 2D situations, such as 2D functional
spaces. However they really come into there own when images are considered. In
a world where digital images are processed by computers ever second, methods
for condensing the information carried in an image are needed. Also with so many
different images in circulation via the internet methods are needed for computational
analysis of the content of these images. 2D wavelets provide ways to tackle both of
these problems.
Before 2D wavelets are introduced, it will be helpful to know about outer products
or tensor products. Outer products work in a similar way to inner products, however
whilst inner products take two vectors and combine to form a scalar, outer products
work the other way, extrapolating a matrix from two vectors. In the case of the
standard outer product this is done in a natural way.
Definition 3.9 Consider two vectors a and b of length n and m respectively. Then
we can define the standard outer product, ⊗, as follows:
a ⊗ b = ab T = C,
(3.7.1)
where C is a matrix with elements determined by:
Ci, j = ai × b j .
(3.7.2)
This concept of outer products can then be used to define 2D discrete wavelets. This
is done by taking the outer products of one-dimensional scaling function and wavelet
function as follows:
Definition 3.10 For 1D wavelets ψ and scaling function ϕ, the 2D scaling function
and wavelet functions are defined by the matrices given by:
(x, y) = ϕ(x)ϕ(y),
(3.7.3)
H (x, y) = ϕ(x)ψ(y),
(3.7.4)
V (x, y) = ψ(x)ϕ(y),
(3.7.5)
D (x, y) = ψ(x)ψ(y),
(3.7.6)
and
where H measures the horizontal variations(horizontal edges), V corresponds
to the vertical variations (vertical edges) and D detects the variations along the
diagonal directions.
3.7 Two-Dimensional Wavelets
67
For each j, m, n ∈ Z, define
j,m,n (x, y) = 2 j (2 j x − m, 2 j y − n) = ϕ j,m (x)ϕ j,n (y),
(3.7.7)
H
(x, y) = 2 j H (2 j x − m, 2 j y − n) = ϕ j,m (x)ψ j,n (y),
j,m,n
(3.7.8)
Vj,m,n (x, y) = 2 j V (2 j x − m, 2 j y − n) = ψ j,m (x)ϕ j,n (y),
(3.7.9)
D
j,m,n
(x, y) = 2 j D (2 j x − m, 2 j y − n) = ψ j,m (x)ψ j,n (y).
(3.7.10)
Theorem 3.7 (i) The collection
{ ij,m,n (x, y)}i=H,V,D, j,m,n∈Z
(3.7.11)
form an orthonormal basis on R2 .
(ii) For each J ∈ Z, the collection
{ J,m,n (x, y)}m,n∈Z ∪ { iJ,m,n (x, y)}i=H,V,D, j≥J,m,n∈Z
(3.7.12)
form an orthonormal basis on R2 .
The proof of the above theorem is beyond the scope of this book.
3.8 2D-discrete Wavelet Transform
Given separable 2D scaling and wavelet functions, extension of the 1D DWT to two
dimensions is straightforward. The two-dimensional (2D) discrete wavelet transform
of function f (x, y) of size M × N is defined as
Wϕ ( j0 , m, n) = √
Wψi ( j, m, n) = √
1
MN
1
MN
M−1 N −1
f (x, y)ϕ j0 ,m,n (x, y),
(3.8.1)
x=0 y=0
M−1 N −1
f (x, y)ψ ij,m,n (x, y),
i = {H, V, D},
(3.8.2)
x=0 y=0
where
ϕ j,m,n (x, y) = 2 j ϕ(2 j x − m, 2 j y − n),
ψ ij,m,n (x, y) = 2 j ψ(2 j x − m, 2 j y − n),
i = {H, V, D}.
(3.8.3)
(3.8.4)
68
3 Wavelets and Wavelet Transform
Fig. 3.5 The 2D FWT analysis filter bank
Fig. 3.6 The 2D inverse FWT synthesis filter bank
As in one dimensional case, j0 is an arbitrary starting scale and the Wϕ ( j0 , m, n)
coefficients define an approximation of f (x, y) at scale j0 . The Wψ ( j, m, n) coefficients add horizontal, vertical, and diagonal details for scales j ≥ j0 . Given the Wϕ
and Wψi , f (x, y) is obtained via the two-dimensional (2D) inverse discrete wavelet
transform
f (x, y) = √
1
Wϕ ( j0 , m, n)ϕ j0 ,m,n (x, y) + √
MN
m
n
1
MN
i=H,V,D
∞
Wψi ( j, m, n)ψ ij,m,n (x, y).
j= j0 m
(3.8.5)
n
Likewise the 1D discrete wavelet transform, the 2D DWT can be implemented
using digital filters and downsamplers. With separable two-dimensional scaling and
wavelet functions, we simply take the 1D FWT of the rows of f (x, y), followed by
the 1D FWT of the resulting columns (see Figs. 3.5 and 3.6).
3.9 Continuous Wavelet Transform
69
3.9 Continuous Wavelet Transform
3.9.1 1D Continuous Wavelet Transform
The continuous wavelet transform (CWT) of one-dimensional signal f (t) is given
by
∞
t −b
1
dt.
(3.9.1)
f (t)ψ ∗
Wψ (a, b) = √
a
|a| −∞
The continuous wavelet transform is a function of two variables a and b, where a
is the scaling parameter and b is the shifting parameter. Here ψ(t) is the mother
wavelet or the basic function. The scale parameter a gives the frequency information
in the wavelet transform. The translating parameter or shifting parameter b gives the
time information in the wavelet transform. It indicates the location of the window
as it is shifted through the signal. A low scale corresponds to wavelets of smaller
width, which gives the detailed information in the signal. A high scale corresponds
to wavelets of larger width which gives the global view of the signal.
The inverse continuous wavelet transform (ICWT) of one-dimensional signal is
given by
∞ ∞
dbda
1
Wψ (a, b)ψa,b (t) 2 ,
(3.9.2)
f (t) =
Cψ 0
a
−∞
where ψa,b (t) =
√1 ψ
|a|
t−b a
.
The difference between the wavelet and windowed Fourier transform or STFT
lies in the shapes of the analyzing functions gτ ,ω and ψa,b . The functions gτ ,ω , all
consist of the same envelope function g, translated to the proper time location, and
“filled in” with higher frequency oscillations. All the gτ ,ω , regardless of the value
of ω, have the same width. In contrast, the ψa,b have time-widths adapted to their
frequency: high frequency ψa,b are very narrow, while low frequency ψa,b are much
broader. As a result, the wavelet transform is better able than the windowed Fourier
transform or STFT to “zoom in” on very short-lived high frequency phenomena,
such as transients in signals (or singularities in functions or integral kernels).
3.9.2 2D Continuous Wavelet Transform
The two-dimensional (2D) continuous wavelet transform (CWT) of signal f (x, y)
is given by
1
Wψ (a, b) = √
|a|
∞
−∞
f (x, y)ψ
∗
x −m y−n
,
a
a
d xd y,
(3.9.3)
70
3 Wavelets and Wavelet Transform
where m, n are shifting parameter or translation parameter and a is the scaling parameter.
3.10 Undecimated Wavelet Transform or Stationary
Wavelet Transform
We know that the classical DWT suffers a drawback that the DWT is not a timeinvariant transform. This means that, even with periodic signal extension, the DWT
of a translated version of a signal x is not, in general, the translated version of the
DWT of x.
How to restore the translation invariance, which is a desirable property lost by
the classical DWT? The idea is to average some slightly different DWT, called decimated DWT, to define the Stationary wavelet transform (SWT) [19].
-decimated DWT: There exist a lot of slightly different ways to handle the discrete wavelet transform. Let us recall that the DWT basic computational step is a
convolution followed by a decimation. The decimation retains even indexed elements.
But the decimation could be carried out by choosing odd indexed elements instead
of even indexed elements. This choice concerns every step of the decomposition process, so at every level we chose odd or even. If we perform all the different possible
decompositions of the original signal, we have 2 J different decompositions, for a
given maximum level J .
Let us denote by j = 1 or 0 the choice of odd or even indexed elements at step j.
Every decomposition is labeled by a sequence of 0 s and 1 s : = 1 , 2 , ···, J . This
transform is called the -decimated DWT or stationary wavelet transform (SWT).
3.11 Biorthogonal Wavelet Transform
A biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal. Designing biorthogonal wavelets allows more
degrees of freedom than orthogonal wavelets. One additional degree of freedom is
the possibility to construct symmetric wavelet functions.
3.11.1 Linear Independence and Biorthogonality
The notion of the linear independence of vectors is an important concept in
the theory of finite-dimensional vector spaces. Specifically, a collection of vectors {x1 , x2 , . . . , xn } in Rn is linearly independent if any collection of scalars
3.11 Biorthogonal Wavelet Transform
71
{a1 , a2 , . . . , an } such that
a1 x 1 + a2 x 2 + · · · + an x n = 0
(3.11.1)
must satisfy a1 = a2 = · · · = an = 0. If in addition m = n that is, if the number of
vectors in the set matches the dimension of the space, then {x1 , x2 , . . . , xn } is called
a basis for Rn . This means that any vector x ∈ Rn has a unique representation as
x = b1 x1 + b2 x2 + · · · + bn xn ,
(3.11.2)
where bi s are real scalars. For computing bi s, there exists a unique collection of
x2 , . . . , xn } called the dual basis that is biorthogonal to the collection
vectors {
x1 , {x1 , x2 , . . . , xn }. That is
(3.11.3)
xi , x j = δ(i − j).
xi , x . In generalizing the notion of a basis to the
Hence, bi s are given by bi = infinite-dimensional setting, we retain the notion of linear independence.
Definition 3.11 A collection of functions {gn }n∈N ∈ L 2 (R) is linearly independent
if given any l 2 -sequence of coefficients a(n) such that
∞
a(n)gn (x) = 0
(3.11.4)
n=1
in L 2 (R), then a(n) = 0 for all n ∈ N.
Definition 3.12 A collection of functions {
gn }n∈N ∈ L 2 (R) is biorthogonal to a
2
collection {gn }n∈N ∈ L (R), if
gm =
gn (x)
gm (x) = δ(n − m).
(3.11.5)
gn ,
R
It is often difficult to verify directly whether a given collection of functions is
linearly independent. The next lemma gives a sufficient condition for linear independence of collection of functions:
Lemma 3.1 Let {gn (x)} be a collection of functions in L 2 (R), suppose that there is
a collection {
gn (x)}n∈N ∈ L 2 (R) biorthogonal to {gn (x)}. Then {gn (x)} is linearly
independent.
Proof Let {a(n)}n∈N be an l 2 -sequence, and satisfy
∞
a(n)gn (x) = 0
n=1
in L 2 (R). Then for each m ∈ N,
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3 Wavelets and Wavelet Transform
∞
0 = 0,
gm =
∞
a(n)gn (x),
gm (x) =
n=1
a(n) gn (x),
gm (x) = a(m)
n=1
by biorthogonality. Therefore {gn (x)} is linearly independent.
3.11.2 Dual MRA
j } j∈Z
Definition 3.13 A pair of MRA’s {V j } j∈Z with scaling function ϕ(x) and {V
with scaling function ϕ
(x) are dual to each other if ϕ(x − n) is biorthogonal to
ϕ
(x − n).
Since there may be more than one function ϕ
(x) such that ϕ(x −n) is biorthogonal
j } j∈Z dual to {V j } j∈Z .
to ϕ
(x − n), there may be more than one MRA {V
Definition 3.14 Let ϕ(x) and ϕ
(x) be scaling function for dual MRA. For each
J , and the detail operators Q J and
j ∈ Z, define an approximation operators PJ , P
2
Q J on L (R) functions f (x) by
PJ f (x) =
f, ϕ
j,k ϕ j,k (x),
(3.11.6)
f, ϕ j,k ϕ
j,k (x),
(3.11.7)
k
J f (x) =
P
k
Q J f (x) = PJ +1 f (x) − PJ f (x),
(3.11.8)
J +1 f (x) − P
J f (x).
J f (x) = P
Q
(3.11.9)
Definition 3.15 Let ϕ(x) and ϕ
(x) be scaling function for dual MRA and let h(n)
and h(n) be the scaling filters corresponding to ϕ(x) and ϕ
(x). Define the filters
g(n) and g (n) by
h(1 − n) and g (n) = (−1)n h(1 − n).
g(n) = (−1)n
(3.11.10)
by
Define the wavelet ψ(x) and the dual wavelet ψ(x)
=
g(n)21/2 ϕ(2x − n) and ψ(x)
ψ(x) =
n
g (n)21/2 ϕ
(2x − n). (3.11.11)
n
The following illustrate some basic properties of the wavelet and its dual. Let ψ(x)
be the wavelet and dual wavelet corresponding to the MRA’s {V j } j∈Z with
and ψ(x)
j } j∈Z with scaling function ϕ
(x). Then the following
scaling function ϕ(x) and {V
holds:
3.11 Biorthogonal Wavelet Transform
73
∈V
1 .
(a) ψ(x) ∈ V1 and ψ(x)
0,n (x)} is biorthogonal to {ψ0,n (x)}.
(b) {ψ
0,n (x)} is a orthogonal
(c) {ψ0,n (x)} is a orthogonal basis for span{ψ0,n (x)} and {ψ
0,n (x)}.
basis for span{ψ
0,n , ϕ0,n = 0.
0,n = ψ
(d) For all n, m ∈ Z, ψ0,n , ϕ
(e) for any f (x) ∈ Cc0 (R),
0,n (x)}
0 f (x) ∈ span{ψ
Q 0 f (x) ∈ span{ψ0,n (x)} and Q
j,k (x)} j,k∈Z defined
It can be easily seen that the collections {ψ j,k (x)} j,k∈Z and {ψ
by Eq. (3.11.11) are orthogonal basis on R.
3.11.3 Discrete Transform for Biorthogonal Wavelets
As with orthogonal wavelets, there is a very simple and fast discrete version of the
biorthogonal wavelet expansion. Recall that by Definition 3.14, for any j ∈ Z,
− j f (x) =
P
f, ϕ− j,n ϕ
− j,n (x) =
n
and
c j (n)
ϕ− j,n (x),
(3.11.12)
− j,n (x).
d j (n)ψ
(3.11.13)
n
− j,n (x) =
f, ψ− j,n ψ
− j f (x) =
Q
n
n
Also by definition 3.14,
− j−1 f (x) + Q
− j−1 f (x).
− j f (x) = P
P
Writing out in terms of equations (3.11.12) and (3.11.13), we have
c j (n)
ϕ− j,n (x) =
n
d j+1 (n)
ψ− j−1,n (x)
c j+1 (n)
ϕ− j−1,n (x) +
n
=
n
h(k − 2n)
ϕ− j,k (x) +
c j+1 (n)
n
c j+1 (n)
h(k − 2n) +
=
k
d j+1 (n)
n
k
n
g(k − 2n)
ϕ− j,k (x)
k
d j+1 (n)
g(k − 2n) ϕ− j,k (x).
n
By matching the coefficients, we conclude that
c j+1 (n)
h(k − 2n) +
c j (k) =
n
d j+1 (n)
g (k − 2n).
n
(3.11.14)
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3 Wavelets and Wavelet Transform
Now, we summarize these results in the following theorem:
Theorem 3.8 Let ϕ(x) and ϕ
(x) be scaling function for dual MRA’s and let h(n) and
h(n) be the scaling filters corresponding to ϕ(x) and ϕ
(x). Define the wavelet filters
g(n) and g (n) by Eq. (3.11.10) and the wavelets ψ(x) and ψ(x)
by Eq. (3.11.11).
Given a function f (x) ∈ L 2 (R), define for k ∈ Z,
c0 (k) = f, ϕ0,k ,
and for every j ∈ N and k ∈ Z, define c j (k) and d j (k) by
c j (k) = f, ϕ− j,k
and d j (k) = f, ψ− j,k .
Then
c j+1 (k) =
c j (n)h(n − 2k)
d j+1 (k) =
n
c j (n)g(n − 2k)
(3.11.15)
n
and
c j+1 (n)
h(k − 2n) +
c j (k) =
n
d j+1 (n)
g (k − 2n).
(3.11.16)
n
The operations in Eq. (3.11.15) are precisely the approximation operator and detail
operator corresponding to the filters h(n) and g(n). Equation (3.11.16) involves the
approximation adjoint and detail adjoint corresponding to the filters h(n) and g (n).
This leads to the following definition:
Definition 3.16 Given a signal c(n) and a pair of filters h(k) and h(k), define g(k) and
g (k) by g(k) = (−1)k
h(1 − k) and
g (k) = (−1)k h(1 − k). Define the corresponding
and detail operators G and G
on signal c(n) by
approximation operators H and H
c(n)h(n − 2k),
(H c)(k) =
(Gc)(k) =
n
c(n)g(n − 2k),
(3.11.17)
n
c)(k) =
(H
c(n)
h(n − 2k),
(Gc)(k)
=
c(n)
g (n − 2k),
n
n
∗ and detail adjoints G ∗ , G
∗ are defined by
and the approximation adjoints H ∗ , H
(H ∗ c)(k) =
c(n)h(k − 2n),
(G ∗ c)(k) =
n
c(n)g(k − 2n),
(3.11.18)
n
∗ c)(k) =
(H
c(n)
h(k − 2n),
n
∗ c)(k) =
(G
c(n)
g (k − 2n).
n
3.11 Biorthogonal Wavelet Transform
75
Theorem 3.9 Keeping the same notation as Theorem 3.8, we have
c j+1 = H c j ,
and
d j+1 = Gc j ,
(3.11.19)
∗ c j+1 + G
∗ d j+1 .
cj = H
(3.11.20)
Next, we will define the analogue of the QMF conditions in the biorthogonal case.
suppose that ϕ(x) and ϕ
(x) are scaling functions for dual GMRA’s, with scaling
filters h(n) and h(n) and g(n) and g (n) are corresponding wavelet filters. Then we
can prove the following analogue of Theorem 3.3.
Theorem 3.10 With h(n), h(n), g(n) and g (n) are defined as above. Then
(a) n h(n)h(n − 2k) = n g(n)
g (n − 2k) = δ(k),
g (n)h(n − 2k) = 0, f or all k ∈ Z, and
(b) n g(n)h(n − 2k) = n (c) k h(m − 2k)
h(n − 2k) + k g(m − 2k)
g (n − 2k) = δ(m − n).
We also have the following analogue of Theorem 3.4.
Theorem 3.11 With h(n), h(n), g(n) and g (n) are defined as above, define
1
m 0 (ξ) = √
2
1
h(n)e−inξ , m
0 (ξ) = √
2
h(n)e−inξ ,
(3.11.21)
0 (ξ + π), m
1 (ξ) = e−i(ξ+π) m 0 (ξ + π).
m 1 (ξ) = e−i(ξ+π) m
(3.11.22)
n
n
m 1 (ξ), and m
1 (ξ), by
∗ , G ∗ and G
∗ by Eqs. (3.11.17) and
, G, G,
H ∗, H
Define the operators H, H
(3.11.18). Then the following are equivalent:
m 0 (ξ) + m 0 (ξ + π)
m 0 (ξ + π) = 1.
(a) m 0 (ξ)
(b) n h(n)h(n − 2k) = δ(k).
∗ G = I.
∗ H + G
(c) H
∗
∗ = I.
(d) H H = G G
This leads to the following definition.
Definition 3.17 Given a filter h(k) and h(k), define m o (ξ) and m
0 (ξ) by
1
m 0 (ξ) = √
2
n
1
h(n)e−inξ , m
0 (ξ) = √
2
h(n)e−inξ .
n
Then h(k) and h(k) form a QMF provided that:
0 (0) = 1 and
(a) m 0 (0) = m
m 0 (ξ) + m 0 ( 2ξ + π)
m 0 (ξ + π) = 1, for all ξ ∈ R.
(b) m 0 ( 2ξ )
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3 Wavelets and Wavelet Transform
conditions (a) and (b) are called the biorthogonal QMF conditions.
Theorem 3.12 Given h(n) and√
h(n) are a QMF pair. Then:
(a) n h(n) = n h(n) = 2.
g (n) = 0.
(b) n g(n) = n g (n − 2k) = δ(k).
(c) n h(n)
h(n − 2k) = n g(n)
(d) n g(n)h(n − 2k) = n g(n)h(n − 2k) = 0, f or all k ∈ Z, and
h(n − 2k) + k g(m − 2k)
g (n − 2k) = δ(m − n).
(e) k h(m − 2k)
3.12 Scarcity of Wavelet Transform
Why have wavelets and multiscale analysis proved so useful in such a wide range
of applications? The primary reason is because they provide an extremely efficient
representation for many types of signals that appear often in practice but are not well
matched by the Fourier basis, which is ideally meant for periodic signals. In particular,
wavelets provide an optimal representation for many signals containing singularities.
The wavelet representation is optimally sparse for such signals, requiring an order of
magnitude fewer coefficients than the Fourier basis to approximate within the same
error. The key to the sparsity is that since wavelets oscillate locally, only wavelets
overlapping a singularity have large wavelet coefficients; all other coefficients are
small.
The sparsity of the wavelet coefficients of many real-world signals enables nearoptimal signal processing based on simple thresholding (keep the large coefficients
and kill the small ones), the core of a host of powerful image compression, denoising,
approximation, deterministic and statistical signal and image algorithms.
In spite of its efficient computational algorithm and sparse representation, the
wavelet transform suffers from four fundamental, intertwined shortcomings.
1. Oscillations
Since wavelets are bandpass functions, the wavelet coefficients tend to oscillate positive and negative around singularities. This considerably complicates wavelet-based
processing, making singularity extraction and signal modeling, in particular, very
challenging. Moreover, since an oscillating function passes often through zero, we
see that the conventional wisdom that singularities yield large wavelet coefficients
is overstated. Indeed, it is quite possible for a wavelet overlapping a singularity to
have a small or even zero wavelet coefficient.
2. Shift Variance
A small shift of the signal greatly perturbs the wavelet coefficient oscillation pattern
around singularities. Shift variance also complicates wavelet-domain processing.
Algorithms must be made capable of coping with the wide range of possible wavelet
3.12 Scarcity of Wavelet Transform
77
coefficient patterns caused by shifted singularities. To better understand wavelet
coefficient oscillations and shift variance, consider a piecewise smooth signal x(t −t0 )
like the step function u(t) = 0, t < 0 and u(t) = 1, t ≥ 0 analyzed by a wavelet basis
having a sufficient number of vanishing moments. Its wavelet coefficients consist of
samples of the step response of the wavelet
d j (n) ≈ 2−3 j/2 2 j t0 −n
ψ(t)dt,
−∞
where is the height of the jump. Since ψ(t) is a bandpass function that oscillates
around zero, so does its step response d j (n) as a function of n. Moreover, the factor
2 j in the upper limit ( j ≥ 0) amplifies the sensitivity of d j (n) to the time shift t0 ,
leading to strong shift variance.
3. Aliasing
The wide spacing of the wavelet coefficient samples, or equivalently, the fact that the
wavelet coefficients are computed via iterated discrete-time downsampling operations interspersed with non ideal low-pass and high-pass filters, results in substantial
aliasing. The inverse DWT cancels this aliasing, but only if the wavelet and scaling
coefficients are not changed. Any wavelet coefficient processing upsets the delicate
balance between the forward and inverse transforms, leading to artifacts in the reconstructed signal.
4. Lack OF Directionality
Finally, while Fourier sinusoids in higher dimensions correspond to highly directional plane waves, the standard tensor product construction of multidimensional
signal (MD) wavelets produces a checkerboard pattern that is simultaneously oriented along several directions. This lack of directional selectivity greatly complicates
modeling and processing of geometric image features like ridges and edges.
Fortunately, there is a simple solution, called complex wavelet transform (CWT),
to these four DWT shortcoming. The key is to note that the Fourier transform does
not suffer from these problems.
(i) The magnitude of the Fourier transform does not oscillate positive and negative
but rather provides a smooth positive envelope in the Fourier domain.
(ii) The magnitude of the Fourier transform is perfectly shift invariant, with a
simple linear phase offset encoding the shift.
(iii) The Fourier coefficients are not aliased and do not rely on a complicated
aliasing cancellation property to reconstruct the signal;
(iv) The sinusoids of the MD Fourier basis are highly directional plane waves.
What is the difference between FT and DWT? Unlike the DWT, which is based on
real-valued oscillating wavelets, the Fourier transform is based on complex-valued
oscillating sinusoids
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3 Wavelets and Wavelet Transform
eiξt = cos(ξt) + isin(ξt) with i =
√
−1.
The oscillating cosine and sine components, respectively, form a Hilbert transform
pair, i.e., they are 90◦ out of phase with each other. Together they constitute an
analytic signal eiξt that is supported on only one-half of the frequency axis.
3.13 Complex Wavelet Transform
Inspired by the Fourier representation, define a complex wavelet transform (CWT) as
similar to the DWT but with a complex-valued scaling function and complex-valued
wavelet
(3.13.1)
ψc (x) = ψr e (x) + iψim (x),
where ψr e (x) is real and even and iψim (x) is imaginary and odd. Moreover, if ψr e (x)
and iψim (x) form a Hilbert transform pair, then ψc (x) is an analytic signal and
supported on only one-half of the frequency axis. projecting the signal onto ψc, j,n ,
we obtain the complex wavelet coefficient
dc, j (k) = dr e, j (k) + i dim, j (k)
with magnitude
|dc, j (k)| =
and phase
[dr e, j (k)]2 + [dim, j (k)]2
dim, j (k)
∠dc, j (k) = arctan
dr e, j (k)
when |dc, j (k)| > 0. As with the Fourier transform, complex wavelets can be
used to analyze and represent both real-valued signals (resulting in symmetries in
the coefficients) and complex-valued signals. In either case, the CWT enables new
coherent multiscale signal processing algorithms that exploit the complex magnitude
and phase. In particular, a large magnitude indicates the presence of a singularity
while the phase indicates its position within the support of the wavelet.
The theory and practice of discrete complex wavelets can be divided into two
class. The first approach seeks a ψc (x) that forms an orthonormal or biorthogonal
basis [1, 4, 10, 20, 32, 37]. This strong constraint prevents the resulting CWT
from overcoming most of the four DWT shortcomings discussed above. The second
approach seeks a redundant representation, with both ψr e (x) and ψim (x) individually
forming orthonormal or biorthogonal bases. The resulting CWT is a 2× redundant
tight frame [7] in 1D, with the power to overcome the four shortcomings.
In this subsection, we will focus on a particularly second approach proposed by
Kingsbury in [15, 16], redundant type of CWT, is also called the dual-tree CWT
3.13 Complex Wavelet Transform
79
approach, which is based on two filter bank trees and thus two bases. Any CWT
based on wavelets of compact support cannot exactly possess the Hilbert transform
and analytic signal properties, and hence any such CWT will not perfectly overcome
the four DWT shortcomings. The key challenge in dual-tree wavelet design is the
joint design of its two filter banks to yield a complex wavelet and scaling function
that are as close as possible to analytic.
As a result, the dual-tree CWT comes very close to mirroring the attractive properties of the Fourier transform, including a smooth, non-oscillating magnitude; a
nearly shift-invariant magnitude with a simple near-linear phase encoding of signal
shifts; substantially reduced aliasing and directional wavelets in higher dimensions.
The only cost for all of this is a moderate redundancy: 2× redundancy in 1D. This is
much less than the log2 N × redundancy of a perfectly shift-invariant DWT, which,
moreover, will not offer the desirable magnitude and phase interpretation of the CWT
nor the good directional properties in higher dimensions.
3.14 Dual-Tree Complex Wavelet Transform
The discrete complex wavelet transform (DCWT) is a form of discrete wavelet transform, which generates complex coefficients by using a dual tree of wavelet filters
to obtain their real and imaginary parts. What makes the complex wavelet basis
exceptionally useful for denoising purposes is that it provides a high degree of shiftinvariance and better directionality compared to the real DWT. The DWT suffers
from the following two problems:
Lack of shift invariance - this results from the downsampling operation at each
level. When the input signal is shifted slightly, the amplitude of the wavelet coefficients varies so much.
Lack of directional selectivity - as the DWT filters are real and separable the DWT
cannot distinguish between the opposing diagonal directions.
These problems hinder the use of wavelets in other areas of image processing. The
first problem can be avoided if the filter outputs from each level are not downsampled
but this increases the computational costs significantly and the resulting undecimated
wavelet transform still cannot distinguish between opposing diagonals since the
transform is still separable. To distinguish opposing diagonals with separable filters
the filter frequency responses are required to be asymmetric for positive and negative
frequencies. A good way to achieve this is to use complex wavelet filters which can
be made to suppress negative frequency components.
One effective approach for implementing an analytic wavelet transform, first introduced by Kingsbury in 1998 [17], is called the dual-tree CWT (DTCWT). Like the
idea of positive and negative post-filtering of real subband signals, the idea behind
the dual-tree approach is quite simple. The dual-tree CWT employs two real DWTs.
The first DWT gives the real part of the transform while the second DWT gives
the imaginary part. The analysis and synthesis filter banks used to implement the
dual-tree CWT and its inverse are illustrated in given Figs. 3.7 and 3.8.
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3 Wavelets and Wavelet Transform
Fig. 3.7 The analysis filter bank for the Dual-Tree CWT
Fig. 3.8 The synthesis filter bank for the Dual-Tree CWT
The two real wavelet transforms use two different sets of filters, with each satisfying the perfect reconstruction conditions. The two sets of filters are jointly designed so
that the overall transform is approximately analytic. Let h 0 (n), h 1 (n) denote the lowpass, high-pass filter pair respectively for the upper filter bank, and let g0 (n), g1 (n)
denote the low-pass, high-pass filter pair respectively for the lower filter bank. We will
denote the two real wavelets associated with each of the two real wavelet transforms
as ψh (x) and ψg (x). In addition to satisfying the perfect reconstruction (PR) conditions, the filters are designed so that the complex wavelet ψ(x) = ψh (x) + iψg (x)
is approximately analytic. Equivalently, they are designed so that ψg (x) is approximately the Hilbert transform of ψh (x) [denoted ψg (x) ≈ H {ψh (x)}].
Note that the filters are themselves real and no complex arithmetic is required for
the implementation of the dual-tree CWT. Also note that the dual-tree CWT is not
a critically sampled transform and it is two times expansive in 1D because the total
output data rate is exactly twice the input data rate.
The inverse of the dual-tree CWT is as simple as the forward transform. To inverse
the transform, the real part and the imaginary part are each inverted, the inverse of
each of the two real DWTs are used to obtain two real signals. These two real signals
are then averaged to obtain the final output. Note that the original signal x(n) can be
3.14 Dual-Tree Complex Wavelet Transform
81
recovered from either the real part or the imaginary part alone, however, such inverse
dual-tree CWTs do not capture all the advantages an analytic wavelet transform
offers.
If the two real DWTs are represented by the square matrices Fh and Fg , then the
dual-tree CWT can be represented by the rectangular matrix
F=
Fh
.
Fg
(3.14.1)
If the vector x represents a real signal, then x h = Fh x represents the real part
and xg = Fg x represents the imaginary part of the dual-tree CWT. The complex
coefficients are given by x h + i xg . A (left) inverse of F is then given by
F −1 =
1 −1 −1 F
Fg ,
2 h
(3.14.2)
as we can easily see that
F −1 · F =
1
1 −1 −1 Fh
Fh Fg
= [I + I ] = I.
·
Fg
2
2
We can just as well share the factor of one half between the forward and inverse
transforms, to obtain
1 Fh
1 F −1 = √ Fh−1 Fg−1 .
F=√
(3.14.3)
F
g
2
2
If the two real DWTs are orthonormal transforms, then the transpose of Fh is its
inverse Fht · Fh = I and similarly for Fg . The dual-tree wavelet transform defined
in (3.14.3) keeps the real and imaginary parts of the complex wavelet coefficients
separate. However, the complex coefficients can be explicitly computed using the
following form:
1 I iI
Fh
.
(3.14.4)
·
Fc =
Fg
2 I −iI
Fc−1 =
1 −1 −1 I I
Fh Fg
·
.
−i I i I
2
(3.14.5)
Note that the complex sum and difference matrix in (3.14.4) is unitary. Therefore,
if the two real DWTs are orthonormal transforms then the dual-tree CWT satisfies
Fc∗ · Fc = I, where ∗ denotes conjugate transpose. If
u
= Fc · x.
v
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3 Wavelets and Wavelet Transform
When x is real, we have v = u ∗ . When the input signal x is complex, then v = u ∗ ,
so both u and v need to be computed.
When the DTCWT is applied to a real signal, the output of the upper and lower
filter banks will be the real and imaginary parts of the complex coefficients and they
can be stored separately as represented by Eq. (3.14.3). However, if the DTCWT is
applied to a complex signal, then the output of both the upper and lower filter banks
will be complex and it is no longer correct to label them as the real and imaginary
parts.
The DTCWT is also very easy to implement because there is no data flow between
the two real DWTs. They can be easily implemented using existing software and
hardware. Moreover, the transform id naturally parallelized for efficient hardware
implementation. In addition, the use of the DTCWT can be informed by the existing
theory and practice of real wavelet transforms because the DTCWT is implemented
using two real wavelet transforms.
The DTCWT has the following properties:
1. nearly shift invariance.
2. good selectivity and directionality in 2D (or higher dimension) with Gabor-like
filters.
3. perfect reconstruction (PR) using short linear phase filters,
4. limited redundancy, independent of the number of scales, for example, a redundancy factor of only 2n for n-dimensional signals,
5. efficient order N computation: only 2n times the simple real DWT for ndimensional signal.
The multidimensional DTCWT is non-separable but is based on a computationally
efficient, separable filter bank. A 2D dual-tree complex wavelet can be defined as
ψ(x, y) = ψ(x)ψ(y) associated with the row-column implementation of the wavelet
transform, where ψ(x) and ψ(y) are two complex wavelets, ψ(x) = ψh (x) + iψg (x)
and ψ(y) = ψh (y) + iψg (y), ψh (·) and ψg (·) are real wavelet transforms of upper
filter bank and lower filter bank, respectively. Then we obtain the following for the
expression:
ψ(x, y) = [ψh (x) + iψg (x)][ψh (y) + iψg (y)]
= [ψh (x)ψh (y) − ψg (x)ψg (y)] + i[ψg (x)ψh (y) + ψh (x)ψg (y)].
(3.14.6)
A 2D DTCWT is oriented and approximately analytic at the cost of being four
times expansive. It also possesses the full shift-invariant properties of the constituent
1D transforms. The real parts of six oriented complex wavelets of DTCWT can be
defined as follows:
1
ϕk (x, y) = √ (ψ1,k (x, y) − ψ2,k (x, y))
2
1
ϕk+3 (x, y) = √ (ψ1,k (x, y) − ψ2,k (x, y))
2
(3.14.7)
3.14 Dual-Tree Complex Wavelet Transform
83
where k = 1, 2, 3 and
ψ1,1 (x, y) = φh (x)ψh (y)
ψ2,1 (x, y) = φg (x)ψg (y)
ψ1,2 (x, y) = ψh (x)φh (y)
ψ1,3 (x, y) = ψh (x)ψh (y)
ψ2,2 (x, y) = ψg (x)φg (y)
ψ2,3 (x, y) = ψg (x)ψg (y).
(3.14.8)
The imaginary parts of six oriented complex wavelets of DTCWT can be defined as
follows:
1
ξk (x, y) = √ (ψ3,k (x, y) − ψ4,k (x, y))
2
(3.14.9)
1
ξk+3 (x, y) = √ (ψ3,k (x, y) − ψ4,k (x, y))
2
where k = 1, 2, 3 and
ψ3,1 (x, y) = φg (x)ψh (y)
ψ4,1 (x, y) = φh (x)ψg (y)
ψ3,2 (x, y) = ψg (x)φh (y)
ψ3,3 (x, y) = ψg (x)ψh (y)
ψ4,2 (x, y) = ψh (x)φg (y)
ψ4,3 (x, y) = ψh (x)ψg (y).
(3.14.10)
In Eqs. (3.14.8) and (3.14.10), h(·) and g(·) are the low-pass functions of upper
filter bank and lower filter bank, respectively along the first dimension. h(·) and g(·)
are the high-pass functions of upper filter bank and lower filter bank, respectively
along the second dimension. A 2D DTCWT produces three sub-bands in each of
spectral quadrants 1 and 2, giving six sub-bands of complex coefficients at each
level, which are strongly oriented at angles of 15◦ , 45◦ , 75◦ .
3.15 Quaternion Wavelet and Quaternion Wavelet
Transform
As a mathematical tool, wavelet transform is a major breakthrough over the Fourier
transform since it has good time-frequency localization and multiple resolution analysis features. Wavelet analysis theory has become one of the most useful tools in
signal analysis, image processing, pattern recognition and many other fields. In image
processing, the basic idea of the wavelet transform is to decompose an image into
multiple resolutions. Specifically, the original image is decomposed into different
space and frequency sub-images, and coefficients of the sub-image are then processed. A small shift in the signal significantly changes the distribution of the real
discrete wavelet transform. Although the dual-tree complex wavelet transform can
overcame the problem, it cannot avoid the phase ambiguity of two-dimensional(2D)
84
3 Wavelets and Wavelet Transform
images features. As an improved alternative, the quaternion wavelet transform is
a new multiscale analysis tool for image processing. It is based on the 2D Hilbert
transform theory, which is shift invariant, and can overcome the drawbacks of real
and complex wavelet transforms.
Quaternion wavelet transform [5, 6, 42] is established based on the quaternion
algebra, quaternion Fourier transform and Hilbert transform. A quaternion, or quaternion analytic signal, has a real part and three imaginary parts. Given a 2D real signal,
its real DWT is the real part of the quaternion wavelet and its three Hilbert transforms become three imaginary part of the quaternion wavelet. It can be understood as
the improved real wavelet and complex wavelet’s promotion, which are shift invariant with abundant phase information and limited redundancy. Since the quaternion
wavelet retains the traditional wavelet’s time-frequency localization ability, it is easy
to design filters Hilbert transform pair of the dual tree structure.
3.15.1 2D Hilbert Transform
Definition 3.18 Let f (x, y) ∈ L 2 (R2 ). Then Hilbert transform of f (x, y), denoted
by f Hx (x, y), f Hy (x, y) and f Hx y (x, y) along x-axis, y-axis and x, y-axis respectively, are defined by
f (ξ1 , y)
1
f Hx (x, y) =
dξ1 ,
(3.15.1)
π R x − ξ1
f Hy (x, y) =
and
f Hx y (x, y) =
1
π2
R
f (x, ξ2 )
dξ2 ,
y − ξ2
(3.15.2)
f (ξ1 , ξ2 )
dξ1 dξ2
(x − ξ1 )(y − ξ2 )
(3.15.3)
1
π
R
and the corresponding frequency domain are
and
f H
(u, v) = − j sgn(u) f
(u, v),
x
(3.15.4)
f H
(u, v) = − j sgn(v) f
(u, v)
y
(3.15.5)
f H
(u, v) = −sgn(u)sgn(v) f
(u, v).
xy
(3.15.6)
3.15 Quaternion Wavelet and Quaternion Wavelet Transform
85
3.15.2 Quaternion Algebra
The quaternion is proposed by W.R. Hamilton in 1843. The quaternion algebra over
R, denoted by H, is an associative non-commutative four-dimensional algebra. Every
element of H is a linear combination of a scalar and three imaginary units i, j, and
k with real coefficients
H = {q : q = q0 + iq1 + jq2 + kq3 ,
q0 , q1 , q2 , q3 ∈ R},
(3.15.7)
where i, j and k obey Hamiltons multiplication rules
i j = − ji = k, jk = −k j = i, ki = −ik = j, i 2 = j 2 = k 2 = i jk = −1.
(3.15.8)
The quaternion conjugate of a quaternion q is given by
q = q0 − iq1 − jq2 − kq3
q0 , q1 , q2 , q3 ∈ R.
(3.15.9)
The quaternion conjugation (3.15.9) is a linear anti-involution
q = q, p + q = p + q, pq = q p,
f or all p, q ∈ H.
(3.15.10)
The multiplication of a quaternion q and its conjugate can be expressed as
qq = q02 + q12 + q22 + q32 .
(3.15.11)
This leads to the modulus |q| of a quaternion q defined as
|q| =
qq = q02 + q12 + q22 + q32 .
(3.15.12)
Using (3.15.9) and (3.15.12), we can defined the inverse of q ∈ H\{0} as
q −1 =
q
,
|q|2
(3.15.13)
which shows that H is a normed division algebra. Furthermore, we get |q −1 | = |q|−1 .
In addition, quaternion q can also be expressed as:
q = |q|eiφ eiθ eiξ ,
(3.15.14)
where |q| is the modulus of q and (φ, θ, ξ) are the three phase angles which is
uniquely defined within the range (φ, θ, ξ) ∈ [−π, π]×[−π/2, π/2]×[−π/4, π/4].
The quaternion module L 2 (R2 , H) is defined as
86
3 Wavelets and Wavelet Transform
L 2 (R2 , H) = { f : R2 → H, || f || L 2 (R2 ,H) < ∞}.
(3.15.15)
For all quaternion functions f, g ∈ R2 → H, the inner product is defined as follows:
( f, g) L 2 (R2 ,H) =
R2
f (x)g(x)d 2 x,
f or x ∈ R2 .
(3.15.16)
If f = g almost everywhere, we obtain the associated norm
|| f || L 2 (R2 ,H) = ( f, g)1/2 =
R2
1/2
| f (x)|2 d 2 x
,
x ∈ R2 .
(3.15.17)
With the usual addition and scalar multiplication of functions together with the inner
product L 2 (R2 , H) becomes a Hilbert space.
Definition 3.19 If f (x, y) is a real two-dimensional signals, then quaternion analytic signal can be defined as f q (x, y) = f (x, y) + i f Hx (x, y) + j f Hy (x, y) +
k f Hx y (x, y), where f Hx (x, y), f Hy (x, y), and f Hx y (x, y) are the Hilbert transform of
f (x, y) along the x-axis, the y-axis and along the x y-axis respectively.
This section is based on the study of quaternion wavelet theory, which proved and
presented the correlative properties and concepts of scale basis and wavelet basis of
quaternion wavelet.
Lemma 3.2 Let {ϕ(x − k)}k∈Z be the space of standard orthogonal basis for V ⊂
L 2 (R), then the system {ϕg (x − k)}k∈Z , where ϕg (x) = H (φ(x)), is the standard
= H V ⊂ L 2 (R).
orthogonal basis of space V
Proof According to Mallat, we know that standard orthogonal basis in the frequency
domain is equivalent to k∈Z |φ̂(ξ+2kπ)|2 = 1, by the property of Hilbert transform,
we have k∈Z |φˆg (ξ + 2kπ)|2 = 1.
According to the lemma, it is not difficult to prove the following.
Theorem 3.13 Let {V j } j∈Z be a one dimensional orthogonal multiresolution analysis (MRA), and the corresponding scale and wavelet functions are ϕh (x) and ψh (x),
respectively, then {ϕg (x − l)ϕh (x − k)}l,k∈Z or {ϕh (x − l)φg (x − k)}l,k∈Z are the
0 and the space V
0 ⊗ V0 , where
standard orthogonal basis of the space V0 ⊗ V
0 = H V0 . Hx (φh (x)φh (y)) = φg (x)φh (y), Hy (φh (x)φh (y)) = φh (x)φg (y)
V
and Hx y (φh (x)φh (y)) = ϕg (x)ϕg (y) respectively denote Hilbert transforms of the
function (ϕh (x)ϕh (y)) along x-axis, y-axis and x y-axis directions.
By above Theorem and the analysis, we have the following.
Let {V j } j∈Z be one dimensional orthogonal MRA and ϕh (x) and ψh (x) are the
q
corresponding scale and wavelet functions respectively. Then { j,k,m (x, y)}k,m∈Z is
the orthogonal basis of the quaternion wavelet scale space in L 2 (R2 , H), where
3.15 Quaternion Wavelet and Quaternion Wavelet Transform
87
q (x, y) = ϕh (x)ϕh (y)+iϕg (x)ϕh (y)+ jϕh (x)ϕg (y)+kϕg (x)ϕg (y), (3.15.18)
q
j,k,m (x, y) = ϕh, j,k (x)ϕh, j,m (y) + iϕg, j,k (x)ϕh, j,m (y) + jϕh, j,k (x)ϕg, j,m (y)
+ kϕg, j,k (x)ϕg, j,m (y),
(3.15.19)
and ϕh, j,k (x) = 2− j/2 φh (2− j x − k), j, k, m ∈ Z. q (x, y) is called a quaternion
q
wavelet scale function in L 2 (R2 , H) and { j,k,m (x, y)}k,m∈Z is called the discrete
quaternion wavelet scale function in L 2 (R2 , H).
Further, Let {V j } j∈Z be one dimensional orthogonal MRA and ϕh (x) and ψh (x) are
the corresponding scale and wavelet functions respectively. Then { q,1 (x, y), q,2
(x, y), q,3 (x, y)} are the quaternion wavelet basis functions in L 2 (R2 , H) and
q,1
q,2
q,3
{ j,k,m , j,k,m , j,k,m } are discrete quaternion wavelet basis functions in L 2 (R2 , H),
where
q,1 (x, y) = ϕh (x)ψh (y) + iϕg (x)ψh (y) + jϕh (x)ψg (y)
+ kϕg (x)ψg (y),
(3.15.20)
q,2 (x, y) = ψh (x)ϕh (y) + iψg (x)ϕh (y) + jψh (x)ϕg (y)
+ kψg (x)ϕg (y),
(3.15.21)
q,2 (x, y) = ψh (x)ψh (y) + iψg (x)ψh (y) + jψh (x)ψg (y)
+ kψg (x)ψg (y),
(3.15.22)
the shift and expand form of { q,1 (x, y), q,2 (x, y) and q,3 (x, y)} are
q,1
j,k,m (x, y) = ϕh, j,k (x)ψh, j,m (y) + iϕg, j,k (x)ψh, j,m (y) + jϕh, j,k (x)ψg, j,m (y) + kϕg, j,k (x)ψg, j,m (y),
(3.15.23)
q,2
j,k,m (x, y) = ψh, j,k (x)ϕh, j,m (y) + iψg, j,k (x)ϕh, j,m (y) + jψh, j,k (x)ϕg, j,m (y) + kψg, j,k (x)ϕg, j,m (y),
(3.15.24)
q,3
j,k,m (x,
y) = ψh, j,k (x)ψh, j,m (y) + iψg, j,k (x)ψh, j,m (y) + jψh, j,k (x)ψg, j,m (y) + kψg, j,k (x)ψg, j,m (y),
and ψh, j,k (x) = 2− j/2 ψh (2− j x − k), j, k, m ∈ Z.
(3.15.25)
Definition 3.20 For all f (x, y) ∈ L 2 (R2 , H), define
q
a j,k,m =
q,i
d j,k,m =
q,i
q
f (x, y), j,k,m (x, y) ,
f (x, y), j,k,m (x, y) , (i = 1, 2, 3, j, k, m ∈ Z).
(3.15.26)
(3.15.27)
88
3 Wavelets and Wavelet Transform
Fig. 3.9 Decomposition filter bank for QWT
q,i
d j,k,m (i = 1, 2, 3), is called the discrete quaternion wavelet transform (DQWT) of
f (x, y).
The above discussion shows that quaternion wavelet transform by using four real
discrete wavelet transforms, the first real discrete wavelet corresponding quaternion
wavelet real part, the other real discrete wavelets are formed by the first real discrete
wavelet transform by Hilbert transform, corresponding to the three imaginary parts
of quaternion wavelet, respectively. The quaternion scale function q (x, y) and the
quaternion wavelet basis functions q,1 (x, y), q,2 (x, y), q,3 (x, y) corresponding real component are taken out to form a matrix:
⎛
ϕh (x)ϕh (y)
⎜ϕg (x)ϕh (y)
G=⎜
⎝ϕh (x)ϕg (y)
ϕg (x)ϕg (y)
ϕh (x)ψh (y)
ϕg (x)ψh (y)
ϕh (x)ψg (y)
ϕg (x)ψg (y)
ψh (x)ϕh (y)
ψg (x)ϕh (y)
ψh (x)ϕg (y)
ψg (x)ϕg (y)
⎞
ψh (x)ψh (y)
ψg (x)ψh (y)⎟
⎟.
ψh (x)ψg (y)⎠
ψg (x)ψg (y)
Then each row of the matrix G corresponds to the one real wavelet of quaternion
wavelet, the first column corresponding to quaternion wavelet scale function, the
other columns are quaternion wavelets three wavelet functions corresponding to
horizontal, vertical and diagonal three subbands. In the space L 2 (R2 ), there are four
standard orthogonal real wavelet bases, by wavelet frame and the concept of 2D real
wavelet, we know that quaternion wavelet base in L 2 (R2 ) form a tight frame with
frame bound 4.
3.15 Quaternion Wavelet and Quaternion Wavelet Transform
89
Fig. 3.10 Reconstruction structure of filter bank
Let h 0 and h 1 are low-pass and high-pass filter of real wavelet, g0 and g1 are lowpass and high-pass filter, corresponding to Hilbert transform of h 0 and h 1 , respectively and let, h̃ 0 , h̃ 1 , g̃0 and g̃1 are synthesis filter. Figures 3.9 and 3.10 show the
decomposition and reconstruction filter bank for quaternion wavelet transform.
In order to calculate the coefficients of QWT, the quaternion wavelet filters system
is similar to dual-tree complex wavelet, quaternion wavelet filters and coefficients
is quaternion, and it is realized by using the dual-tree algorithm, using an analytic
quaternion wavelet bases in order to satisfy the Hilbert transform. Quaternion wavelet
filters are dual-tree filters and each filters subtree part comprises 2 analysis filters
and 2 synthesis filters, respectively.
3.15.3 Quaternion Multiresolution Analysis
For a two dimensional image function f (x, y), a quaternion multiresolution analysis
(QMRA) can be expressed as
90
3 Wavelets and Wavelet Transform
n
f (x, y) = Aqn f (x, y) +
"
#
q,1
q,2
q,3
D j f (x, y) + D j f (x, y) + D j f (x, y) .
j=1
(3.15.28)
q
We can characterize each approximation function A j f (x, y) and the difference comq,i
ponents D j f (x, y) for i = 1, 2, 3, by means of a 2D scaling function q (x, y)
and its associated wavelet function q,i (x, y) as follows:
q
q
A j f (x, y) =
a j,k,m j,k,m (x, y),
(3.15.29)
q,i
(3.15.30)
k∈Z m∈Z
q,i
D j f (x, y) =
d ij,k,m j,k,m (x, y),
k∈Z m∈Z
where
q
a j,k,m =
q,i
d j,k,m =
q,i
q
f (x, y), j,k,m (x, y) ,
f (x, y), j,k,m (x, y) , (i = 1, 2, 3, j, k, m ∈ Z).
(3.15.31)
(3.15.32)
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Chapter 4
New Multiscale Constructions
4.1 Overview
Although the wavelet transform has been proven to be powerful in many signal
and image processing applications such as compression, noise removal, image edge
enhancement, and feature extraction; wavelets are not optimal in capturing the twodimensional singularities found in images. Therefore, several transforms have been
proposed for image signals that have incorporated directionality and multiresolution
and hence, could more efficiently capture edges in natural images. Multiscale methods have become very popular, especially with the development of the wavelets in
the last decade. Despite the success of the classical wavelet viewpoint, it was argued
by Candes and Donoho [13, 14] that the traditional wavelets present some strong
limitations that question their effectiveness in higher-dimension than 1. Wavelets rely
on a dictionary of roughly isotropic elements occurring at all scales and locations,
do not describe well highly anisotropic elements, and contain only a fixed number
of directional elements, independent of scale. Despite the fact that wavelets have
had a wide impact in image processing, they fail to efficiently represent objects with
highly anisotropic elements such as lines or curvilinear structures (e.g. edges). The
reason is that wavelets are non-geometrical and do not exploit the regularity of the
edge curve.
Following this reasoning, new constructions have been proposed such as the
ridgelets by Candes and Donoho [13], Candes [15] and the curvelets by Candes
and Donoho [11, 14], Donoho and Duncan [26], Strack et al. [59]. The Ridgelet
and the Curvelet transforms were developed by Candes and Donoho [13, 14] as an
answer to the weakness of the separable wavelet transform in sparsely representing
what appears to be simple building atoms in an image, that is lines, curves and edges.
Curvelets and ridgelets take the form of basis elements which exhibit high directional
sensitivity and are highly anisotropic [14, 15, 26, 59]. These, very recent geometric image representations, are built upon ideas of multiscale analysis and geometry.
They have had an important success in a wide range of image processing applications
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_4
93
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4 New Multiscale Constructions
including denoising [41, 55, 59], deconvolution [33, 60], contrast enhancement [60],
texture analysis [1, 62], detection [43], watermarking [68], component separation
[61], inpainting [32, 34] or blind source separation [5, 6]. Curvelets have also proven
useful in diverse fields beyond the traditional image processing application: for example seismic imaging [30, 41, 42], astronomical imaging [49, 57, 60], scientific computing and analysis of partial differential equations [9, 10]. Another reason for the
success of ridgelets and curvelets is the availability of fast transform algorithms
which are available in non-commercial software packages following the philosophy
of reproducible research (BeamLab [4], The curvelet toolbox [66]).
Ridgelets and curvelets are special members of the family of multiscale orientationselective transforms, which has recently led to a flurry of research activity in the field
of computational and applied harmonic analysis. Many other constructions belonging
to this family have been investigated in the literature, and go by the name contourlets
[20], directionlets [67], bandlets [53] and shearlets [48] etc.
In this chapter, we shall mainly focus on ridgelet transform, curvelet transform,
contourlet transform and shearlet transform.
4.2 Ridgelet Transform
4.2.1 The Continuous Ridgelet Transform
The two-dimensional continuous ridgelet transform in R2 is defined by Candes [17].
Let ψ : R → R be a smooth univariate function with sufficient decay and satisfying
the admissibility condition
|ψ̂(ξ)|2
dξ ≤ ∞,
(4.2.1)
|ξ|2
which holds if ψ has a vanishing mean ψ(x)d x = 0. We will assume a special
∞
normalization for ψ so that 0 |ψ̂(ξ)|2 |ξ|−2 dξ = 1.
For each scale a > 0, each position b ∈ R and each orientation θ ∈ [0, 2π), we
define the bivariate ridgelet ψa,b,θ : R2 → R by
ψa,b,θ (x) = ψa,b,θ (x1 , x2 ) = a −1/2 · ψ((x1 cosθ + x2 sinθ − b)/a)
(4.2.2)
A ridgelet is constant along lines x1 cosθ + x2 sinθ = constant. Transverse to these
ridges it is a wavelet.
Figure 4.1 depicts few examples of ridgelets. The second to fourth panels are
obtained after simple geometric manipulations of the ridgelet (left panel), namely
rotation, rescaling, and shifting.
4.2 Ridgelet Transform
95
Fig. 4.1 Few Examples of Ridgelets
Given an integrable bivariate function f (x), we define its ridgelet coefficients by
f (a, b, θ) := f, ψa,b,θ =
R2
f (x)ψ a,b,θ (x)d x.
We have the exact reconstruction formula
2π ∞ ∞
da dθ
f (a, b, θ)ψa,b,θ (x) 3 db ,
f (x) =
a
4π
0
−∞ 0
(4.2.3)
(4.2.4)
valid almost everywhere for functions which are both integrable and square integrable. This formula is stable and so we have a Parseval’s relation:
| f (x)| d x =
2π
2
0
∞
−∞
∞
| f (a, b, θ)|2
0
da dθ
db .
a 3 4π
(4.2.5)
This approach generalizes to any dimension.
Given a ψ obeying |ψ̂(ξ)|2 ξ −n
√
dξ = 1, define ψa,b,θ (x) = ψ((θ x − b)/a)/ a and f (a, b, θ) = f, ψa,b,θ . Then
there is an n-dimensional reconstruction formula
da
f = cn
(4.2.6)
f (a, b, θ)ψa,b,θ (x) n+1 dbdθ,
a
with dθ the uniform measure on the sphere and a Parseval’s relation
||
f ||2L 2 (Rn )
=
| f (a, b, θ)|2
da
dbdθ.
a n+1
(4.2.7)
The CRT appears similar to the 2D CWT except that the point parameters (k1 , k2 )
are replaced by the line parameters (b, θ). In brief, those 2D transform are related
by:
Wavelets:−→ ψscale, point position
Ridgelets:−→ ψscale, line position
96
4 New Multiscale Constructions
The consequence of this is: as wavelet analysis is very effective at representing
objects with isolated point singularities and ridgelet analysis can be very effective
at representing objects with singularities along lines. In fact, one can loosely view
ridgelets as a way of concatenating 1D wavelets along lines. Hence the motivation
for using ridgelets in image processing tasks is very appealing as singularities are
often joined together along edges or contours in images.
In the Radon domain, ridgelet analysis may be constructed as wavelet analysis.
The rationale behind this is that the Radon transform translates singularities along
lines into point singularities, for which the wavelet transform is known to provide a
sparse representation. The Radon transform of an object f is the collection of line
integrals indexed by (θ, t) ∈ [0, 2π) × R given by
R f (θ, t) =
R2
f (x1 , x2 )δ(x1 cosθ + x2 sinθ − t)d x1 d x2 ,
(4.2.8)
where δ is the Dirac distribution. Then the ridgelet transform is precisely the application of a 1-dimensional wavelet transform to the slices of the Radon transform
where the angular variable θ is constant and t is varying. Thus, the basic strategy for
calculating the continuous ridgelet transform is first to compute the Radon transform
R f (θ, t) and second, to apply a one-dimensional wavelet transform to the slices
R f (, θ).
Several digital ridgelet transforms (DRTs) have been proposed, and we will
describe some of them in this section, based on different implementations of the
Radon transform.
1. The RectoPolar Ridgelet Transform
A fast implementation of the Radon transform based on the projection-slice-theorem
can be proposed in the Fourier domain. First, the 2D FFT of the given image is
computed and then the resulting function in the frequency domain is to be used to
evaluate the frequency values in a polar grid of rays passing through the origin and
spread uniformly in angle. This conversion, from Cartesian to Polar grid, could be
obtained by interpolation, and this process is known by the Gridding in tomography.
Given the polar grid samples with the number of rays corresponds to the number of
projections and the number of samples on each ray corresponds to the number of shifts
per such angle. The Radon projections are obtained by applying one-dimensional
inverse Fourier transform for each ray.
Due to the sensitivity to the interpolation involved, the above described process
is known to be inaccurate. This implies that for a better accuracy, the first 2D FFT
employed should be done with high-redundancy. An alternative solution for the
Fourier-based Radon transform exists, where the polar grid is replaced with a pseudopolar one.
Concentric circles of linearly growing radius in the polar grid are replaced by
concentric squares of linearly growing sides. The rays are spread uniformly not in
angle but in slope. These two changes give a grid vaguely resembling the polar
one, but for this grid a direct FFT can be implemented with no interpolation. When
4.2 Ridgelet Transform
97
applying 1D FFT for the rays, we get a variant of the Radon transform, where the
projection angles are not spaced uniformly. For the pseudo-polar FFT to be stable,
it was shown that it should contain at least twice as many samples, compared to
the original image we started with. A by-product of this construction is the fact that
the transform is organized as a 2D array with rows containing the projections as a
function of the angle. Thus, processing the Radon transform in one axis is easily
implemented (See Strack et al. [59] for more details).
2. One-Dimensional Wavelet Transform
To complete the ridgelet transform, we must take a one-dimensional wavelet transform along the radial variable in Radon space. We now discuss the choice of the digital
1D-WT. As we know that compactly supported wavelets can lead to many visual artifacts when used in conjunction with nonlinear processing, such as hard-thresholding
of individual wavelet coefficients, particularly for decimated wavelet schemes used
at critical sampling. Also, because of the lack of localization of such compactly supported wavelets in the frequency domain, fluctuations in coarse-scale wavelet coefficients can introduce fine-scale fluctuations. A frequency-domain approach must be
taken, where the discrete Fourier transform is reconstructed from the inverse Radon
transform. These considerations lead to use band-limited wavelet, whose support is
compact in the Fourier domain rather than the time-domain. A specific over-complete
wavelet transform has been used by Strack et al. [58] and Strack et al. [63]. The
wavelet transform algorithm is based on a scaling function ϕ such that ϕ̂ vanishes
outside of the interval [ξc , ξc ]. We define the Fourier transform of the scaling function
as a re-normalized B3 -spline
3
(4.2.9)
ϕ̂(ξ) = B3 (4ξ),
2
and ψ̂ as the difference between two consecutive resolutions
ψ̂(2ξ) = ϕ̂(ξ) − ϕ̂(2ξ)
(4.2.10)
Because ψ̂ is compactly supported, the sampling theorem shows than one can easily
build a pyramid of n + n/2 + · · · + 1 = 2n elements (see Strack et al. [63] for more
details). This 1D-WT transform possesses the following useful properties:
(i) The wavelet coefficients are directly calculated in the Fourier space. This
allows avoiding the computation of the one-dimensional inverse Fourier transform
along each radial line in the context of the ridgelet transform.
(ii) Each sub-band is sampled above the Nyquist rate, hence, avoiding aliasing
phenomenon typically encountered by critically sampled orthogonal wavelet transforms.
(iii) The reconstruction is trivial. The wavelet coefficients simply need to be coadded to reconstruct the input signal at any given point.
This implementation would be useful to the practitioner whose focuses on
data analysis, for which it is well-known that over-completeness through almost
98
4 New Multiscale Constructions
translation-invariance can provide substantial advantages. The discrete ridgelet transform (DRT) of an image of size n × n is an image of size 2n × 2n, introducing a
redundancy factor equal to 4.
We note that, because this transform is made of a chain of steps, each one of which
is invertible, the whole transform is invertible, and so has the exact reconstruction
property. For the same reason, the reconstruction is stable under perturbations of the
coefficients. This discrete transform is also computationally attractive.
4.2.2 Discrete Ridgelet Transform
For applications point of view, it is important that one obtains a discrete representation
using ridgelets. Typical discrete representations include expansions in orthonormal
basis. Here we describe an expansion only in two dimensions by frames. For higher
dimensions, see Candes [8].
The formula for the continuous ridgelet transform (CRT) using Fourier domain
is defined by
1
(4.2.11)
ψ̂ a,b,θ (ξ) fˆ(ξ)dξ,
f (a, b, θ) :=
2π
where ψ̂a,b,θ (ξ) is interpreted as a distribution supported on the radial line in the
frequency plane. Letting ξ(λ, θ) = (λ · cos(θ), λ · sin(θ)) we can write
f (a, b, θ) :=
1
2π
∞
−∞
a 1/2 ψ̂(aλ)e−iλb fˆ(ξ(λ, θ))dλ.
(4.2.12)
This says that the CRT is obtainable by integrating the weighted Fourier transform
ωa,b fˆ(ξ) along a radial line in the frequency domain, with weight ωa,b given by
a 1/2 ψ̂(a|ξ|) times a complex exponential in e−iλb . Or, we can see that the function
of b, with a and θ considered fixed, ρa,θ (b) = f (a, b, θ), satisfies
ρa,θ (b) = {ρ̂a,θ (λ)}ˇ,
and
ρ̂a,θ (λ) = a 1/2 ψ̂(aλ) fˆ(ξ(λ, θ))
−∞<λ<∞
is the restriction of ωa,0 fˆ(ξ) to the radial line.
Hence the CRT at a certain scale a and angle θ can be obtained by first taking 2D
Fourier transform for obtaining fˆ(ξ), take radial windowing for obtaining ωa,0 fˆ(ξ)
and finally take 1D inverse Fourier transform along radial lines for obtaining ρa,θ (b),
for all b ∈ R.
Now we are providing a method for sampling (a j , b j,k , θ j,l ) so that we obtain
frame bounds, i.e. so we have equivalence
4.2 Ridgelet Transform
99
| f (a j , bk , θ j,l )|2
| f (a, b, θ)|2 da/a 3 dbdθ.
(4.2.13)
j,k,l
To simplify exposition, assume ψ̂(λ) = 1{1≤|ξ|≤2} , a j = a0 2 j and b j,k = 2πk2− j .
The ridgelet coefficients may be written as by using Eq. (4.2.12)
1 − j/2
f (a j , b j,k , θ) =
2
2π
2 j ≤|λ|≤2 j+1
−j
e−iλ2π2 fˆ(ξ(λ, θ))dλ,
(4.2.14)
|ω2 j ,0 |2 | fˆ(ξ(λ, θ))|2 dλ.
(4.2.15)
and hence, the Plancherel theorem gives
k
1
f (a j , b j,k , θ) = √
2π
2 j ≤|λ|≤2 j+1
Discretizing the angular variable θ amounts to performing a sampling of such
segment-integrals from which the integral of | fˆ(ξ)|2 over the whole frequency
domain needs to be inferred. This is not possible without support constraints on f ,
as functions f can always be constructed with f (x) having slow decay as |x| → ∞
so that fˆ will vanish on a collection of disjoint segments without being identically
zero. However, under a support restriction, so that f is supported inside the unit disk
(or any other compact
set), the integrals over the segments can provide sufficient
information to infer | fˆ(ξ)|2 dξ.
Indeed, under a support constraint, the Fourier transform fˆ(ξ) is a band-limited
function, and over ‘cells’ of appropriate size can only display very banal behavior.
If we sample once per cell, we will capture sufficiently much of the behavior of this
object that we will be in a position to infer the size of the function from those samples.
The solution found by Candes is to sample with increasing angular resolution at
increasingly fine scales, something like the following:
θ j,l = 2πl2− j .
This strategy gives the equivalence in Eq. (4.2.13). It then follows that the collection
{2 j/2 ψ(2 j (x1 cos(θ j,l ) + x2 sin(θ j,l ) − 2πk2− j ))}( j≥ j0 ,l,k)
is a frame for the unit disk. For any f supported in the disk with finite L 2 norm,
|ψa j ,b j,k ,θ j,l , f |2
|| f ||2 .
j,k,l
The existence of frame bounds implies, by soft analysis, that there are dual ridgelets
ψ̃ j,k,l so that
f, ψ̃ j,k,l ψ j,k,l
f =
j,k,l
100
4 New Multiscale Constructions
and
f =
f, ψ j,k,l ψ̃ j,k,l
j,k,l
with equality in an L 2 -sense, and so that
j,k,l
| f, ψ̃ j,k,l |2
| f, ψ j,k,l |2
|| f ||2L 2 .
j,k,l
4.2.3 The Orthonormal Finite Ridgelet Transform
The orthonormal finite ridgelet transform (OFRT) has been proposed by Do and
Vetterli [24] for image compression and filtering. This transform is based on the finite
Radon transform (see Matus and Flusser [51]) and a 1D orthogonal wavelet transform.
OFRT is not redundant and reversible. It would have been a great alternative to the
previously described ridgelet transform if the OFRT were not based on an awkward
definition of a line. In fact, a line in the OFRT is defined algebraically rather that
geometrically, and so the points on a line can be arbitrarily and randomly spread out
in the spatial domain.
Because of this specific definition of a line, the thresholding of the OFRT coefficients produces strong artifacts. Finally, the OFRT presents another limitation: the
image size must be a prime number. This last point is however not too restrictive,
because we generally use a spatial partitioning when denoising the data, and a prime
number block size can be used. The OFRT is interesting from the conceptual point
of view, but still requires work before it can be used for real applications such as
denoising.
4.2.4 The Fast Slant Stack Ridgelet Transform
The Fast Slant Stack (FSS) (see Averbuch et al. [2]) is a Radon transform of data
on a Cartesian grid, which is algebraically exact and geometrically more accurate
and faithful than the previously described methods. The back-projection of a point
in Radon space is exactly a ridge function in the spatial domain.
The transformation
of an n × n image is a 2n × 2n image. n line integrals with
angle between − π4 , π4 are calculated from the zero padded image on the y-axis, and
n line integrals with angle between π4 , 3π
are computed by zero padding the image
4
π π
on the x-axis. For a given angle inside − 4 , 4 , 2n line integrals are calculated by
first shearing the zero-padded image, and then integrating
the pixel values along all
. The shearing is performed
horizontal lines (resp. vertical lines for angles in π4 , 3π
4
one column at a time (resp. one line at a time) by using the 1D FFT. A DRT based on
4.2 Ridgelet Transform
101
the FSS transform has been proposed in (Donoho and Flesia, [27]). The connection
between the FSS and the Linogram has been investigated by Averbuch et al. [2]. A
FSS algorithm is also proposed by Averbuch et al. [2], based on the 2D Fast Pseudopolar Fourier transform which evaluates the 2D Fourier transform on a non-Cartesian
grid, operating in O(n 2 log n) flops.
4.2.5 Local Ridgelet Transform
The ridgelet transform is optimal for finding global lines of the size of the image. To
detect line segments, a partitioning must be introduced by Candes [15]. The image
can be decomposed into overlapping blocks of side-length b pixels in such a way
that the overlap between two vertically adjacent blocks is a rectangular array of size
b by b/2; we use overlap to avoid blocking artifacts. For an n by n image, we count
2n/b such blocks in each direction, and thus the redundancy factor grows by a factor
of 4.
The partitioning introduces redundancy, as a pixel belongs to 4 neighboring
blocks. We present two competing strategies to perform the analysis and synthesis:
1. The block values are weighted by a spatial window w (analysis) in such a
way that the co-addition of all blocks reproduce exactly the original pixel value
(synthesis).
2. The block values are those of the image pixel values (analysis) but are weighted
when the image is reconstructed (synthesis).
4.2.6 Sparse Representation by Ridgelets
The continuous ridgelet transform provides sparse representation of both smooth
functions (in the Sobolev space W22 ) and of perfectly straight lines (Candes [16],
Donoho [28]). As we know that there are also various DRTs, i.e. expansions with
countable discrete collection of generating elements, which correspond either to
frames or orthonormal bases. It has been shown for these schemes that the DRT
achieves near optimal M-term approximation, that is the non-linear approximation
of f using the M highest ridgelet coefficients in magnitude, to smooth images with
discontinuities along straight lines [13, 28]. In short, ridgelets provide sparse presentation for piecewise smooth images away from global straight edges.
102
4 New Multiscale Constructions
4.3 Curvelets
The ridgelet transform is optimal at representing straight-line singularities. This
transform with arbitrary directional selectivity provides a key to the analysis of higher
dimensional singularities. Unfortunately, the ridgelet transform is only applicable
to objects with global straight-line singularities, which are rarely observed in real
applications. In the area of image processing, most of the image edges are curved
rather than straight lines. Hence, ridgelets are not able to efficiently represent such
images. In order to analyze local line or curve singularities, a natural idea is to
consider a partition for the image, and then to apply the ridgelet transform to the
obtained sub-images. This block ridgelet based transform, which is named curvelet
transform, was first proposed by Candes and Donoho [11]. This is also called firstgeneration curvelet transform or CurveletG1 (Candes and Donoho [11]).
Despite these interesting properties, however, the application of or CurveletG1 is
limited because the geometry of ridgelets is itself unclear, as they are not true ridge
functions in digital images. First, the construction involves a complicated sevenindex structure among which we have parameters for scale, location and orientation.
In addition, the parabolic scaling ratio width ≈ length2 is not completely true. In fact,
CurveletG1 assumes a wide range of aspect ratios. These facts make mathematical
and quantitative analysis especially delicate. Second, the spatial partitioning of the
CurveletG1 transform uses overlapping windows to avoid blocking effects. This
leads to an increase of the redundancy of the discrete curveletG1 (DCTG1). The
computational cost of the DCTG1 algorithm may also be a limitation for large-scale
data, especially if the FSS-based DRT implementation is used.
Later, the second-generation curvelet transform or CurveletG2, introduced by
Cand‘es and Donoho [12], exhibit a much simpler and natural indexing structure with
three parameters: scale, orientation or angle and location, hence simplifying mathematical analysis. The CurveletG2 transform also implements a tight frame expansion and has a much lower redundancy. Unlike the DCTG1, the discrete CurveletG2
(DCTG2) implementation will not use ridgelets yielding a faster algorithm (Candes
et al. [18]).
4.3.1 The First Generation Curvelet Transform
The CurveletG1 transform [11, 26] opens the possibility to analyze an image with
different block sizes, but with a single transform. The main idea is to first decompose
the image into a set of wavelet bands, and then analyze each band by using a local
ridgelet transform. The block size can be changed at each scale level. Different
levels of the multiscale ridgelet pyramid are used to represent different sub-bands
of a filter bank output. At the same time, this sub-band decomposition imposes a
relationship between the width and length of the important frame elements so that
4.3 Curvelets
103
they are anisotropic and obey approximately the parabolic scaling law width ≈
length2 .
The first-generation discrete curvelet transform (DCTG1) of a continuum function
f (x) makes use of a dyadic sequence of scales, and a bank of filters with the property
that the bandpass filter j is concentrated near the frequencies [22 j , 22 j+2 ], for
example
ˆ 2 j (ξ) = (2
ˆ −2 j ξ).
(4.3.1)
j ( f ) = 2 j ∗ f, In wavelet theory, one uses a decomposition into dyadic sub-bands [2 j , 2 j+1 ]. In contrast, the subbands used in the discrete curvelet transform of continuum functions
have the nonstandard form [22 j , 22 j+2 ]. This is nonstandard feature of the DCTG1
well worth remembering (this is where the approximate parabolic scaling law comes
into play).
The DCTG1 decomposition is the sequence of the following steps:
1. Sub-band Decomposition: The object f is decomposed into sub-bands.
2. Smooth Partitioning: Each sub-band is smoothly windowed into squares of an
appropriate scale (of side-length ∼ 2 j ).
3. Ridgelet Analysis: Each square is analyzed via the DRT.
In this definition, the two dyadic sub-bands [22 j , 22 j+1 ] and [22 j+1 , 22 j+2 ] are
merged before applying the ridgelet transform.
4.3.2 Sparse Representation by First Generation Curvelets
The CurveletG1 elements can form either a frame or a tight frame for L 2 (R2 ) (Candes and Donoho [14]), depending on the 2D-WT used and the DRT implementation
(rectopolar or FSS Radon transform). The frame elements are anisotropic by construction and become successively more anisotropic at progressively higher scales.
These curvelets also exhibit directional sensitivity and display oscillatory components across the ridge. A central motivation leading to the curvelet construction was
the problem of non-adaptively representing piecewise smooth (e.g. C 2 ) images f
which have discontinuity along a C 2 curve. Such a model is called cartoon model
of non-textured images. With the CurveletG1 tight frame construction, it was shown
by Candes and Donoho [14] that for such f , the M-term non-linear approximations
f M of f obey, for each κ > 0,
|| f − f M ||2 ≤ Cκ M −2+κ , M → +∞.
The M-term approximations in the CurveletG1 are almost rate optimal, much better
than M-term Fourier or wavelet approximations for such images.
104
4 New Multiscale Constructions
4.3.3 The Second-Generation Curvelet Transform
The second-generation curvelet transform has been shown to be a very efficient
tool for many different applications in image processing, seismic data exploration,
fluid mechanics, and solving PDEs. The second generation curvelets or CurveletG2
j,l
(2− j k1 , 2− j/2 k2 ) by
are defined at scale 2− j , orientation θl and position xk = Rθ−1
l
translation and rotation of a mother curvelet ϕ j as
j,l
ϕ j,l,k (x) = ϕ j (Rθl (x − xk )),
(4.3.2)
where Rθl is the rotation by θl radians. θl is the equispaced sequence of rotation
angles θl = 2π2−| j/2| l, with
√integer l such that 0 ≤ θl ≤ 2π (note that the number
of orientations varies as 1/ scale. k = (k1 , k2 ) ∈ Z2 is the sequence of translation
parameters. The waveform ϕ j is defined by means of its Fourier transform ϕ̂ j (ξ),
written in polar coordinates in the Fourier domain
ϕ̂ j (r, θ) = 2−3 j/4 ω̂(2− j r )v̂
2
j/2
θ
.
2π
(4.3.3)
The support of ϕ̂ j is a polar parabolic wedge defined by the support of ω̂ and v̂. The
radial and angular windows, both are smooth, nonnegative and real-valued, applied
with scale-dependent window widths in each direction. ω̂ and v̂ must also satisfy the
partition of unity property (See Candes et al. [18] for more details).
In continuous frequency ξ, the CurveletG2 coefficients of data f (x) are defined
as the inner product
c j,l,k := f, ϕ j,l,k =
fˆ(ξ)ϕ̂ j (Rθl ξ)ei xk
j,l
R2
·ξ
dξ.
(4.3.4)
This construction implies a few properties:
(i) the CurveletG2 defines a tight frame of L 2 (R2 ),
(ii) the effective length and width of these curvelets obey the parabolic scaling
relation (2 j = width) = (length = 2 j/2 )2 ,
(iii) the curvelets exhibit an oscillating behavior in the direction perpendicular to
their orientation.
Curvelets as just constructed are complex-valued. It is very easy to obtain realvalued curvelets by working on the symmetrized version ϕ̂ j (r, θ) + ϕ̂ j (r, θ + π).
The discrete transform takes as input data defined on a Cartesian grid and outputs
a collection of coefficients. The continuous-space definition of the CurveletG2 uses
coronae and rotations that are not especially adapted to Cartesian arrays. It is then
convenient to replace these concepts by their Cartesian counterparts. That is, we use
concentric squares instead of concentric circles and shears instead of rotations.
The Cartesian equivalent to the radial window ω̂ j (ξ) = ω̂ j (2− j ξ) would be a
bandpass frequency-localized window which can be derived from the difference of
4.3 Curvelets
105
separable low-pass windows H j (ξ) = ĥ(2− j ξ1 )ĥ(2− j ξ2 ), where h is a 1D low-pass
filter, we have:
ω̂ j (ξ) =
2
H j+1
(ξ) − H j2 (ξ) ∀ j ≥ 0, and, ω̂0 (ξ) = ĥ(ξ1 )ĥ(ξ2 )
(4.3.5)
Another possible choice is to select these windows inspired by the construction
of Meyer wavelets, for more details see Candes and Donoho [12]. For more details
about the construction of the Cartesian ω̂ j ’s see Candes et al. [18].
Each cartesian coronae has four quadrants: east, west, north and south. Each
quadrant is separated into 2 j/2 wedges with the same area. For example take the
east quadrant −π/4 ≤ θl < π/4 and by symmetry around the origin, we would
proceed for west quadrant and finally for north and south quadrant by exchanging
the roles of ξ1 and ξ2 . Define the angular window for the l-th direction as
j/2 ξ2 − ξ1 tanθl
,
v̂ j,l (ξ) = v̂ 2
ξ1
(4.3.6)
with the sequence of equispaced slopes (not angles) tanθl = 2− j/2 l, for l =
−2 j/2 , . . . , 2 j/2 − 1. Now, we are defining the window which is the Cartesian
analog of φ̂ j above,
û j,l (ξ) = ω̂ j (ξ)v̂ j,l (ξ) = ω̂ j (ξ)v̂ j,0 (Sθl ξ),
(4.3.7)
where Sθl is the shear matrix. From the definition, it can be seen that û j,l is supported
near the trapezoidal wedge {ξ = (ξ1 , ξ2 ) : 2 j ≤ ξ1 ≤ 2 j1 , −2− j/2 ≤ ξ2 /ξ1 −tanθl ≤
2− j/2 }. The collection of û j,l (ξ) gives rise to the frequency tiling. From û j,l (ξ), the
digital CurveletG2 construction suggests cartesian curvelets that are translated and
sheared versions of a mother cartesian curvelet
φ̂ Dj (ξ) = û j,0 (ξ), φ Dj,k,l (x) = 23 j/4 φ Dj SθTl x − m
where m = (k1 2− j , k2 2− j/2 ).
4.3.4 Sparse Representation by Second Generation Curvelets
It has been shown by Candes and Donoho [12] that with the CurveletG2 tight frame
construction, the M-term non-linear approximation error of C 2 images except at
discontinuities along C 2 curves obey
|| f − f M ||2 ≤ C M 2 (logM)3 .
106
4 New Multiscale Constructions
This is an asymptotically optimal convergence rate (up to the (logM)3 factor), and
holds uniformly over the C 2 -C 2 class of functions. This is a remarkable result since
the CurveletG2 representation is non-adaptive. However, the simplicity due to the
non-adaptivity of curvelets has a cost: curvelet approximations loose their near optimal properties when the image is composed of edges which are not exactly C 2 .
Additionally, if the edges are C α -regular with α > 2, then the curvelets convergence
rate exponent remain 2. Other adaptive geometric representations such as bandlets
are specifically designed to reach the optimal decay rate O(M α ) (Peyre and Mallat
[54]).
4.4 Contourlet
In the field of geometrical image transforms, there are many 1D transforms designed
for detecting or capturing the geometry of image information, such as the Fourier
transform and wavelet transform. However, the ability of 1D transform processing of
the intrinsic geometrical structures, such as smoothness of curves, is limited in one
direction, then more powerful representations are required in higher dimensions. The
contourlet transform which was proposed by Do and Vetterli [20, 21], is a new twodimensional transform method for image representations. The contourlet transform
has properties of multiresolution, localization, directionality, critical sampling and
anisotropy. Its basic functions are multiscale and multidimensional. The contours of
original images, which are the dominant features in natural images, can be captured
effectively with a few coefficients by using contourlet transform.
The contourlet transform is inspired by the human visual system and curvelet
transform which can capture the smoothness of the contour of images with different
elongated shapes and in variety of directions. However, it is difficult to sampling on
a rectangular grid for curvelet transform since curvelet transform was developed in
continuous domain and directions other than horizontal and vertical are very different
on rectangular grid. Therefore, the contourlet transform was proposed initially as a
directional multiresolution transform in the discrete domain and then studies its
convergence to an expansion in the continuous domain.
The contourlet transform is one of the new geometrical image transforms, which
can efficiently represent images containing contours and textures (Mallat [50], Skodras et al. [56]). This transform uses a structure similar to that of curvelets (Donoho
and Vetterli [29], that is, a stage of subband decomposition followed by a directional
transform. In the contourlet transform, a Laplacian pyramid is employed in the first
stage, while directional filter banks (DFB) are used in the angular decomposition
stage. Due to the redundancy of the Laplacian pyramid, the contourlet transform has
a redundancy factor of 4/3 and hence, it may not be the optimum choice for image
coding applications. The discrete contourlet transform has a fast iterated filter bank
algorithm that requires an order N operations for N -pixel images.
4.5 Contourlet Transform
107
4.5 Contourlet Transform
Wavelets in 2D are good at isolating the discontinuities at edge points, but will not
see the smoothness along the contours. In addition, separable wavelets can capture
only limited directional information i.e. an important and unique feature of multidimensional signals. These disappointing behaviors indicate that more powerful
representations are needed in higher dimensions. Consider the following scenario to
see how one can improve the 2D separable wavelet transform for representing images
with smooth contours. Imagine that there are two painters, one with a wavelet style
and the other with a new style, both wishing to paint a natural scene. Both painters
apply a refinement technique to increase resolution from coarse to fine. Here, efficiency is measured by how quickly, that is with how few brush strokes, one can
faithfully reproduce the scene. Consider the situation when a smooth contour is
being painted. Because 2D wavelets are constructed from tensor products of 1D
wavelets, the wavelet style painter is limited to using square-shaped brush strokes
along the contour, using different sizes corresponding to the multiresolution structure
of wavelets. As the resolution becomes finer, we can clearly see the limitation of the
wavelet style painter who needs to use many fine dots to capture the contour. The new
style painter, on the other hand, exploits effectively the smoothness of the contour by
making brush strokes with different elongated shapes and in a variety of directions
following the contour. This intuition was formalized by Candes and Donoho in the
curvelet construction.
Curvelet transform can capture the smoothness of the contour of images with
different elongated shapes and in variety of directions. However, it is difficult to
sampling on a rectangular grid for curvelet transform since it was developed in continuous domain and directions other than horizontal and vertical are very different
on rectangular grid. Therefore, Do and Vetterli in 2001 [23] proposed a double filter
bank structure for obtaining sparse expansions for typical images having smooth
contours. In this double filter bank, the Laplacian pyramid constructed by Burt and
Adelson in 1983 [7], is first used to capture the point discontinuities, and then followed by a directional filter bank provided by Bamberger and Smith in 1992 [3], to
link point discontinuities into linear structures. The overall result is an image expansion using basic elements like contour segments and thus are named contourlets.
In particular, contourlets have elongated supports at various scales, directions, and
aspect ratios. This allows contourlets to efficiently approximate a smooth contour at
multiple resolutions in much the same way as the new scheme shown in Fig. 4.2.
The contourlet transform was proposed initially as a directional multiresolution
transform in the discrete domain and then studies its convergence to an expansion in
the continuous domain. In the frequency domain, the contourlet transform provides
a multiscale and directional decomposition. In this section, we will provide only the
discrete domain construction using filter banks. For detail study of its convergence
to an expansion in the continuous domain, see Do and Vetterli [21].
108
4 New Multiscale Constructions
Fig. 4.2 Wavelet versus new scheme: illustrating the successive refinement by the two systems
near a smooth contour, which is shown as a thick curve separating two smooth regions
4.5.1 Multiscale Decomposition
For obtaining a multiscale decomposition, we use a Laplacian pyramid (LP) as introduced by Burt and Adelson in1983 [7]. The LP decomposition at each step generates
a sampled lowpass version of the original and the difference between the original
and the prediction, resulting in a bandpass image. The process can be iterated on the
coarse version.
A drawback of the LP is the implicit oversampling. However, in contrast to the
critically sampled wavelet scheme, the LP has the distinguishing feature that each
pyramid level generates only one bandpass image [even for multidimensional cases]
which does not have scrambled frequencies. This frequency scrambling happens in
the wavelet filter bank when a highpass channel, after downsampling, is folded back
into the low frequency band, and thus its spectrum is reflected. In the LP, this effect
is avoided by downsampling the lowpass channel only.
Do and Vellerli in 2003 [20] had studied the LP using the theory of frames and
oversampled filter banks and they showed that the LP with orthogonal filters (that is,
h[n] = g[−n] and g[n] is orthogonal to its translates with respect to the subsampling
lattice by M) is a tight frame with frame bounds equal to 1. In this case, they suggested
the use of the optimal linear reconstruction using the dual frame operator, which is
symmetrical with the forward transform. Note that this new reconstruction is different
from the usual reconstruction and is crucial for our contourlet expansion.
Figure 4.3a shows the decomposition process and Fig. 4.3b shows the new reconstruction scheme for the Laplacian pyramid, where H and G are the analysis and
synthesis filters, respectively and M is the sampling matrix.
4.5 Contourlet Transform
109
Fig. 4.3 Laplacian Pyramid a decomposition scheme b reconstruction scheme
Fig. 4.4 The multichannel view of an l-level tree structured directional filter bank
4.5.2 Directional Decomposition
Bamberger and Smith in 1992 [3] introduced a 2D directional filter bank (DFB) that
can be maximally decimated while achieving perfect reconstruction. The DFB is
efficiently implemented using a l-level tree-structured decomposition that leads to
2l subbands with wedge-shaped frequency partition.
The original construction of the DFB involves modulating the input signal and
using quincunx filter banks with diamond-shaped filters. An involved tree expanding
rule has to be followed to obtain the desired frequency partition. As a result, the
frequency regions for the resulting subbands do not follow a simple ordering as
shown in Fig. 4.4 based on the channel indices, see Park et al. [52] for more details.
By using the multirate identities, we can transform a l-level tree-structured DFB
into a parallel structure of 2l channels with directional filters and overall sampling
matrices. Denote these directional synthesis filters as Dk(l) , 0 ≤ k < 2l , which correspond to the subbands indexed. The oversampling matrices have diagonal form as:
Sk(l) =
diag(2l−1 , 2), for 0 ≤ k < 2l−1 ,
diag(2, 2l−1 ), for 2l−1 ≤ k < 2l ,
(4.5.1)
which correspond to the basically horizontal and basically vertical subbands, respectively.
With this, it is easy to see that the family
{gk(l) [n − Sk(l) m]}0≤k<2l ,m∈Z2
(4.5.2)
110
4 New Multiscale Constructions
obtained by translating the impulse responses of the directional synthesis filters
Dk(l) over the sampling lattices Sk(l) , is a basis for discrete signals in l 2 (Z2 ). This
basis exhibits both directional and localization properties. These basis functions
have linear supports in space and span all directions. Therefore Eq. (4.5.2) resembles
a local Radon transform and the basis functions are referred to as Radonlets.
Furthermore, Do [25], in his Ph.D. thesis, had shown that if the building block
filter uses orthogonal filters, then the resulting DFB is orthogonal and Eq. (4.5.2)
becomes an orthogonal basis.
4.5.3 The Discrete Contourlet Transform
The directional filter bank (DFB) is designed to capture the high frequency components of images and is representing directionality. Therefore, low frequency components are handled poorly by the DFB. In fact, low frequencies would leak into
several directional subbands, hence DFB does not provide a sparse representation
for images. To improve the situation, low frequencies should be removed before
the DFB. This provides another reason to combine the DFB with a multiresolution
scheme.
Therefore, the LP permits further subband decomposition to be applied on its
bandpass images. Those bandpass images can be fed into a DFB so that directional
information can be captured efficiently. The scheme can be iterated repeatedly on the
coarse image (see Fig. 4.5). The end result is a double iterated filter bank structure,
named pyramidal directional filter bank (PDFB) or contourlet filter bank, which
decomposes images into directional subbands at multiple scales. The scheme is
flexible since it allows for a different number of directions at each scale.
Fig. 4.5 Pyramidal directional filter bank
4.5 Contourlet Transform
111
Precisely, let a0 [n] be the input image. After the LP stage the output is a lowpass
image a J [n] and J bandpass images b j [n], j = 1, 2, ···, J in the fine-to-coarse order.
In detail, the j-th level of the LP decomposes the image a j−1 [n] into a coarser image
a j [n] and a detail image b j [n]. Each bandpass image b j [n] is further decomposed by
(l )
an l j -level DFB into 2l j bandpass directional images c j,kj [n], k = 0, 1, · · ·, 2l j − 1.
The main properties of the discrete contourlet transform are stated as follows:
1. If, the LP and the DFB, both use perfect-reconstruction filters, then the discrete
contourlet transform succeeds perfect reconstruction, i.e. it provides a frame operator.
With orthogonal filters, the LP is a tight frame with frame bounds equal to
1 (Do and Vetterli [24]), which means it preserves the l 2 -norm, or ||a0 ||22 =
J
2
2
j=1 ||b j ||2 + ||a J ||2 . Similarly, with orthogonal filters the DFB is an orthogolj
(l )
2 −1
||c j,kj ||22 . Combining these
nal transform (Do [23]), which means ||b j ||22 = k=0
two stages, the discrete contourlet transform satisfies the norm preserving or tight
frame condition. We have the following:
2. If, the LP and the DFB, both use orthogonal filters, then the discrete contourlet
transform forms a tight frame with frame bounds equal to 1.
Since the DFB is critically sampled, the redundancy of the discrete contourlet
transform is equal to the redundancy of the LP, which is 1 + Jj=1 (1/4) j < 4/3.
Hence, we have the following result:
3. The discrete contourlet transform has a redundancy ratio that is less than 4/3.
By using multirate identities, the LP bandpass channel corresponding to the
pyramidal level j is approximately equivalent to filtering by a filter of size about
C1 2 j × C1 2 j , followed by downsampling by 2 j−1 in each dimension. For the DFB,
from (3) we see that after l j levels (l j ≥ 2) of tree-structured decomposition, the
equivalent directional filters have support of width about C2 2 and length about
C2 2l j −1 . Combining these two stages, again using multirate identities, into equivalent
contourlet filter bank channels, we see that contourlet basis images have support of
width about C2 j and length about C2 j+l j −2 . Hence, we have the following result:
4. Suppose an l j -level DFB is applied at the pyramidal level j of the LP, then the basis
images of the discrete contourlet transform (i.e. the equivalent filters of the contourlet
filter bank) have an essential support size of width ≈ C2 j and length ≈ C2 j+l j −2 .
Let L p and L d be the number of taps of the pyramidal and directional filters used
in the LP and the DFB, respectively (without loss of generality we can suppose that
lowpass, highpass, analysis and synthesis filters have same length). With a polyphase implementation, the LP filter bank requires L p /2 + 1 operations per input
sample. Thus, for an N -pixel image, the complexity of the LP stage in the contourlet
filter bank is:
J
j=1
N
j−1 Lp
Lp
4
1
+1 < N
+ 1 (operations).
4
2
3
2
(4.5.3)
112
4 New Multiscale Constructions
For the DFB, its building block two-channel filter banks requires L d operations per
input sample. With an l-level full binary tree decomposition, the complexity of the
DFB multiplies by l. This holds because the initial decomposition block in the DFB
is followed by two blocks at half rate, four blocks at quarter rate and so on. Thus,
the complexity of the DFB stage for an N -pixel image is:
J
j=1
j−1
1
4
N
L d l j < N L d max{l j }(operations).
4
3
(4.5.4)
Combining (4.5.3) and (4.5.4) we obtain the following result:
5. Using FIR filters, the computational complexity of the discrete contourlet transform is O(N ) for N -pixel images.
Since the multiscale and directional decomposition stages are decoupled in the
discrete contourlet transform, we can have a different number of directions at different scales, thus offering a flexible multiscale and directional expansion. Moreover,
the full binary tree decomposition of the DFB in the contourlet transform can be
generalized to arbitrary tree structures, similar to the wavelet packets generalization
of the wavelet transform (Coifman et al. [19]). The result is a family of directional
multiresolution expansions, which we call contourlet packets.
Furthermore, similar to the wavelet filter bank, the contourlet filter bank has an
associated continuous domain expansion in L 2 (R2 ) using contourlet functions and
the new elements in this framework are multidirection and its combination with
multiscale.
4.6 Shearlet
One of the most useful features of wavelets is their ability to efficiently approximate signals containing pointwise singularities. However, wavelets fail to capture
the geometric regularity along the singularities of surfaces, because of their isotropic
support. To exploit the anisotropic regularity of a surface along edges, the basis
must include elongated functions that are nearly parallel to the edges. Several image
representations have been proposed to capture the geometric regularity of a given
image. They include ridgelet, curvelets, contourlets and bandelets etc. In particular,
the construction of curvelets is not built directly in the discrete domain and they do
not provide a multiresolution representation of the geometry. In consequence, the
implementation and the mathematical analysis are more involved and less efficient.
Contourlets are bases constructed with elongated basis functions using the combination of a multiscale and a directional filter bank. However, contourlets have less clear
directional features than curvelets, which leads to artifacts in denoising and compression. Bandelets are bases adapted to the function that is represented. Asymptotically,
the resulting bandelets are regular functions with compact support, which is not the
case of contourlets. However, in order to find basis optimally adapted to an image
4.6 Shearlet
113
of size N , the bandelet transform searches for the optimal geometry. For an image
of N pixels, the complexity of this best bandelet basis algorithm is O(N 3/2 ) which
requires extensive computation.
Recently, a new representation scheme has been introduced by Labate et al. in
2005 [46] is called the shearlet representation which yields nearly optimal approximation properties (Guo and Labate [36]). Shearlets are frame elements used in this
representation scheme. This new representation is based on a simple and rigorous
mathematical framework which not only provides a more flexible theoretical tool for
the geometric representation of multidimensional data, but is also more natural for
implementation. As a result, the shearlet approach can be associated to a multiresolution analysis (MRA) and this leads to a unified treatment of both the continuous and
discrete world (Labate et. al. [46]). However, all known constructions of shearlets so
far are band-limited functions which have an unbounded support in space domain. In
fact, in order to capture the local features of a given image efficiently, representation
elements need to be compactly supported in the space domain. Furthermore, this
property often leads to more convenient framework for practically relevant discrete
implementation.
Before defining the system of shearlets in a formal way, let us introduce intuitively
the ideas which are at the core of its construction. in order to achieve optimally sparse
approximations of signals exhibiting anisotropic singularities such as cartoon-like
images, the analyzing elements must consist of waveforms ranging over several
scales, orientations, and locations with the ability to become very elongated. This
requires a combination of an appropriate scaling operator to generate elements at
different scales, an orthogonal operator to change their orientations, and a translation
operator to displace these elements over the 2D plane.
A family of vectors {ψn }n∈ form a frame for a Hilbert space H if there exist two
positive constants A, B such that for each f ∈ H we have
A|| f ||2 ≤
| f, ψn |2 ≤ B|| f ||2 .
(4.6.1)
n∈
If A = B, the frame is called tight frame. The constants A and B are called lower
and upper frame bounds, respectively. If A = B = 1, the frame is called normalized
tight frame or Parseval frame.
Let f ∈ L 2 (Rd ) and G L d (R) be the set of all d × d invertible matrices with real
entries. The dilation operator D A , for A ∈ G L d (R) is defined by
D A f (x) = |det A|−1/2 f (A−1 x),
(4.6.2)
and the translation operator Tt for t ∈ Rd is defined by
Tt f (x) = f (x − t).
(4.6.3)
Since the scaling operator is required to generate waveforms with anisotropic support,
we utilize the family of dilation operators D Aa , a > 0, based on parabolic scaling
114
4 New Multiscale Constructions
matrices Aa of the form
Aa =
a
0
0
a 1/2
.
This type of dilation resembles to the parabolic scaling, which has a elongated history
in the literature of harmonic analysis and can be outlined back to the second dyadic
decomposition from the theory of oscillatory integrals. It should be point out that,
rather than Aa , the more general matrices diag(a, a α ) with the parameter α ∈ (0, 1)
controlling the degree of anisotropy could be used. Though, the value α = 1/2 shows
a special role in the discrete setting. In fact, parabolic scaling is essential in order
to obtain optimally sparse approximations of cartoon-like images, since it is best
adapted to the C 2 -regularity of the curves of discontinuity in this model class.
Now, we need an orthogonal transformation to change the orientations of the
waveforms. The best obvious choice seems to be the rotation operator. Though,
rotations destroy the structure of the integer lattice Z2 whenever the rotation angle is
. This issue becomes a thoughtful problem for the transition
different from 0, π2 , π, 3π
2
from the continuum to the digital setting. we choose the shearing operator D Ss , s ∈
R, where the shearing matrix Ss as an alternative orthogonal transformation, is given
by
1 s
.
Ss =
0 1
The shearing matrix parameterizes the orientations using the variable s associated
with the slopes rather than the angles, and has the advantage of leaving the integer
lattice invariant, provided s is an integer. Now, we define continuous shearlet systems
by combining these three operators.
Definition 4.1 Let ψ ∈ L 2 (R2 ). The continuous shearlet system S H (ψ) is defined
by
S H (ψ) = {ψa,s,t = Tt D Aa D Ss ψ : a > 0, s ∈ R, t ∈ R2 }.
Similar to the relation of wavelet systems to group representation theory, the
theory of continuous shearlet systems can also be developed within the theory of
unitary representations of the affine group and its generalizations. For this, we define
the shearlet group S, as the semi-direct product
(R+ × R) × R2 ,
equipped with group multiplication given by
(a, s, t) · (a , s , t ) = (aa , s + s
√
a, t + Ss Aa t ).
A left-invariant Haar measure of this shearlet group is
sentation σ : S → U(L 2 (R2 )) is defined by
da
dsdt.
a3
The unitary repre-
4.6 Shearlet
115
σ(a, s, t)ψ = Tt D Aa D Ss ψ,
where U(L 2 (R2 )) denotes the group of unitary operators on L 2 (R2 ). A continuous
shearlet system S H (ψ) can be re-written as
S H (ψ) = {σ(a, s, t)ψ : (a, s, t) ∈ S}.
The representation σ is unitary but not irreducible. If this additional property is
desired, the shearlet group needs to be extended to (R∗ ×R)×R2 , where R∗ = R−{0},
yielding the continuous shearlet system
S H (ψ) = {σ(a, s, t)ψ : a ∈ R∗ , s ∈ R, t ∈ R2 }.
4.7 Shearlet Transform
4.7.1 Continuous Shearlet Transform
Similar to the continuous wavelet transform, the continuous shearlet transform
defines a mapping of f ∈ L 2 (R2 ) to the components of f associated with the
elements of S.
Definition 4.2 Let ψ ∈ L 2 (R2 ). The continuous shearlet transform (CST) of f ∈
L 2 (R2 ) is defined by
f −→ S H ψ ( f )(a, s, t) = f, σ(a, s, t)ψ
=
f (x)σ(a, s, t)ψ(x)d x,
(a, s, t) ∈ S.
Thus, S H ψ maps the function f to the coefficients S H ψ ( f )(a, s, t) associated
with the scale variable a > 0, the orientation variable s ∈ R, and the location
variable t ∈ R2 .
Of specific importance are the conditions on ψ under which the continuous shearlet transform is an isometry, since this is automatically associated with a reconstruction formula. For this, we define the notion of an admissible shearlet is also called
continuous shearlet.
Definition 4.3 If ψ ∈ L 2 (R2 ) satisfies
R2
|ψ̂(ξ1 , ξ2 )|2
dξ2 dξ1 < ∞,
ξ12
(4.7.1)
116
4 New Multiscale Constructions
it is called an admissible shearlet.
It is necessary to mention that any function ψ such that ψ̂ is compactly supported
away from the origin is an admissible shearlet.
Definition 4.4 Let ψ ∈ L 2 (R2 ) be defined by
ψ̂(ξ) = ψ̂(ξ1 , ξ2 ) = ψ̂1 (ξ1 )ψ̂2
ξ2
ξ1
,
where ψ1 ∈ L 2 (R2 ) is a discrete wavelet in the sense that it satisfies the discrete
Calderón condition, given by
|ψ̂1 (2− j ξ)|2 = 1 for a.e. ξ ∈ R,
(4.7.2)
j∈Z
1
with ψ̂1 ∈ C ∞ (R) and supp ψ̂1 ⊆ − 21 , − 16
and ψ̂2 ∈ L 2 (R) is a bump function in
the sense that
1
|ψ̂2 (ξ + k)|2 = 1 for a.e. ξ ∈ [−1, 1],
(4.7.3)
k=−1
satisfying ψ̂2 ∈ C ∞ (R) and supp ψ̂2 ⊆ [−1, 1]. Then ψ is called a classical shearlet.
Thus, a classical shearlet ψ is a function which is wavelet-like along one axis and
bump-like along another one. Note that there exist several choices of ψ1 and ψ2
satisfying conditions (4.7.2) and (4.7.3). One possible choice is to set ψ1 to be a
Lemarié-Meyer wavelet and ψ̂2 to be a spline (Easley et al. [31], Guo and Labate
[36]). This example was originally introduced in (Guo et al. [40]) and later slightly
modified by Guo et al. [36] and Labate et al. [46].
Let ψ ∈ L 2 (R2 ) be an admissible shearlet. Define
Cψ+ =
∞
0
R
|ψ̂(ξ1 , ξ2 )|2
dξ2 dξ1 and Cψ− =
ξ12
If Cψ− = Cψ+ = 1, then S H ψ is an isometry.
0
−∞
R
|ψ̂(ξ1 , ξ2 )|2
dξ2 dξ1 .
ξ12
(4.7.4)
4.7.2 Discrete Shearlet Transform
The discrete shearlet systems are formally defined by sampling continuous shearlet
systems on a discrete subset of the shearlet group S.
Definition 4.5 Let ψ ∈ L 2 (R2 ) and ⊆ S. An irregular discrete shearlet system
associated with ψ and , denoted by S H (ψ, ), is defined by
4.7 Shearlet Transform
117
S H (ψ, ) = {ψa,s,t = a −3/4 ψ(Aa−1 Ss−1 (· − t)) : (a, s, t) ∈ }.
A (regular) discrete shearlet system associated with ψ, denoted by S H (ψ), is defined
by
S H (ψ) = {ψ j,k,m = 23 j/4 ψ(Sk A2 j · −m) : j, k ∈ Z, m ∈ Z2 }.
By choosing = {(2 j , k, Sk A2 j m) : j, k ∈ Z, m ∈ Z2 }, we can find the regular
versions of discrete shearlet systems from the irregular systems. Furthermore, in the
definition of a regular discrete shearlet system, the translation parameter is sometimes
chosen to belong to c1 Z × c2 Z for some (c1 , c2 ) ∈ (R+ )2 . This provides some
additional flexibility which is very useful for some other constructions.
We are particularly interested not only in finding generic generator functions ψ
but also in selecting a generator ψ with special properties, e.g., regularity, vanishing
moments, and compact support similar to the wavelet case, so that the corresponding
basis or frame of shearlets has satisfactory approximation properties.
Theorem 4.1 Let ψ ∈ L 2 (R2 ) be a classical shearlet. Then S H (ψ) is a Parseval
frame for L 2 (R2 ).
Proof For a.e. ξ ∈ R2 , Using the properties of classical shearlets, we have,
j∈Z k∈Z
ξ2
|ψ̂1 (2− j ξ1 )|2 |ψ̂2 2 j/2 − k |2
ξ1
j∈Z k∈Z
−j
2
j/2 ξ2
=
|ψ̂1 (2 ξ1 )|
|ψ̂2 2
+ k |2 = 1.
j∈Z
k∈Z
ξ1
T
|ψ̂(S−k
A2− j ξ)|2 =
result follows immediately from this observation and the fact that supp ψ̂ ⊂
The
− 21 , 21 .
Since a classical shearlet ψ is a well-localized function, by Theorem 4.1, there
exit Parseval frames S H (ψ) of well-localized discrete shearlets. The well localization
property is critical for deriving superior approximation properties of shearlet systems
and it will be required for deriving optimally sparse approximations of cartoon-like
images. Without the assumption that ψ is well localized, one can construct discrete
shearlet systems which form not only tight frames but also orthonormal bases. Thus,
a well localized discrete shearlet system can form a frame or a tight frame but not an
orthonormal basis.
Now, we define a discrete shearlet transform as follows. We state the definition
only for regular case with obvious extension to the irregular shearlet systems.
Definition 4.6 Let ψ ∈ L 2 (R2 ). The discrete shearlet transform of f ∈ L 2 (R2 ) is
the mapping defined by
f −→ S H ψ f ( j, k, m) = f, ψ j,k,m ( j, k, m) ∈ Z × Z × Z2 .
118
4 New Multiscale Constructions
Thus, S H ψ maps the function f to the coefficients S H ψ f ( j, k, m) associated
with the scale index j, the orientation index k, and the position index m.
4.7.3 Cone-Adapted Continuous Shearlet Transform
Even though the continuous shearlet systems defined as above display an elegant
group structure, they do have a directional bias, which can be easily seen in Fig. 4.6.
For better understanding, consider a function which is mostly concentrated along
the ξ2 axis in the frequency domain. Then the energy of f is more and more concentrated in the shearlet components S Hψ f (a, s, t) as s → ∞. In the limiting case,
f is a delta distribution supported along the ξ2 axis, the typical model for an edge
along the x1 axis in spatial domain, f can only be detected in the shearlet domain as
s → ∞. It is clear that this behavior can be a serious limitation for some applications.
Fig. 4.6 Fourier domain support of several elements of the shearlet system
4.7 Shearlet Transform
119
Fig. 4.7 Resolving the problem of biased treatment of directions by continuous shearlet systems
One way to address this problem is to partition the Fourier domain into four cones,
while separating the low-frequency region by cutting out a square centered around
the origin. This yields a partition of the frequency plane as shown in Fig. 4.7. Notice
that, within each cone, the shearing variable s is only allowed to vary over a finite
range, hence producing elements whose orientations are distributed more uniformly.
Now, we are defining the following variant of continuous shearlet systems, known
as cone-adapted continuous shearlet system.
∈ L 2 (R2 ). The cone-adapted continuous shearlet system,
Definition 4.7 Let φ, ψ, ψ
denoted by S H (φ, ψ, ψ), is defined by
= (φ) ∪ (ψ) ∪ (ψ),
S H (φ, ψ, ψ)
120
4 New Multiscale Constructions
where
(φ) = {φt = φ(· − t) : t ∈ R2 },
(ψ) = {ψa,s,t = a −3/4 φ(Aa−1 Ss−1 (· − t)) : a ∈ (0, 1], |s| ≤ 1 + a 1/2 , t ∈ R2 },
= {ψ
a,s,t = a −3/4 ψ(
A
a−1 Ss−1 (· − t)) : a ∈ (0, 1], |s| ≤ 1 + a 1/2 , t ∈ R2 },
(ψ)
1/2
a = diag(a , a).
and A
In the following definition, the function φ will be chosen to have compact frequency support near the origin, which ensures that the system (φ) is associated with
the low-frequency region R = {(ξ1 , ξ2 ) : |ξ1 |, |ξ2 | ≤ 1}. By choosing ψ to satisfy the
condition of definition 4.4, the system (ψ) is associated with the horizontal cones
can be chosen likewise
C1 ∪ C3 = {(ξ1 , ξ2 ) : |ξ2 /ξ1 | ≤ 1, |ξ1 | > 1}. The shearlet ψ
(ψ)
with the roles of ξ1 and ξ2 reversed, i.e., ψ(ξ1 , ξ2 ) = ψ(ξ1 , ξ2 ). Then the system is associated with the vertical cones C2 ∪ C4 = {(ξ1 , ξ2 ) : |ξ2 /ξ1 | > 1, |ξ2 | > 1}.
Similar to the situation of continuous shearlet systems, an associated cone-adapted
continuous shearlet transform can be defined.
Definition 4.8 Let
Scone = {(a, s, t) : a ∈ (0, 1], |s| ≤ 1 + a 1/2 , t ∈ R2 }.
∈ L 2 (R2 ), the cone-adapted continuous shearlet transform of
Then, for φ, ψ, ψ
2
2
f ∈ L (R ) is the mapping
a ,s,t )
a ,
s, t)) = ( f, φt , f, ψa,s,t , f, ψ
f −→ S H φ,ψ,
ψ f (t , (a, s, t), (
where
a ,
s, t)) ∈ R2 × S2cone .
(t , (a, s, t), (
and φ can be formulated for
Similar to the situation above, conditions on ψ, ψ
which the mapping S H φ,ψ,
ψ is an isometry.
∈ L 2 (R2 ) be admissible shearlets satisfying C + = C − = 1
Theorem 4.2 Let ψ, ψ
ψ
ψ
+
−
and C
= C
= 1, respectively, and let ∈ L 2 (R2 ) be such that, for a.e. ξ =
ψ
ψ
(ξ1 , ξ2 ) ∈ R2 ,
1
|φ̂(ξ)| + χC1 ∪C3 (ξ)
2
|ψ̂1 (aξ1 )|
2 da
a
0
1
+ χC2 ∪C4 (ξ)
|ψ̂1 (aξ2 )|2
0
da
= 1.
a
Then, for each f ∈ L 2 (R2 )
|| f ||2 =
R
| f, Tt φ|2 dt +
+
Scone
Scone
|( fˆχC1 ∪C3 ) ˇ, ψa,s,t |2
a ,s,t |2
|( fˆχC2 ∪C4 ) ˇ, ψ
d
a
d
sd
t
a3
da
dsdt
a3
4.7 Shearlet Transform
121
ˆ can in fact be chosen to be in C ∞ (R2 ). In
In this result, the function φ̂, ψ̂ and ψ
0
addition, the cone-adapted shearlet system can be designed so that the low-frequency
and high-frequency parts are smoothly combined. A more detailed analysis of the
continuous shearlet transform and cone-adapted continuous shearlet transform and
its generalizations can be found in Grohs [35].
4.7.4 Cone-Adapted Discrete Shearlet Transform
Similar to the situation of continuous shearlet systems, discrete shearlet systems
also suffer from a biased treatment of directions. For the sake of generality, let us
start by defining cone-adapted discrete shearlet systems with respect to an irregular
parameter set.
∈ L 2 (R2 ), ⊂ R2 and , ⊆ Scone . The irregular
Definition 4.9 Let φ, ψ, ψ
) is defined by
cone-adapted discrete shearlet system S H (φ, ψ, ψ; , , , , ) = (φ; ) ∪ (ψ; ) ∪ (ψ;
),
S H (φ, ψ, ψ;
where
(φ; ) = {φt = φ(· − t) : t ∈ },
(ψ; ) = {ψa,s,t = a −3/4 ψ(Aa−1 Ss−1 (· − t)) : (a, s, t) ∈ },
a,s,t = a −3/4 ψ(
A
a−1 Ss−1 (· − t)) : (a, s, t) ∈ (ψ;
) = {ψ
}.
The regular variant of the cone-adapted discrete shearlet systems is much more
frequently used. To allow more flexibility and enable changes to the density of the
translation grid, we introduce a sampling factor c = (c1 , c2 ) ∈ (R+ )2 in the translation index. Hence, we have the following definition.
∈ L 2 (R2 ), ⊂ R2 and c = (c1 , c2 ) ∈ (R+ )2 , the
Definition 4.10 For φ, ψ, ψ
c) is defined by
regular cone-adapted discrete shearlet system S H (φ, ψ, ψ;
c),
c) = (φ; c1 ) ∪ (ψ; c) ∪ (ψ;
S H (φ, ψ, ψ;
where
(φ; c1 ) = {φm = φ(· − c1 m) : m ∈ Z2 },
(ψ; c) = {ψ j,k,m = 23 j/4 ψ(Sk A2 j · −Mc m) : j ≥ 0, |k| ≤ 2 j/2 , m ∈ Z2 },
j/2
c) = {ψ
j,k,m = 23 j/4 ψ(S
TA
(ψ;
, m ∈ Z2 }
k 2 j · − Mc m) : j ≥ 0, |k| ≤ 2
with
c1 0
c2 0
and Mc =
.
Mc =
0 c2
0 c1
If c = (1, 1), the parameter c is omitted in the formulae above.
122
4 New Multiscale Constructions
The generating functions φ will be called shearlet scaling functions and the gener are called shearlet generators. Notice that the system (φ; c1 )
ating functions ψ, ψ
c)
(ψ;
is associated with the low-frequency region, and the systems (ψ; c) and are associated with the conic regions C1 ∪ C3 and C2 ∪ C4 , respectively.
Since the construction of discrete shearlet orthonormal basis is impossible, one
aims to derive Parseval frames. For that we first see that a classical shearlet is a
shearlet generator of a Parseval frame for the subspace of L 2 (R2 ) of functions whose
frequency support lies in the union of two cones C1 ∪ C3 .
Theorem 4.3 Let ψ ∈ L 2 (R2 ) be a classical shearlet. Then the shearlet system
(ψ) = {ψ j,k,m = 23 j/4 ψ(Sk A2 j · −Mc m) : j ≥ 0, |k| ≤ 2 j/2 , m ∈ Z2 }
is a Parseval frame for L 2 (C1 ∪ C3 )ˇ = { f ∈ L 2 (R2 ) : supp fˆ ⊂ C1 ∪ C3 }.
Proof Since ψ be a classical shearlet, Eq. (4.7.3) implies that, for any j ≥ 0,
|ψ̂2 (2 j/2 ξ + k)|2 = 1,
|ξ| ≤ 1.
|k|≤2 j/2 By using this observation together with Eq. (4.7.2), a direct computation gives that,
for a.e. ξ = (ξ1 , ξ2 ) ∈ C1 ∪ C3 ,
ξ2
|ψ̂1 (2− j ξ1 )|2 |ψ̂2 2 j/2 − k |2
ξ1
j≥0 |k|≤2 j/2 −j
2
j/2 ξ2
=
|ψ̂1 (2 ξ1 )|
|ψ̂2 2
+ k |2 = 1.
ξ
1
j/2
j≥0
T
|ψ̂(S−k
A2− j ξ)|2 =
j≥0 |k|≤2 j/2 |k|≤2
claim follows immediately from this observation and the fact that suppψ̂ ⊂
The
− 21 , 21 .
A result very similar to Theorem 4.3 holds for the subspace of L 2 (C2 ∪ C4 )ˇ if
This indicates that one can build up a Parseval frame for the
ψ is a replaced by ψ.
2
2
whole space L (R ) by piecing together Parseval frames associated with different
cones on the frequency domain together with a coarse scale system which takes care
of the low-frequency region. Using this idea, we have the following result.
Theorem 4.4 Let ψ ∈ L 2 (R2 ) be a classical shearlet, and let φ ∈ L 2 (R2 ) be chosen
so that, for a.e. ξ ∈ R2 ,
|φ̂(ξ)|2 +
j≥0 |k|≤2 j/2 T
|ψ̂(S−k
A2− j ξ)|2 χC +
ˆ
2
|ψ(S
−k A2− j ξ)| χC̃ = 1.
j≥0 |k|≤2 j/2 Let PC (ψ) denote the set of shearlet elements in (ψ) after projecting their Fourier
transforms onto C = {(ξ1 , ξ2 ) ∈ R2 : |ξ2 /ξ1 | ≤ 1}, with a similar definition holding
4.7 Shearlet Transform
123
where C
= R2 \ C. Then, the modified cone-adapted discrete shearlet
(ψ)
for PC
is a Parseval frame for L 2 (R2 ).
(ψ)
system (φ) ∪ PC (ψ) ∪ PC
Notice that, despite its simplicity, the Parseval frame construction above has one
the
drawback. When the cone-based shearlet systems are projected onto C and C,
shearlet elements overlapping the boundary lines ξ1 = ±ξ2 in the frequency domain
are cut so that the boundary shearlets lose their regularity properties. To avoid this
problem, it is possible to redefine the boundary shearlets in such a way that their regularity is preserved. This requires to slightly modifying the definition of the classical
shearlet. Then the boundary shearlets are obtained, essentially, by piecing together
the shearlets overlapping the boundary lines ξ1 = ±ξ2 which have been projected
This modified construction yields smooth Parseval frames of bandonto C and C.
limited shearlets and can be found in Guo and Labate [37], where also the higher
dimensional versions are discussed. The shearlet transform associated with regular
cone-adapted discrete shearlet systems is defined as follows:
∈ L 2 (R2 ),
Definition 4.11 Set = N0 × {−2 j/2 , . . . , 2 j/2 } × Z2 . For φ, ψ, ψ
2
2
the cone-adapted discrete shearlet transform of f ∈ L (R ) is the mapping defined
by
)) = ( f, φm , f, ψ j,k,m , f, ψ
f −→ S H φ,ψ,
ψ f (m , ( j, k, m), ( j , k, m
j̃ ,k̃,m̃ )
where
j, k, m
)) ∈ Z2 × × .
(m , ( j, k, m), (
4.7.5 Compactly Supported Shearlets
The shearlet systems, which are generated by classical shearlets, are band-limited,
i.e., they have compact support in the frequency domain and, hence, cannot be compactly supported in the spatial domain. Thus, a different approach is needed for the
construction of compactly supported shearlet systems.
Before stating the main result, let us first introduce the following notation.
∈ L 2 (R2 ). Define : R2 × R2 → R by
Let φ, ψ, ψ
(ξ, ω) = |φ̂(ξ)||φ̂(ξ + ω)| + 1 (ξ, ω) + 2 (ξ, ω),
where
1 (ξ, ω) =
|ψ̂(SkT A2− j ξ)||ψ̂(SkT A2− j ξ + ω)|
(4.7.5)
(4.7.6)
j≥0 |k|≤2 j/2 and
2 (ξ, ω) =
j≥0 |k|≤2 j/2 |ψ̂(SkT A2− j ξ)||ψ̂(SkT A2− j ξ + ω)|.
(4.7.7)
124
4 New Multiscale Constructions
Also, for c = (c1 , c2 ) ∈ (R+ )2 , let
R(c) =
(0 (c1−1 m)0 (−c1−1 m))1/2 + (1 (Mc−1 m)1 (−Mc−1 m))1/2
m∈Z2 \{0}
c−1 m)1 (− M
c−1 m))1/2 ,
+ (2 ( M
(4.7.8)
where
0 (ω) = ess supξ∈R2 |φ̂(ξ)||φ̂(ξ + ω)| and i (ω) = ess supξ∈R2 |i (ξ, ω)| for i = 1, 2.
Using these notation, we can now state the following theorem from Kittipoom et
al. [44].
Theorem 4.5 Let φ, ψ ∈ L 2 (R2 ) be such that
φ̂(ξ1 , ξ2 ) ≤ C1 · min{1, |ξ1 |−γ } · min{1, |ξ2 |−γ }
and
|ψ̂(ξ1 , ξ2 )| ≤ C2 · min{1, |ξ1 |α } · min{1, |ξ1 |−γ } · min{1, |ξ2 |−γ }
for some positive constants C1 , C2 < ∞ and α > γ > 3. Define ψ̂(x1 , x2 ) =
ψ(x1 , x2 ), and let L in f , L sup be defined by
L in f = essinfξ∈R2 (ξ, 0) and L sup = esssupξ∈R2 (ξ, 0).
Then there exists a sampling parameter c = (c1 , c2 ) ∈ (R+ )2 with c1 = c2 such that
S H (φ, ψ, ψ̃; c) forms a frame for L 2 (R2 ) with frame bounds A and B satisfying
0<
1
1
[L in f − R(c)] ≤ A ≤ B ≤
[L sup + R(c)] < ∞.
|det Mc |
|det Mc |
It can be easily verified that the conditions imposed on φ and ψ by Theorem 4.5
are satisfied by many suitably chosen scaling functions and classical shearlets. In
addition, one can construct various compactly supported separable shearlets that
satisfy these conditions. The difficulty, however, arises when aiming for compactly
supported separable functions φ and ψ which ensure that the corresponding coneadapted discrete shearlet system is a tight or almost tight frame. Separability is useful
to achieve fast algorithmic implementations. In fact, it was shown by Kittipoom et
al. [44] that there exists a class of functions generating almost tight frames, which
have (essentially) the form
ψ̂(ξ) = m 1 (4ξ1 )φ̂(ξ1 )φ̂(2ξ2 ), ξ = (ξ1 , ξ2 ) ∈ R2 ,
(4.7.9)
4.7 Shearlet Transform
125
where m 1 is a carefully chosen bandpass filter and φ an adaptively chosen scaling
function. The proof of this fact is highly technical and will be omitted.
4.7.6 Sparse Representation by Shearlets
One of the main motivations for the introduction of the shearlet framework is the
derivation of optimally sparse approximations of multivariate functions. Before stating the main results, it is enlightening to present a heuristic argument in order to
describe how shearlet expansions are able to achieve optimally sparse approximations of cartoon-like images.
For this, consider a cartoon-like function f and let S H (φ, ψ, ψ̃; c) be a shearlet
system. Since the elements of S H (φ, ψ, ψ̃; c) are effectively or in case of compactly
supported elements, exactly supported inside a box of size 2− j/2 × 2− j/2 , it follows that at scale 2− j there exist about O(2 j/2 ) such waveforms whose support is
tangent to the curve of discontinuity. Similar to the wavelet case, for j sufficiently
large, the shearlet elements associated with the smooth region of f , as well as the
elements whose overlap with the curve of discontinuity is non-tangential, yield negligible shearlet coefficients f, ψ j,k,m (or f, ψ̃ j,k,m ). Each shearlet coefficient can
be controlled by
| f, ψ j,k,m | ≤ || f ||∞ ||ψ j,k,m || L 1 ≤ C2−3 j/4 ,
similarly for f, ψ̃ j,k,m . Using this estimate and the observation that there exist at
most O(2 j/2 ) significant coefficients, we can conclude that the N th largest shearlet
coefficient, which we denote by |s N ( f )|, is bounded by O(N 3/2 ). This implies that
|| f − f N ||2L 2 ≤
|sl ( f )|2 ≤ C N −2 ,
l>N
where f N denotes the N -term shearlet approximation using the N largest coefficients
in the shearlets expansion. Indeed, the following result holds.
be a Parseval frame for L 2 (R2 ) as
(ψ)
Theorem 4.6 Let (φ) ∪ PC (ψ) ∪ PC
2
2
defined in Theorem 4.4, where ψ ∈ L (R ) is a classical shearlet and φ̂ ∈ C0∞ (R2 ).
Let f ∈ ε2 (R2 ) and f N be its nonlinear N -term approximation obtained by selecting
the N largest coefficients in the expansion of f with respect to this shearlet system.
Then there exists a constant C > 0, independent of f and N , such that
|| f − f N ||22 ≤ C N −2 (log N )3 as N → ∞.
Since a log-like factor is negligible with respect to the other terms for large N , the
optimal error decay rate is essentially achieved. It is remarkable that an approximation
rate which is essentially as good as the one obtained using an adaptive construction
126
4 New Multiscale Constructions
can be achieved using a nonadaptive system. The same approximation rate with the
same additional log-like factor is obtained using a Parseval frame of curvelets.
Interestingly, the same error decay rate is also achieved using approximations
based on compactly supported shearlet frames, as stated below.
Theorem 4.7 Let S H (φ, ψ, ψ̃; c) be a frame for L 2 (R2 ), where c > 0, and
φ, ψ, ψ̃ ∈ L 2 (R2 ) are compactly supported functions such that, for all ξ = (ξ1 , ξ2 ) ∈
R2 , the shearlet ψ satisfies
α
−γ
−γ
(i) |ψ̂(ξ)| ≤ C1 min{1, |ξ
−γ |ξ1 | } min{1, |ξ2 | } and
1 | } min{1,
|ξ
|
(ii) | ∂∂ξ ψ̂(ξ)| ≤ |h(ξ1 )| 1 + |ξ21 |
,
where α > 5, γ ≥ 4, h ∈ L 1 (R), C1 is a constant and the shearlet ψ̃ satisfies (i)
and (ii) with the roles of ξ1 and ξ2 reversed.
Let f ∈ ε2 (R2 ) and f N be its nonlinear N -term approximation obtained by
selecting the N largest coefficients in the expansion of f with respect to this shearlet
frame S H (φ, ψ, ψ̃; c). Then there exists a constant C > 0, independent of f and N ,
such that
|| f − f N ||22 ≤ C N −2 (log N )3 as N → ∞.
Conditions (i) and (ii) are rather mild conditions and might be regarded as a weak
version of directional vanishing moment conditions.
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Part III
Application of Multiscale Transforms
to Image Processing
Chapter 5
Image Restoration
There are various applications of image restoration in todays world. Image restoration
is an important process in the field of image processing. It is a process to recover
original image from distorted image. Image restoration is a task to improve the
quality of image via estimating the amount of noises and blur involved in the image.
To restore image its too important to know a prior knowledge about an image i.e.
the knowledge about how an image was degraded or distorted. It is must to find out
that which type of noise is added in an image and how image gets blurred. So the
prior knowledge about an image is a one of the important part in image restoration.
Image gets degraded due to different conditions such as atmospheric conditions and
environmental conditions, so it is required to restore the original image by using
different image restoration algorithms. The main application of image restoration
i.e. image reconstruction is in radio astronomy, radar imaging and tomography in
medical imaging.
The main aim of restoration process is to remove the degradation from the image
and obtain the image F(x, y) which is close to the original image f (x, y). This process
is processed in two domains: spatial domain and frequency domain.
The main aim of this chapter is to focus on image restoration techniques using various transforms such as Fourier transform, wavelet transform, undecimated wavelet
transform, complex wavelet transform, quaternion wavelet transform, ridgelet transform, curvelet transform, contourlet transform and shearlet transform for natural
images.
5.1 Model of Image Degradation and Restoration Process
First of all we will see that how an image f (x, y) gets degraded and then how it can
be restored by using different image restoration algorithms. Consider the original
image f (x, y). If noise η(x, y) operates on original input image then a degraded
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_5
133
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5 Image Restoration
image g(x, y) is produced. The main objective of image restoration is that the output
to be as same as possible to the original image. In mathematical form, the degraded
image g(x, y) can be represented as:
g(x, y) = h(x, y) ∗ f (x, y) + η(x, y),
(5.1.1)
where h(x, y) is the degradation function and the symbol ∗ represents convolution of
h(x, y) with f (x, y). From Eq. (5.1.1), it is clear that the original image gets convolved
with the degraded image i.e. the original image f (x, y) gets convolved with the
degradation function h(x, y).
As we know, taking convolution of two functions in spatial domain is equivalent
to the product of the Fourier transform of the two functions in frequency domain.
Hence, to convert the convolutions into multiplication take DFT of above equation
in frequency domain. We have:
G(u, v) = H (u, v)F(u, v) + N (u, v)
(5.1.2)
where the terms in capital letters are the Fourier transforms of the corresponding terms
in Eq. (5.1.1). To reduce the effect of noise from degraded image inverse filtering
or pseudo inverse filtering can be used. In the next section, we are providing brief
discussion about noise models, types of noise and image denoising techniques.
5.2 Image Quality Assessments Metrics
In the digital image processing, the image restoration and enhancement techniques are
performed to improve the quality of the degraded image. In addition, since the original
image corresponding to an degraded image is unknown, the quality assessment is
performed by simulating the degradation model to the original (reference) image. To
demonstrate the performance of the image restoration and enhancement techniques,
objective image quality assessments metrics have been used in the literature.
The fundamental quality assessment metric is the mean squared error (MSE),
which represents the loss between the observed and predicted in statistics and regression analysis. In digital image processing, the MSE means that how close the restored
image to the latent image. The MSE is defined as
MSE =
M
N
2
1 ˆ
f (x, y) − f (x, y) ,
MN x=1 y=1
(5.2.1)
where fˆ (x, y) and f (x, y) respectively represent the restored and reference image
of the size M × N . The MSE is computed by averaging the square of the errors
between the restored and reference images. The small MSE value represents better
performance of the techniques and high-quality of the restored image. The root mean
5.2 Image Quality Assessments Metrics
135
square error (RMSE) is the square root of the MSE and is defined as
M N 2
1 RMSE =
fˆ (x, y) − f (x, y) .
MN x=1 y=1
(5.2.2)
The mean absolute error (MAE) is the average of the absolute of the difference
between the observed and predicted signals. The MAE between the original and
restored images is computed as
MAE =
M
N
1 ˆ
f (x, y) − f (x, y) .
MN x=1 y=1
(5.2.3)
The peak signal-to-noise ratio (PSNR) represents the ratio between the maximum
power of an image (signal) and the power of the errors between the reference and
restored images. The PSNR is written using the MSE as
L2
,
MSE
L
,
= 20 log10 √
MSE
= 20 log10 (L) − 10 log10 (MSE) ,
PSNR(dB) = 10 log10
(5.2.4)
where L is the maximum intensity level in the original image f (x, y). If the image is
quantized by 8 bits, the L is 255. If the MSE value is close to 0, the PSNR value is
infinite. The higher PSNR value represents better image quality.
The a measure of structural similarity (SSIM) is computed based on the properties
of the human visual system (HVS) [183]. Since the HVS is sensitive to the variation
in the luminance, contrast, and structural information of a scene, the SSIM assesses
the image quality by using the similarity measurement in luminance, contrast, and
structural information on the local patterns between the two images. The SSIM is
defined as
SSIM (p, q) =
2μp μq + C1
μ2p + μ2q + C1
α
×
2σp σq + C2
σp2 + σq2 + C2
β
×
σpq + C3
σp σq + C3
γ
(5.2.5)
where p and q respectively represent the window of the size n × n in the original
and restored images. μ and σ are the average and variance of the window, and
σpq the covariance between the window p and q. The first term compares the local
luminance change relative to background luminance, which is related to Weber’s law.
The second terms is a function to compare the change in the low-contrast region than
high-contrast region. The third term compares the structural similarity by computing
the correlation coefficient between p and q.
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5 Image Restoration
The overall SSIM index of the entire image is evaluated by averaging the SSIM
index of each local window as
SSIM (P, Q) =
N
1 SSIM (pk , qk ),
N
(5.2.6)
k=1
where P and Q respectively represent the reference and restored images, pk and qk
the k-th window, and N the total number of windows.
5.3 Image Denoising
Digital images play an important role both in daily life applications such as satellite
television, magnetic resonance imaging, computer tomography as well as in areas of
research and technology such as geographical information systems and astronomy.
Data sets collected by image sensors are generally contaminated by noise. Imperfect
instruments, problems with the data acquisition process, and interfering natural phenomena can all degrade the data of interest. Furthermore, noise can be introduced by
transmission errors and compression. Thus, denoising is often a necessary and the
first step to be taken before the images data is analyzed. It is necessary to apply an
efficient denoising technique to compensate for such data corruption.
Removing noise from the original signal is still a challenging problem for
researchers. There have been several published algorithms and each approach has its
assumptions, advantages, and limitations. Image denoising still remains a challenge
for researchers because noise removal introduces artifacts and causes blurring of the
images.
Noise modeling in images is greatly affected by capturing instruments, data transmission media, image quantization and discrete sources of radiation. Different algorithms are used depending on the noise model. Most of the natural images are assumed
to have additive random noise which is modeled as a Gaussian. Speckle noise is
observed by Guo et al. [72] in 1994 in ultrasound images whereas Jain [80] in 2006
observed Rician noise affects MRI images.
5.4 Noise Models
In order to restore an image we need to know about the degradation functions.
Different models for the noise are described in this section. The set of noise models
are defined by specific probability density functions (PDFs). Noise can be present in
image in two ways; either in additive or multiplicative form.
5.4 Noise Models
137
5.4.1 Additive Noise Model
Noise signal that is additive in nature gets added to the original signal to generate a
corrupted noisy signal and follows the following rule:
g(x, y) = f (x, y) + η(x, y),
(5.4.1)
where f (x, y) is the original image intensity and η(x, y) denotes the noise introduced
to produce the corrupted signal g(x, y) at (x, y) pixel location.
5.4.2 Multiplicative Noise Model
In this model, noise signal gets multiplied to the original signal. The multiplicative
noise model follows the following rule:
g(x, y) = f (x, y) × η(x, y).
(5.4.2)
5.5 Types of Noise
There are various types of noise. They have their own characteristics and are inherent
in images through different ways. Some commonly found noise models and their
corresponding PDFs are given below.
5.5.1 Amplifier (Gaussian) Noise
The typical model of amplifier noise is additive, independent at each pixel and independent of the signal intensity, called Gaussian noise. In color cameras, blue color
channels are more amplified than red or green channel, hence, more noise can be
present in the blue channel. Amplifier noise is a major part of the read noise of an
image sensor, that is, of the consistent noise level in dark areas of the image. This type
of noise has a Gaussian distribution, which has a bell shaped probability distribution
function (PDF) is given by
P(z) = √
1
2πσ 2
e−(z−μ)
2
/2σ 2
(5.5.1)
where z is the Gaussian random variable representing noise, μ is the mean or average of the function and σ is the standard deviation of the noise. Graphically, it is
represented as shown in Fig. 5.1.
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5 Image Restoration
Fig. 5.1 Gaussian distribution function
5.5.2 Rayleigh Noise
The PDE function of Rayleigh noise is given by
P(z) =
2
(z
b
− a)e−(z−a)
2
/b
if z ≥ a,
if z ≤ a.
0
The mean is given by
z=
and variance by
σ2 =
(5.5.2)
a+b
,
2
(b − a)2
.
12
5.5.3 Uniform Noise
It is another commonly found image noise i.e. uniform noise. Here the noise can take
on values in an interval [a, b] with uniform probability. The PDF is given by
P(z) =
1
(b−a)
if a ≤ z ≤ b,
0
otherwise.
(5.5.3)
5.5 Types of Noise
139
5.5.4 Impulsive (Salt and Pepper) Noise
Impulsive noise is sometimes called as salt and pepper noise or spike noise. This
kind of noise is usually seen on images. It represents itself as arbitrarily occurring
white and black pixels. An image that contains impulsive noise will have dark pixels
in bright regions and bright pixels in dark regions. It can be caused by dead pixels,
analog-to-digital converter errors and transmitted bit errors. The PDF of impulse
noise is given by
⎧
⎨ Pa for z = a,
P(z) = Pb for z = b,
(5.5.4)
⎩
0 otherwise.
If b > a, intensity b will appear as a light dot in the image and the vice-versa, level
a will appear like a dark dot. If Pa or Pb is zero, the impulse noise is called unipolar.
5.5.5 Exponential Noise
The PDF of exponential noise is given by
P(z) =
ae−az if z ≥ 0,
0
if z < 0,
(5.5.5)
where a > 0. The mean is given by
z=
1
,
a
σ2 =
1
.
a2
and variance by
5.5.6 Speckle Noise
Speckle noise is considered as multiplicative noise. It is a granular noise that degrades
the quality of images obtained by active image devices such as active radar and synthetic aperture radar (SAR) images. Due to random fluctuations in the return signal
from an object in conventional radar that is not big as single image processing element, speckle noise occurs. It increases the mean grey level of a local area. Speckle
noise makes image interpretation difficult in SAR images caused mainly due to coherent processing of backscattered signals from multiple distributed targets. Speckle
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5 Image Restoration
Fig. 5.2 Gamma distribution function
noise follows a gamma distribution and its probability distribution function (PDF) is
given by:
z α−1
e−z/α
(5.5.6)
P(z) =
(α − 1)!αα
where σ 2 is variance and z is the gray level. The gamma distribution is given in
Fig. 5.2.
5.6 Image Deblurring
Image deblurring (or restoration) is an old problem in image processing, but it continues to attract the attention of researchers and practitioners alike. A number of
real-world problems from astronomy to consumer imaging find applications for
image restoration algorithms and image restoration is an easily visualized exam-
5.6 Image Deblurring
141
ple of a larger class of inverse problems that arise in all kinds of scientific, medical,
industrial and theoretical problems.
Sometimes blur may be produced by the photographer to strengthen photos expressiveness, but unintentional blur will decrease the image quality, which is caused by
incorrect focus, object motion, hand shaking and so on. Convolution of the ideal
image with 2D point-spread function (PSF) h(x, y) in Eq. (5.1.1) is blurring of the
image. Different types of blurs are also responsible for degradation, blurred images
are restored by restoration model in the frequency domain. There are mainly four
types of deblurring problem:
5.6.1 Gaussian Blur
It is a filter that blends a specific number of pixels incrementally, following a bellshaped curve. Blurring is dense in the center and feathers at the edge. In remote
sensing, atmospheric turbulence is a severe limitation. The occurrence of Gaussian
blur depends on a variety of factors such as temperature, wind speed, exposure
time. It is type of image blurring filter which use Gaussian function for calculating
transformation applied on each pixel. The equation of Gaussian function is
G(x) = √
1
2πσ
e
(−x)2
2σ 2
,
(5.6.1)
where x is distance from origin in horizontal axis and σ is standard deviation of
Gaussian distribution.
5.6.2 Motion Blur
There is relative motion between camera and object due to the image capturing called
motion blur. Many types of motion blur can be distinguished all of which are due to
relative motion between the recording device and the scene. This can be in the form
of a translation, a rotation, a sudden change of scale, or some combinations of these.
5.6.3 Rectangular Blur
This is blurring in image with specific rectangular area. Blur in image can be identified
at any part based on this it can be circular and rectangular.
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5 Image Restoration
5.6.4 Defocus Blur
Defocus blur occurs in image when camera is improperly focused on image. The
resolution of image medium depends on amount of defocus. If there is more tolerance
of image there is low resolution in image. For good resolution of image defocus in
image should be minimize.
5.7 Superresolution
In most digital imaging applications, high resolution images or videos are usually
desired for later image processing and analysis. The desire for high image resolution
stems from two principal application areas: improvement of pictorial information
for human interpretation; and helping representation for automatic machine perception. Image resolution describes the details contained in an image, the higher the
resolution, the more image details. The resolution of a digital image can be classified in many different ways: pixel resolution, spatial resolution, spectral resolution,
temporal resolution, and radiometric resolution.
Superresolution image restoration is the process of producing a high- resolution
image (or a sequence of high-resolution images) from a set of low-resolution images.
The process requires an image acquisition model that relates A high-resolution image
to multiple low resolution images and involves solving the resulting inverse problem.
In addition to the degradation processes in single image restoration, super resolution
image restoration incorporates motion and downsampling operations into the imaging
process.
Superresolution (SR) are techniques that construct high-resolution (HR) images
from several observed low-resolution (LR) images, thereby increasing the high frequency components and removing the degradations caused by the imaging process of
the low resolution camera. The basic idea behind SR is to combine the non-redundant
information contained in multiple low-resolution frames to generate a high-resolution
image.
5.8 Classification of Image Restoration Algorithms
There are various restoration techniques as well as spatial domain filter for noise
removal. The spatial domain methods are used for removing additive noise only.
Sometimes blur helps to increase photos expressiveness but it decreases the quality
of image unintentionally. In image restoration, the improvement in the quality of the
restored image over the recorded blurred one is measured by the signal-to-noise ratio
improvement. Image restoration techniques are used to make the corrupted image as
similar as that of the original image.
5.8 Classification of Image Restoration Algorithms
143
Image Restoration in Presence of Noise
The noise removal is done by filtering of the degraded image. There are two basic
approaches to image denoising; (a) spatial filtering methods and (b) transform domain
filtering methods. The Objective of any filtering approach are:
i. To suppress the noise effectively in uniform regions.
ii. To preserve edges and other similar image characteristics.
iii. To provide a visually natural appearance.
5.8.1 Spatial Filtering
The most widely used filtering techniques in Image Processing are the spatial domain
filtering techniques. Spatial filtering is the method of choice in situations when only
additive noise is present in the image. The main idea behind Spatial Domain Filtering
is to convolve a mask with the image. Spatial filters can be further classified into
mean filters, order-statistics filters, Weiner filters and adaptive filters.
I. Mean Filters
A mean filter is the optimal linear filter for Gaussian noise in the sense of mean
square error. Mean filtering is a simple, intuitive and easy to implement method
of smoothing images i.e. reducing the amount of intensity variation between one
pixel and the next. It is often used to reduce noise in images. It reduces the intensity
variations between the adjacent pixels. Mean filter is nothing just a simple sliding
window spatial filter that replaces the centre value of the window with the original
signal and it works well only if the underlying signal is smooth.
The idea of mean filtering is simply to replace each pixel value in an image
with the mean (average) value of its neighbors, including itself. This has the effect
of eliminating pixel values which are unrepresentative of their surroundings. Mean
filtering is usually thought of as a convolution filter. Like other convolutions it is
based around a kernel, which represents the shape and size of the neighborhood
to be sampled when calculating the mean. Often a 3 × 3 square kernel is used, as
although larger kernels (e.g. 5 × 5 squares) can be used for more severe smoothing.
(Note that a small kernel can be applied more than once in order to produce a similar
but not identical effect as a single pass with a large kernel.) averaging kernel often
used in mean filtering.
An image with the mean (average) value of its neighbors, including itself. This has
the effect of eliminating pixel values which are unrepresentative of their surroundings.
next. It is often used to reduce noise in images. The idea of mean filtering is simply
to replace each pixel value in amount of intensity variation between one pixel and
the The two main problems with mean filtering, which are:
i. A single pixel with a very unrepresentative value can significantly affect the
mean value of all the pixels in its neighborhood.
144
5 Image Restoration
ii. When the filter neighborhood straddles an edge, the filter will interpolate new
values for pixels on the edge and so will blur that edge. This may be a problem if
sharp edges are required in the output.
There are four types of mean filters:
A. Arithmetic Mean Filter
In this type of mean filter the middle pixel value of the mask is replaced with the
arithmetic mean of all the pixel values within the filter window. A mean filter simply
smoothes local variations in an image. Noise is reduced and as a result the image
smoothens, but edges within the image get blurred.
If Sxy represent a rectangular subimage window of size m × n, centered at point
(x, y), then the value of restored image fˆ at the point (x, y) is defined as
1 g(s, t),
fˆ (x, y) =
mn (s,t)∈S
(5.8.1)
xy
where g(x, y) is the corrupted image.
B. Geometric Mean Filter
The working of a geometric mean filter is same as the arithmetic mean filter; the only
difference is that instead of taking the arithmetic mean the geometric mean is taken.
The restored image is given by the expression
⎡
fˆ (x, y) = ⎣
⎤ mn1
g(s, t)⎦
,
(5.8.2)
(s,t)∈Sxy
Value of each restored pixel is the product of pixels in the mask, raised to a power
1
.
mn
C. Harmonic Mean Filter
In the harmonic mean method, the gray value of each pixel is replaced with the
harmonic mean of gray values of the pixels in a surrounding region. The harmonic
mean is defined as
n
H= 1
.
1
+ x2 + · · · + x1n
x1
The restored image is given by the function:
fˆ (x, y) = mn
1
(s,t)∈Sxy g(s,t)
.
(5.8.3)
5.8 Classification of Image Restoration Algorithms
145
D. Contra-Harmonic Mean Filter
The restored image is given by the equation
(s,t)∈Sxy
fˆ (x, y) = g(s, t)Q−1
(s,t)∈Sxy
g(s, t)Q
.
(5.8.4)
where Q is the order of the system. If Q = 0, it behaves as arithmetic mean filter and
if Q = 1, it behaves as harmonic mean filter.
II. Order-Statistic Filter
order-statistic filters are spatial filters whose response is based on ordering (ranking)
the values of the pixels contained in the image area encompassed by the filter. The
ranking result determines the response of the filter. There are four types of orderstatistic filters.
A. Median Filter
The best-known order-statistic filter is the median filter and it is also belongs to the
class of non-linear filter. The median filter is normally used to reduce noise in an
image, somewhat like the mean filter. However, it often does a better job than the
mean filter of preserving useful detail in the image. Median filter replaces the value
of a pixel with the median value of the gray levels within the filter window. Median
filters are very effective for impulse noise.
fˆ (x, y) = median(s,t)∈Sxy {g(s, t)}.
(5.8.5)
Median filters are popular because they provide excellent noise-reduction capabilities
for certain types of random noise, with considerably less blurring than linear smoothing filters of similar size. Median filters are particularly effective in the presence of
bipolar and unipolar impulse noise.
B. Max and Min Filters
The maximum filter is defined as the maximum of all pixels within a local region of
an image. So it replaces the center pixel value with the maximum value of pixel in
the subimage window. Similarly the minimum filter is defined as the minimum of all
pixels within a local region of an image and the center pixel value is replaced with
the minimum value of pixel in the subimage window.
fˆ (x, y) = max(s,t)∈Sxy {g(s, t)} for max filter
(5.8.6)
fˆ (x, y) = min(s,t)∈Sxy {g(s, t)} for min filter
(5.8.7)
C. Midpoint Filter
This filter computes the midpoint between the maximum and minimum values in the
area encompassed by the filter.
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1
fˆ (x, y) = [max(s,t)∈Sxy {g(s, t)} + min(s,t)∈Sxy {g(s, t)}].
2
(5.8.8)
Note that this filter combines order statistics and averaging. It works best for randomly
distributed noise, like Gaussian or uniform noise.
D. Alpha-Trimmed Mean Filter
In Alpha-trimmed filter the d /2 lowest and d /2 highest intensity values of g(s, t) in
the neighborhood Sxy are deleted and the remaining (mn − d ) pixels are averaged.
The center pixel value is replaced with this averaged value.
The estimation function for the restored image is given by
fˆ (x, y) =
1
gr (s, t).
(mn − d ) (s,t)∈S
(5.8.9)
xy
III. Wiener Filter
Wiener filter incorporate both the degradation function and statistical characteristics
of noise in to the restoration process. The scientist Wiener proposed this the concept
in the year 1942. The filter, which consists of the terms inside the brackets, also is
commonly referred as the minimum mean square error filter or least square error
filter The Wiener filter is used to signal estimation problem for stationary signals.
The Wiener filter is the MSE-optimal stationary linear filter for images degraded by
additive noise and blurring. In analysis of the Wiener filter requires the assumption
that the signal and noise processes are second order stationary (in the random process
sense). Wiener filters are also applied in the frequency domain.
IV. Adaptive Filters
The behavior of the Adaptive filters changes with the statistical characteristics of the
image inside the filter window. Therefore the performance of Adaptive filters is much
better in comparison with the non-adaptive filters. But the improved performance is
at the cost of added filter complexity.
Mean and variance are two important statistical measures on which the adaptive filtering is depends upon. For example if the local variance is high compared
to the overall image variance, the filter should return a value close to the present
value. Because high variance is usually associated with edges and edges should be
preserved.
5.8.2 Frequency Domain Filtering
Image smoothing and image sharpening can be achieved by frequency domain filtering. Smoothing is done by high frequency attenuation i.e. by low pass filtering.
Sharpening is done by high pass filtering which attenuates the low frequency components without disturbing the high frequency components.
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The frequency domain filtering methods can be subdivided according to the choice
of the basis functions. The basis functions can be further classified as data adaptive
and non-adaptive. Non-adaptive transforms are discussed first since they are more
popular.
I. Spatial-Frequency Filtering
Spatial-frequency filtering refers use of low pass filters using fast Fourier transform
(FFT). In frequency smoothing methods [51] the removal of the noise is achieved
by designing a frequency domain filter and adapting a cut-off frequency when the
noise components are decorrelated from the useful signal in the frequency domain.
These methods are time consuming and depend on the cut-off frequency and the
filter function behavior. Furthermore, they may produce artificial frequencies in the
processed image.
II. Wavelet Domain
Filtering operations in the wavelet domain can be subdivided into linear and nonlinear
methods.
A. Linear Filters
Linear filters such as Wiener filter in the wavelet domain yield optimal results when
the signal corruption can be modeled as a Gaussian process and the accuracy criterion is the mean square error (MSE) [36, 165]. However, designing a filter based on
this assumption frequently results in a filtered image that is more visually displeasing than the original noisy signal, even though the filtering operation successfully
reduces the MSE. In a wavelet-domain spatially adaptive FIR Wiener filtering for
image denoising is proposed by Zhang et al. [202] in 2000, where wiener filtering is
performed only within each scale and intrascale filtering is not allowed.
B. Non-Linear Threshold Filters
The most investigated domain in denoising using Wavelet Transform is the non-linear
coefficient thresholding based methods. The procedure exploits sparsity property of
the wavelet transform and the fact that the Wavelet Transform maps white noise in
the signal domain to white noise in the transform domain. Thus, while signal energy
becomes more concentrated into fewer coefficients in the transform domain, noise
energy does not. It is important principle that enables the separation of signal from
noise.
The procedure in which small coefficients are removed while others are left
untouched is called Hard Thresholding introduced by Donoho [53]. But the method
generates spurious blips, better known as artifacts, in the images as a result of unsuccessful attempts of removing moderately large noise coefficients. To overcome the
demerits of hard thresholding, wavelet transform using soft thresholding was also
introduced by Donoho in 1995 [53]. In this scheme, coefficients above the threshold
are shrunk by the absolute value of the threshold itself. Similar to soft thresholding, other techniques of applying thresholds are semi-soft thresholding and Garrote
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thresholding. Most of the wavelet shrinkage literature is based on methods for choosing the optimal threshold which can be adaptive or non-adaptive to the image.
i. Non-adaptive Thresholds
VISUShrink is non-adaptive universal threshold introduced by Donoho and
Jhonstone [51] in 1994, which depends only on number of data points. It has asymptotic equivalence suggesting best performance in terms of MSE when the number
of pixels reaches infinity. VISUShrink is known to yield overly smoothed images
because its threshold choice can be unwarrantably large due to its dependence on the
number of pixels in the image.
ii. Adaptive Thresholds
SUREShrink uses a hybrid of the universal threshold and the SURE [Steins Unbiased
Risk Estimator] threshold and performs better than VISUShrink, is also introduced
by Donoho and Jhonstone [51] in 1994. BayesShrink, introduced by Simoncelli and
Adelson [157] in 1996 and by Chipmann et al. in 1997 [32], minimizes the Bayes
Risk Estimator function assuming Generalized Gaussian prior and thus yielding data
adaptive threshold. BayesShrink outperforms SUREShrink most of the times. Cross
validation [81] replaces wavelet coefficient with the weighted average of neighborhood coefficients to minimize generalized cross validation (GCV) function providing
optimum threshold for every coefficient.
The assumption that one can distinguish noise from the signal solely based on
coefficient magnitudes is violated when noise levels are higher than signal magnitudes. Under this high noise circumstance, the spatial configuration of neighboring
wavelet coefficients can play an important role in noise-signal classifications. Signals
tend to form meaningful features (e.g. straight lines, curves), while noisy coefficients
often scatter randomly.
C. Non-orthogonal Wavelet Transforms
Undecimated Wavelet Transform (UDWT) has also been used for decomposing the
signal to provide visually better solution. Since UDWT is shift invariant it avoids
visual artifacts such as pseudo-Gibbs phenomenon. Though the improvement in
results is much higher, use of UDWT adds a large overhead of computations thus
making it less feasible. Lang et al. [99] extended normal hard/soft thresholding
to shift invariant discrete wavelet transform. Cohen [38] exploited Shift Invariant
Wavelet Packet Decomposition (SIWPD) to obtain number of basis functions. Then
using Minimum Description Length principle the best basis function was found out
which yielded smallest code length required for description of the given data. Then,
thresholding was applied to denoise the data.
In addition to UDWT, use of multi-wavelets is explored which further enhances the
performance but further increases the computation complexity. The multi-wavelets
are obtained by applying more than one mother function (scaling function) to given
dataset. Multi-wavelets possess some important properties such as short support,
symmetry, and the most importantly higher order of vanishing moments.
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D. Wavelet Coefficient Model
This approach focuses on exploiting the multi-resolution properties of Wavelet Transform. This technique identifies close correlation of signal at different resolutions by
observing the signal across multiple resolutions. This method produces excellent
output but is computationally much more complex and expensive. The modeling of
the wavelet coefficients can either be deterministic or statistical.
i. Deterministic Modeling of Wavelet Coefficients
The Deterministic method of modeling involves creating tree structure of wavelet
coefficients with every level in the tree representing each scale of transformation and
nodes representing the wavelet coefficients. This approach is adopted by Baraniuk
[14] in 1999. The optimal tree approximation displays a hierarchical interpretation
of wavelet decomposition. Wavelet coefficients of singularities have large wavelet
coefficients that persist along the branches of tree. Thus if a wavelet coefficient has
strong presence at particular node then it being signal, its presence should be more
pronounced at its parent nodes. If it is noisy coefficient, for instance spurious blip,
then such consistent presence will be missing. Lu et al. [106] tracked wavelet local
maxima in scale space, by using a tree structure. Other denoising method based on
wavelet coefficient trees was proposed by Donoho [55] in 1997.
ii. Statistical Modeling of Wavelet Coefficients
This approach focuses on some more interesting and appealing properties of the
wavelet transform such as multiscale correlation between the wavelet coefficients,
local correlation between neighborhood coefficients etc. This approach has an inherent goal of perfecting the exact modeling of image data with use of wavelet transform.
A good review of statistical properties of wavelet coefficients can be found by Buecigrossi and Smoncelli [18] and by Romberg et al. [146]. The following two techniques
exploit the statistical properties of the wavelet coefficients based on a probabilistic
model.
a. Marginal Probabilistic Model
A number of researchers have developed homogeneous local probability models for
images in the wavelet domain. Specifically, the marginal distributions of wavelet coefficients are highly kurtosis, and usually have a marked peak at zero and heavy tails.
The Gaussian mixture model (GMM) [32] and the generalized Gaussian distribution (GGD) [120] are commonly used to model the wavelet coefficients distribution.
Although GGD is more accurate, GMM is simpler to use. Mihcak et al. [115] in 1999
proposed a methodology in which the wavelet coefficients are assumed to be conditionally independent zero-mean Gaussian random variables, with variances modeled
as identically distributed, highly correlated random variables. An approximate Maximum A Posteriori (MAP) Probability rule is used to estimate marginal prior distribution of wavelet coefficient variances. All these methods mentioned above require
a noise estimate, which may be difficult to obtain in practical applications. Simoncelli and Adelson [157] used a two parameter generalized Laplacian distribution for
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the wavelet coefficients of the image, which was estimated from the noisy observations. Chang et al. [25] proposed the use of adaptive wavelet thresholding for image
denoising, by modeling the wavelet coefficients as a generalized Gaussian random
variable, whose parameters are estimated locally i.e. within a given neighborhood.
b. Joint Probabilistic Model
Hidden Markov Models (HMM) [146] models are efficient in capturing inter-scale
dependencies, whereas Random Markov Field [110] models are more efficient to capture intrascale correlations. The complexity of local structures is not well described
by Random Markov Gaussian densities whereas Hidden Markov Models can be used
to capture higher order statistics. The correlation between coefficients at same scale
but residing in a close neighborhood are modeled by Hidden Markov Chain Model
where as the correlation between coefficients across the chain is modeled by Hidden
Markov Trees. Once the correlation is captured by HMM, Expectation Maximization
is used to estimate the required parameters and from those, denoised signal is estimated from noisy observation using well known MAP estimator. Portilla et al. [137]
described a model in which each neighborhood of wavelet coefficients is described
as a Gaussian scale mixture (GSM) which is a product of a Gaussian random vector,
and an independent hidden random scalar multiplier. Strela et al. [165] described
the joint densities of clusters of wavelet coefficients as a Gaussian scale mixture,
and developed a maximum likelihood solution for estimating relevant wavelet coefficients from the noisy observations. Another approach that uses a Markov random
field model for wavelet coefficients was proposed by Jansen and Bulthel [81] in
2001. A disadvantage of HMT is the computational burden of the training stage. In
order to overcome this computational problem, a simplified HMT, named as UHMT,
was proposed by Romberg et al. [146] in 2001.
III. New Multiscale Transforms
Wavelet denoising is performed by taking the wavelet transform of the noisy image
and then removing out the detail (typically high-pass) coefficients that fall below a
certain threshold. The thresholding can be either soft or hard. An inverse wavelet
transform is then applied to the thresholded wavelet coefficients to yield the final
reconstructed image. As in classical low-pass filtering, zeroing out detail coefficients
removes high-frequency noise. However, in wavelet denoising, if the signal itself has a
high-pass feature, such as a sharp discontinuity, the corresponding detail coefficient
will not be removed out in the thresholding. In this way, wavelet denoising can
low-pass filter the signal while preserving the high-frequency components. Same
is the case with other types of Xlet transforms. Recently, a new method known as
cycle spinning has been proposed as an improvement on threshold based on wavelet
denoising.
The wavelet transform is not time-invariant. So the estimate of signal which is
shifted and then denoised is different than that obtained by without shifting. The
cycle spinning estimate is obtained by linearly averaging these shifted estimates. The
errors in the individual estimates will not be statistically dependent, and therefore,
the averaging will reduce the noise power but will introduce blur and artifacts to
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some extent. Wavelet transform in two dimensions is obtained by a tensor product
of one dimensional wavelet transform. Wavelet is good at isolating discontinuities
across horizontal or vertical edges but it will not achieve the smoothness along the
curved edges.
To overcome the drawback of the wavelet transform, Candes [22] presented a
new technique named as ridgelet transform. The basis of ridgelet transform is radon
transform. Radon transform has been widely used for tomographic reconstruction
using Radon projections taken at an angle θ ∈ [0, 2π) introduce a redundancy of four.
Hence the projection in one quadrant θ ∈ [0, 2π) are used for tomographic representation to reduce computational overheads. Though projections at an angle θ ∈ [0, 2π)
do not introduce any redundancy, the achieved tomographic representation is very
poor. The conventional discrete wavelet transform (DWT) introduces artifacts during
processing of curves. Finite Ridgelet Transform (FRIT) solved this problem by mapping the curves in terms of ridges. However, blind application of FRIT all over the
image is computationally very heavy. Finite Curvelet Transform (FCT) selectively
applies FRIT only on the tiles containing small portions of a curve. This work aims
at presenting denoising techniques for digital images using different transforms such
as wavelet, ridgelet, curvelet transform contourlet transform and shearlet transform;
jointly represented as Xlet transforms or new multiscale transforms.
Other Image Restoration Techniques
5.8.3 Direct Inverse Filtering
The blurring function of the corrupted image is known or can be developed then it
has been proved as quickest and easiest way to restore the distorted image. Blurring
can be considered as low pass filtering in inverse filtering approach and use high
pass filtering action to reconstruct the blurred image without much effort. Suppose
first that the additive noise is negligible. A problem arises if it becomes very small
or zero for some point or for a whole region in the plane then in that region inverse
filtering cannot be applied.
5.8.4 Constraint Least-Square Filter
Regularized filtering is used in a better way when constraints like smoothness are
applied on the recovered image and very less information is known about the additive
noise. The blurred and noisy image is regained by a constrained least square restoration algorithm that uses a regularized filter. Regularized restoration provides almost
similar results as the wiener filtering but viewpoint of both the filtering techniques
are different. In regularized filtering less previous information is required to apply
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restoration. The regularization filter is frequently chosen to be a discrete Laplacian.
This filter can be understood as an approximation of a Weiner filter. It is also called
Regularized filter, which is a vector matrix form of linear degradation model.
5.8.5 IBD (Iterative Blind Deconvolution)
Iterative Blind Deconvolution (IBD) was put forward by Ayers and Dainty and it is
one of the methods used in blind Deconvolution. This method is based on Fourier
Transformation causes less computation. Iterative Blind Deconvolution has good
anti-noise capability. In this method image restoration is difficult process where
image recovery is performed with little or no prior knowledge of the degrading PSF.
The Iterative Blind Deconvolution algorithm has higher resolution and better quality.
The main drawback of this method is that convergence of the iterative process is not
guaranteed. But the original image can have effect on the final result.
5.8.6 NAS-RIF (Nonnegative and Support Constraints
Recursive Inverse Filtering)
The aim of blind Deconvolution is to reconstruct a reliable estimated image from a
blurred image. D. Kundur put forward NAS-RIF algorithm (Nonnegative and Support
Constraints Recursive Inverse Filtering) to achieve this aim.
NAS-RIF algorithm based on given image make an estimation of target image.
The estimation is made by minimizing an error function which contains the domain of
image and nonnegative information of pixels of image. There is a feasible solution that
makes the error function globally optimized. In theory, the estimation is equivalent to
the real image. The advantage of this algorithm is that we don’t need to know about
the parameters of PSF and the priori information of original image, all we have to
determine support domain of target area and to make sure the estimation of image
is nonnegative. Another advantage is that this algorithm contains a process which
makes sure the function can convergent to global least. The disadvantage of NASRIF is that it is sensitive to noise, so it is only proper for images with symmetrical
background.
5.8.7 Superresolution Restoration Algorithm Based
on Gradient Adaptive Interpolation
The basic idea of the gradient-based adaptive interpolation is that the interpolated
pixel value is affected by the local gradient of a pixel, mainly in edge areas of the
image. The more influence it should have on the interpolated pixel the smaller the
local gradient of a pixel. The method involves three subtasks: registration, fusion and
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deblurring. Firstly we utilize the frequency domain registration algorithm to estimate
the motions of the low resolution images. According to the motion the low resolution
images are mapped to the uniform high resolution grid, and then the gradient based
adaptive interpolation is used to form a high resolution image. Finally, wiener filter
is applied to reduce the effects of blurring and noise caused by the system. The main
advantage of this algorithm is low computation complexity.
5.8.8 Deconvolution Using a Sparse Prior
This algorithm formulates the Deconvolution problem as given the observation determining the maximum a-posterior estimate of the original image. Furthermore, the
algorithm exploits a prior enforcing spatial-domain sparsely of the image derivatives. The resulting non-convex optimization problem is solved using an iterative
re-weighted least square method. Although this algorithm has not been natively
devised for Poisoning observations, it has been rather successfully applied to raw
images. By the selection of the smoothness-weight parameter allowing a sufficient
number of iterations we can get better result.
5.8.9 Block-Matching
Block-matching is employed to find blocks that contain high correlation because
its accuracy is significantly impaired by the presence of noise. We utilize a blocksimilarity measure which performs a coarse initial denoising in local 2D transform
domain. In this method image is divided into blocks and noise or blur is removed
from each block.
5.8.10 LPA-ICI Algorithm
The LPA-ICI algorithm is nonlinear and spatially-adaptive with respect to the
smoothness and irregularities of the image and blurs operators. Simulation experiments demonstrate efficiency and good performance of the proposed Deconvolution
technique.
5.8.11 Deconvolution Using Regularized Filter (DRF)
Deconvolution by Regularized filtering is another category of Non-Blind Deconvolution technique. When constraints like smoothness are applied on the recovered
image and limited information about the noise is known, then regularized Decon-
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volution is used effectively. The degraded image is actually restored by constrained
least square restoration by using a regularized filter. In regularized filtering less prior
information is required to apply the restoration. Regularization can be a useful tool,
when statistical data is unavailable. Moreover, this framework can be extended to
adapt to edges of image, noise that is varying spatially and other challenges.
5.8.12 Lucy-Richardson Algorithm
Image restoration method is divided into two types one is blind and other is non
blind deconvolution. The non-blind deconvolution is one in which the PSF is known.
The RichardsonLucy deconvolution algorithm has become popular in the fields of
medical imaging and astronomy. Initially it was found in the early 1970s from byes
theorem by Lucy and Richardson. In the early 1980s it was redeliver by Verdi as
an algorithm to solve emission tomography imaging problems, in which Poisoning
statistics are dominant. Lucy Richardson is nonlinear iterative method. During the
past two decades, this method have been gaining more acceptance as restoration
tool that result in better than those obtained with linear methods. Thus for restored
image of good quality the Number of iterations is determined manually fore very
image as per the PSF size. The RichardsonLucy algorithm is an iterative procedure
for recovering a latent image that has been the blurred by Known PSF.
5.8.13 Neural Network Approach
Neural network is a form of multiprocessor computer system, with simple processing elements, interconnected group of nodes. These Interconnected components are
called neurons, which send message to each other. When an element of the neural
network fails, it can continue without any problem by their parallel nature.
ANN provides a robust tool for approximating a target function given a set input
output example and for the reconstruction function from a class images. Algorithm
such as the Back propagation and the Perception use gradient-decent techniques to
tune the network parameters to best-fit a training set of input output examples. Back
propagation neural network approach for image restoration is capable of learning
complex non-linear function this method calculate gradient of function with respect
to all weight in function.
5.9 Application of Multiscale Transform in Image
Restoration
In this section we will detail review of applications of multiscale transform one by
one in image restoration.
5.9 Application of Multiscale Transform in Image Restoration
155
5.9.1 Image Restoration Using Wavelet Transform
Image restoration from corrupted image is a classical problem in the field of image
processing. Mainly, image denoising has remained a basic problem in the field of
image processing. Wavelets give a superior performance in image denoising due
to properties such as sparsity and multiresolution structure. With Wavelet Transform gaining popularity in the last two decades various algorithms for denoising in
wavelet domain were introduced. The focus was shifted from the Spatial and Fourier
domain to the Wavelet transform domain. Ever since Donohos [53] Wavelet based
thresholding approach was published in 1995, there was a surge in the denoising
papers being published. Although Donohos concept was not revolutionary, his methods did not require tracking or correlation of the wavelet maxima and minima across
the different scales as proposed by Mallat. Thus, there was a renewed interest in
wavelet based denoising techniques since Donoho demonstrated a simple approach
to a difficult problem. Researchers published different ways to compute the parameters for the thresholding of wavelet coefficients. Data adaptive thresholds were
introduced to achieve optimum value of threshold. Later efforts found that substantial improvements in perceptual quality could be obtained by translation invariant
methods based on thresholding of an Undecimated Wavelet Transform. These thresholding techniques were applied to the nonorthogonal wavelet coefficients to reduce
artifacts. Multiwavelets were also used to achieve similar results. Probabilistic models using the statistical properties of the wavelet coefficient seemed to outperform the
thresholding techniques. Recently, much effort has been devoted to Bayesian denoising in Wavelet domain. Hidden Markov Models and Gaussian Scale Mixtures have
also become popular and more research continues to be published. Tree Structures
ordering the wavelet coefficients based on their magnitude, scale and spatial location have been researched. Data adaptive transforms such as independent component
analysis (ICA) have been explored for sparse shrinkage. The trend continues to focus
on using different statistical models to model the statistical properties of the wavelet
coefficients and its neighbors. Future trend will be towards finding more accurate
probabilistic models for the distribution of non-orthogonal wavelet coefficients.
The multi-resolution analysis performed by the wavelet transform has proved
to be particularly efficient in image denoising. Since the early use of the classical
orthonormal wavelet transform for removing additive white Gaussian noise through
thresholding, a lot of work has been done leading to some important observations:
1. Better performances can be achieved with shift invariant transformations.
2. Directionality of the transform is important in processing geometrical images.
3. Further improvements can be obtained with more sophisticated thresholding
functions which incorporate inter-scale and intra-scale dependencies.
Various methods have been attempted to take advantage of these observations such
as the undecimated discrete wavelet transform (UDWT) (becomes shift-invariant
through the removal of the down sampling found in the DWT), the double density
discrete wavelet transform (DDDWT) (which uses oversampled filters), the dual tree
discrete wavelet transform (DT-DWT) (uses two sets of critically sampled filters that
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form Hilbert transform pairs) and the double density dual tree wavelet transform
(DDDTDWT) (uses two sets of oversampled filters forming complex sub bands to
give spatial feature information along multiple directions). For estimating the optimal
threshold many procedures have been developed, such as VisuShrink, BayesShrink,
SUREshrink, NeighSURE etc. These methods model the denoising problem assuming various distributions for the signal coefficients and noise components. The optimal threshold is determined based on estimators such as maximum apriority (MAP),
maximum absolute deviation (MAD), maximum likelihood estimation (MLE) etc.
Mallat and Hwang [111], in 1992, demonstrated that wavelet transform is particularly suitable for the applications of non-stationary signals which may spontaneously
vary in time. They developed an algorithm that removes white noises from signals
by analyzing the evolution of the wavelet transform maxima across scales. In twodimensions, the wavelet transform maxima indicate the location of edges in images.
In addition, they also extended the denoising algorithm for image enhancement.
Donoho [53], in 1995, recovered objects from noisy and incomplete data. The
method utilized nonlinear operation in wavelet domain and they concluded that
the recovery of the signals with noisy data is not possible by traditional Fourier
approaches. He proposed the heuristic principles, theoretical foundations and possible application area for denoising.
Xu et al. [192], in 1994, introduced a spatially selective noise filtration technique
based on the direct spatial correlation of the wavelet transform at several adjacent
scales. They used a high correlation to infer that there was a significant feature at the
position that should be passed through the filter.
Donoho and Stone [51, 52] established the thresholding by coining soft-threshold
and hard-threshold wavelet denoising methods. The basic idea was comparison
between different scale coefficients module a certain threshold and obtained the
de-noised signal by the inverse transform. Although the implementation of thresholding method was simple, it did not take into account the correlation between wavelet
coefficients.
Donoho [53], in 1995, proposed a method for signal recovery from noisy data.
The reconstruction is defined in the wavelet domain by translating
all the empirical
√
wavelet coefficients of noisy data towards 0 by an amount σ · (2ln(n)/n). However,
the method lacks the smoothing property. The method of adaption and the method of
proof are bothmore technically discussed by Donoho using soft thresholding which
uses a pyramid filtering. It acts as an unconditional basis for a very wide range of
smoothness spaces.
Malfait and Roose [110], in 1997, proposed a new method for the suppression
of noise in images via the wavelet transform. The method had relies on two measures. The first is a classic measure of smoothness of the image and is based on an
approximation of the local Holder exponent via the wavelet coefficients. The second,
novel measure takes into account geometrical constraints, which are generally valid
for natural images. The smoothness measure and the constraints are combined in a
Bayesian probabilistic formulation, and are implemented as a Markov random field
(MRF) image model. The manipulation of the wavelet coefficients is consequently
based on the obtained probabilities. A comparison of quantitative and qualitative
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157
results for test images demonstrates the improved noise suppression performance
with respect to previous wavelet-based image denoising methods.
Wood and Johnson [186], in 1998, did denoising of synthetic, phantom, and volunteer cardiac images either in the magnitude or complex domains. Authors suggested
denoising prior to rectification for superior edge resolution of real and imaginary
images. Magnitude and complex denoising significantly improved SNR.
Crouse et al. [42], in 1998, developed a novel algorithm for image denoising. The
algorithm used wavelet domain hidden Markov models (HMM) for statistical signal
processing. This algorithm was compared with Sureshrink and Bayesian algorithm
and showed better result on 1024 length test signal. Least MSE obtained was 0.081.
This algorithm was not validated on image but rather on a 1-D test signal so the MSE
was very less.
Strela and Walden [166], in 1998, studied wavelet thresholding in the context of
scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. This led to a careful formulation of the universal threshold
for scalar thresholding and vector thresholding. Multi-wavelets generally outperformed scalar wavelets for image denoising for all four noisy 1D test images and
the results were visually very impressive. Chui-Lian scaling functions and wavelets
combined with repeated row preprocessing appears to be a good general method. For
both 1D and 2D cases, the reconstructed signals derived from such a good general
method demonstrated much reduced noise levels, typically 50 percent of the standard
deviation of the original noise.
Mihcak et al. [115], in 1999, introduced a simple spatially adaptive statistical
model for wavelet image coefficients, called LAWML and LAWMAP methods,
and applied it to image denoising. Their model was inspired by a recent wavelet
image compression algorithm, the estimation-quantization (EQ) coder. They produced model wavelet image coefficients as zero-mean Gaussian random variables
with high local correlation. They assumed a marginal prior distribution on wavelet
coefficients variances and estimate them using an approximate maximum a posteriori probability rule. Then they applied an approximate minimum mean squared error
estimation procedure to restore the noisy wavelet image coefficients. Despite the
simplicity of their method, both in its concept and implementation, their denoising
results were among the best reported in the literature. However, retained too small
wavelet coefficients, severe burr phenomenon appeared in the reconstructed image.
Chang et al. [25], in 2000, organized the paper in two parts. The first part of this
paper proposed an adaptive, data-driven threshold for image denoising via wavelet
soft-thresholding. The threshold was derived in a Bayesian framework, and the prior
used on the wavelet coefficients was the generalized Gaussian distribution (GGD)
widely used in image processing applications. The proposed threshold was simple
and closed-form, and it was adaptive to each subband because it depends on datadriven estimates of the parameters. Experimental results had shown that the proposed
method, called BayesShrink, was typically within 5 percent of the MSE of the best
soft-thresholding benchmark with the image assumed known. It also outperformed
Donoho and Johnstones SureShrink most of the time.
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The second part of the paper attempted to further validate recent claimed that
lossy compression could be used for denoising. The BayesShrink threshold could
aided in the parameter selection of a coder designed with the intention of denoising,
and thus achieving simultaneous denoising and compression. Specifically, the zerozone in the quantization step of compression was analogous to the threshold value
in the thresholding function. The remaining coder design parameters were chosen
based on a criterion derived from Rissanens minimum description length (MDL)
principle. Experiments show that this compression method did indeed remove noise
significantly, especially for large noise power. However, it introduced quantization
noise and should be used only if bitrate were an additional concern to denoising.
Rosenbaum et al. [147], in 2000, used wavelet shrinkage denoising algorithms
and Nowak’s algorithm for denoising the magnitude images. The wavelet shrinkage
denoising methods were performed using both soft and hard thresholding and it was
suggested that changes in mean relative SNR are statistically associated with type
of threshold and type of wavelet. The data-adaptive wavelet filtering was found to
provide the best overall performance as compared to direct wavelet shrinkage.
Chang et al. [26, 27] recovered image from multiple noisy copies by combining the
two operations of averaging and thresholding. Averaging followed by thresholding or
thresholding followed by averaging produces different estimators. The signal wavelet
coefficients are modeled as Laplacian and the noise is modeled as Gaussian. Four
standard images are used for performance comparison with other standard denoising
techniques. Barbara image denoised by the proposed algorithm have least MSE of
51.27. Along with Barbara image, three other test images are used to evaluate the
performance of this algorithm.
Pizurica et al. [134], in 2002, presented a new wavelet-based image denoising
method, which extended a recently emerged geometrical Bayesian framework. The
new method combined three criteria for distinguishing supposedly useful coefficients from noise: coefficient magnitudes, their evolution across scales and spatial
clustering of large coefficients near image edges. These three criteria were combined in a Bayesian framework. The spatial clustering properties were expressed
in a prior model. The statistical properties concerning coefficient magnitudes and
their evolution across scales were expressed in a joint conditional model. The three
main novelties with respect to related approaches were (1) the interscale-ratios of
wavelet coefficients were statistically characterized and different local criteria for
distinguishing useful coefficients from noise were evaluated, (2) a joint conditional
model was introduced, and (3) a novel anisotropic Markov random field prior model
was proposed. The results demonstrated an improved denoising performance over
related earlier techniques.
Sender and Selesnick [152], in 2002, proposed new non-Gaussian bivariate distributions only for the dependencies between the coefficients and their parents in
detail since most simple nonlinear thresholding rules for wavelet-based denoising
assume that the wavelet coefficients are independent, however, wavelet coefficients
of natural images have significant dependencies and corresponding nonlinear threshold functions (shrinkage functions) were derived from the models using Bayesian
estimation theory. The new shrinkage functions did not assume the independence of
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wavelet coefficients. They showed three image denoising examples in order to show
the performance of these new bivariate shrinkage rules. In the second example, a
simple subband-dependent data-driven image denoising system was described and
compared with effective data-driven techniques in the literature, namely VisuShrink,
SureShrink, BayesShrink, and hidden Markov models. In the third example, the same
idea was applied to the dual-tree complex wavelet coefficients.
Regularization is achieved by promoting a reconstruction with low-complexity,
expressed in the wavelet coefficients, taking advantage of the well-known sparsity of
wavelet representations. Previous works have investigated wavelet-based restoration
but, except for certain special cases, the resulting criteria are solved approximately
or require demanding optimization methods.
High-resolution image reconstruction refers to the reconstruction of
high-resolution images from multiple low-resolution, shifted, degraded samples of
a true image. Chan et al. [28], in 2003 analyzed this problem from the wavelet point
of view. By expressing the true image as a function in L(R2 ), they derived iterative algorithms which recover the function completely in the L sense from the given
low-resolution functions. These algorithms decomposed the function obtained from
the previous iteration into different frequency components in the wavelet transform
domain and add them into the new iterate to improve the approximation. We apply
wavelet (packet) thresholding methods to denoised the function obtained in the previous step before adding it into the new iterate. Their numerical results showed that
the reconstructed images from our wavelet algorithms are better than that from the
Tikhonov least-squares approach.
Figueiredo and Nowak [64], in 2003, introduced an expectation maximization
(EM) algorithm for image restoration based on a penalized likelihood formulated in
the wavelet domain. The proposed EM algorithm combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization
of the convolution operator obtained in the Fourier domain. The algorithm substitutes between an E-step based on the fast Fourier transform (FFT) and a DWT-based
M-step, resulting in an efficient iterative process requiring O(N logN ) operations per
iteration. This approach performed competitively with, in some cases better than, the
best existing methods in benchmark tests.
Argenti and Torricelli [10], in 2003, assumed Wiener-like filtering and performed
in a shift-invariant wavelet domain by means of an adaptive rescaling of the coefficients of undecimated octave decomposition calculated from the parameters of
the noise model, and the wavelet filters. The proposed method found in excellent
background smoothing as well as preservation of edge sharpness and well details.
LLMMSE evaluation in an undecimated wavelet domain tested on both ultrasonic
images and synthetically speckled images demonstrated an efficient rejection of the
distortion due to speckle.
Portilla et al. [137], in 2003, proposed a method for removing noise from digital
images, based on a statistical model of the coefficients of an over-complete multiscale
oriented basis. Neighborhoods of coefficients at adjacent positions and scales were
modeled as the product of two independent random variables: a Gaussian vector and
a hidden positive scalar multiplier. The latter modulated the local variance of the
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coefficients in the neighborhood, and was thus able to account for the empirically
observed correlation between the coefficients amplitudes. Under this model, the
Bayesian least squares estimate of each coefficient reduced to a weighted average of
the local linear estimates over all possible values of the hidden multiplier variable.
They demonstrated through simulations with images contaminated by additive white
Gaussian noise that the performance of this method substantially surpassed that of
previously published methods, both visually and in terms of mean squared error.
Synthetic aperture radar (SAR) images are inherently affected by multiplicative
speckle noise, which is due to the coherent nature of the scattering phenomenon.
Achim et al. [3], in 2003, proposed a novel Bayesian-based algorithm within the
framework of wavelet analysis, which reduces speckle in SAR images while preserving the structural features and textural information of the scene. Firstly they showed
that the subband decompositions of logarithmically transformed SAR images were
accurately modeled by alpha-stable distributions, a family of heavy-tailed densities.
Consequently, they exploited this a priori information by designing a maximum a
posteriori (MAP) estimator. They used the alpha-stable model to develop a blind
speckle-suppression processor that performed a nonlinear operation on the data and
they related this nonlinearity to the degree of non-Gaussianity of the data. Finally,
they compared their proposed method to current state-of-the-art soft thresholding
techniques applied on real SAR imagery and they quantified the achieved performance improvement.
Xie et al. [189], in 2004, using the MDL principle, provided the denoising method
based on a doubly stochastic process model of wavelet coefficients that gave a new
spatially varying threshold. This method outperformed the traditional thresholding
method in both MSE error and compression gain.
Wink and Roerdink [185], in 2004, estimated two denoising methods for the
simulation of an fMRI series with a time signal in an active spot by the average
temporal SNR inside the original activated spot and by the shape of the spot detected
by thresholding the temporal SNR maps. These methods were found to be better
suited for low SNRs but they were not preferred for reasonable quality images as
they introduced heavy decompositions. Therefore, wavelet based denoising methods
were used since they preserved sharpness of the images, from the original shapes of
active regions as well and produced a smaller total number of errors than Gaussian
noise. But both Gaussian and wavelet based smoothing methods introduced severe
deformations and blurred the edges of the active mark. For low SNR both techniques
are found to be on similarity. For high SNR Wavelet methods are better than Gaussian
method, giving a maximum output of above 10 db.
Choi and Baranuik [35], in 2004, defined Besov Balls (a convex set of images
whose Besov norms are bounded from above by their radii) in multiple wavelet
domains and projected them onto their intersection using the projection onto convex
sets (POCS) algorithm. It resembled to a type of wavelet shrinkage for image denoising. This algorithm provided significant improvement over conventional wavelet
shrinkage algorithm, based on a single wavelet domain such as hard thresholding in
a single wavelet domain.
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Yoon and Vaidyanathan [199], in 2004, offered the custom thresholding scheme
and demonstrated that it outperformed the traditional soft and hard-thresholding
schemes.
Hsung et al. [75], in 2005, improved the traditional wavelet method by applying Multivariate Shrinkage on multiwavelet transform coefficients. Firstly a simple
2nd order orthogonal pre filter design method was used for applying multiwavelet
of higher multiplicities (preserving orthogonal pre-filter for any multiplicity). Then
threshold selections were studied using Steins unbiased risk estimator (SURE) for
each resolution point, provided the noise constitution is known. Numerical experiments showed that a multivariate shrinkage of higher multiplicity usually gave better
performance and the proposed LSURE substantially outperformed the traditional
SURE in multivariate shrinkage denoising, mainly at high multiplicity.
Zhang et al. [201], in 2005, proposed a wavelet-based multiscale linear minimum
mean square-error estimation (LMMSE) scheme for image denoising and the determination of the optimal wavelet basis with respect to the proposed scheme was also
discussed. The overcomplete wavelet expansion (OWE), which is more effective than
the orthogonal wavelet transform (OWT) in noise reduction, was used. To explore the
strong interscale dependencies of OWE, they combined the pixels at the same spatial
location across scales as a vector and apply LMMSE to the vector. Compared with
the LMMSE within each scale, the interscale model exploited the dependency information distributed at adjacent scales. The performance of the proposed scheme was
dependent on the selection of the wavelet bases. Two criteria, the signal information
extraction criterion and the distribution error criterion, were proposed to measure the
denoising performance. The optimal wavelet that achieves the best tradeoff between
the two criteria could be determined from a library of wavelet bases. To estimate
the wavelet coefficient statistics precisely and adaptively, they classified the wavelet
coefficients into different clusters by context modeling, which exploited the wavelet
intrascale dependency and yields a local discrimination of images. Experiments show
that the proposed scheme outperforms some existing denoising methods.
Selesnick et al. [151], in 2005, developed a dual-tree complex wavelet transform
with important additional properties such as shift invariant and directional selectivity at higher dimensions. The dual-tree complex wavelet transform is non-separable
but is based on computationally efficient separable filter bank. Kingsbury proved
how complex wavelets with good properties illustrate the range of applications such
as image denoising, image rotation, estimating image geometrical structures, estimating local displacement, image segmentation, image sharpening and many more
applications.
Cho and Bui [34], in 2005, proposed the multivariate generalized Gaussian distribution model, which adjusts different parameters and can include Gaussian, generalized Gaussian, and non-Gaussian model, but parameters estimation was more
complex in image denoising process.
Deconvolution of images is an ill-posed problem, which is very often tackled
by using the diagonalization property of the circulant matrix in the discrete Fourier
transform (DFT) domain. On the other hand, the discrete wavelet transform (DWT)
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has shown significant success for image denoising because of its space-frequency
localization.
Koc and Ercelebi [89], in 2006, proposed a method of applying a lifting based
wavelet domain e-median filter (LBWDEMF) for image restoration. The proposed
method transforms an image into the wavelet domain using lifting-based wavelet
filters, then applies an e-median filter in the wavelet domain, transforms the result
into the spatial domain, and finally goes through one spatial domain e-median filter
to produce the final restored image. They compared the result obtained using the
proposed method to those using a spatial domain median filter (SDMF), spatial
domain e-median filter (SDEMF), and wavelet thresholding method. Experimental
results showed that the proposed method was superior to SDMF, SDEMF, and wavelet
thresholding in terms of image restoration.
Sudha et al. [167], in 2007, described a new method for suppression of noise
in image by fusing the wavelet Denoising technique with optimized thresholding
function, improving the denoised results significantly. Simulated noise images were
used to evaluate the denoising performance of proposed algorithm along with another
wavelet-based denoising algorithm. Experimental result showed that the proposed
denoising method outperformed standard wavelet denoising techniques in terms of
the PSNR and the preservation of edge information. Thy had compared this with
various denoising methods like wiener filter, Visu Shrink, Oracle Shrink and Bayes
Shrink.
Luisier et al. [108], in 2007, introduced a new approach to orthonormal wavelet
image denoising. Instead of postulating a statistical model for the wavelet coefficients, we directly parametrize the denoising process as a sum of elementary nonlinear processes with unknown weights. We then minimize an estimate of the mean
square error between the clean image and the denoised one. The key point was
that we had at our disposal a very accurate, statistically unbiased, MSE estimate,
Steins unbiased risk estimate, that depends on the noisy image alone, not on the
clean one. Like the MSE, this estimate was quadratic in the unknown weights, and
its minimization amounts to solving a linear system of equations. The existence
of this a priori estimate made it unnecessary to devise a specific statistical model
for the wavelet coefficients. Instead, and contrary to the custom in the literature,
these coefficients were not considered random anymore. We described an interscale
orthonormal wavelet thresholding algorithm based on this new approach and showed
its near-optimal performance, both regarding quality and CPU requirement, by comparing it with the results of three state-of-the-art nonredundant denoising algorithms
on a large set of test images. An interesting fallout of this study was the development
of a new, group-delay-based, parentchild prediction in a wavelet dyadic tree.
Giaouris and Finch [68], in 2008, presented that the denoising scheme based on
the Wavelet Transform did not distort the signal and the noise component after the
process was found to be small.
Rahman et al. [138], in 2008, proposed a hybrid-type image restoration algorithm
that takes the advantage of the diagonalization property of the DFT of a circulant
matrix during the deconvolution process and space-frequency localization property
of the DWT during the denoising process. The restoration algorithm was operated
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iteratively and switched between two different domains, viz., the DWT and DFT.
For the DFT-based deconvolution, they used the least-squares method, wherein regularization parameter was estimated adaptively. For the DWT-based denoising, a
MAP estimator that modeled the local neighboring wavelet coefficients by the GramCharlier PDF, had been used. The proposed algorithm showed a well convergence
and experimental results on standard images showed that the proposed method provides better restoration performance than that of several existing methods in terms
of signal-to-noise ratio and visual quality.
Poornachandra [136], in 2008, used the wavelet-based denoising for the recovery
of signal contaminated by white additive Gaussian noise and investigated the noise
free reconstruction property of universal threshold.
Gnanadurai and Sadasivam [70], in 2008, proposed method in which the choice of
the threshold estimation was carried out by analysing the statistical parameters of the
wavelet subband coefficients like standard deviation, arithmetic mean and geometrical mean. This frame work described a computationally more efficient and adaptive
threshold estimation method for image denoising in the wavelet domain based on
Generalized Gaussian Distribution (GGD) modeling of subband coefficients. The
noisy image was first decomposed into many levels to obtain different frequency
bands. Then soft thresholding method was used to remove the noisy coefficients, by
fixing the optimum thresholding value by the proposed method. Experimental results
on several test images by using this method showed that this method yields significantly superior image quality and better Peak signal-to-noise ratio (PSNR). To prove
the efficiency of this method in image denoising, they compared this with various
denoising methods like wiener filter, Average filter, VisuShrink and BayesShrink.
Anbarjafari and Demirel [6] proposed a new super-resolution technique based
on interpolation of the high-frequency subband images obtained by discrete wavelet
transform (DWT) and the input image. The proposed technique used DWT to decompose an image into different subband images and then the high-frequency subband
images and the input low-resolution image had been interpolated, followed by combining all these images to generate a new super-resolved image by using inverse
DWT. The proposed technique had been tested on Lena, Elaine, Pepper, and Baboon
and the quantitative peak signal-to-noise ratio (PSNR) and visual results showed the
superiority of the proposed technique over the conventional and state-of-art image
resolution enhancement techniques. For Lenas image, the PSNR was 7.93 dB higher
than the bicubic interpolation.
Firoiu et al. [65] presented a Bayesian approach of wavelet based image denoising. They proposed the denoising strategy in two steps. In the first step, the image is
denoised using association of bishrink filter hyper analyticwavelet transform (HWT)
computed with intermediate wavelets, for example Daubechies-12. In the second
step, the same denoising approach is followed with only the difference in the application of the bishrink filter HWT using the mother wavelets Daubechies-10. Orthonormal bases of compactly supported wavelets, with arbitrarily high regularity are constructed. The order of high regularity increases linearly with the support width given
by Daubechies (1998). PSNR improvement of 6.67 dB is observed for Lena image.
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Motwani et al. [119] described different methodologies for noise reduction (or
denoising) giving an insight as to which algorithm should be used to find the most
reliable estimate of the original image data given its degraded version.
Mathew and Shibu [114] developed a technique for Reconstruction of super resolution image using low resolution natural color image. The presented technique
identified local features of low resolution image and then enhanced its resolution
appropriately. It is noticed that the higher PSNR was observed for the developed
technique than the existing methods.
Vohra and Tayal [177] analyzed the image de-noising using discrete wavelet transform. The experiments were conducted to study the suitability of different wavelet
bases and also different window sizes. Among all discrete wavelet bases, coiflet performs well in image de-noising. Experimental results also showed that Sureshrink
provided better result than Visushrink and Bayesshrink as compared to Weiner filter.
Later [2010, 2011], many effective image denoising methods based on this model
combining different transforms are obtained.
Wang et al. [180] in 2011 proposed a simple model of pixel pattern classifier with
orientation estimation module to strengthen the robustness of denoising algorithm.
Moreover, instead of determining the transform strategy, sub-blocks, robust adaptive directional lifting (RADL) algorithm is performed at each pixel level to pursue
better denoising results. RADL is performed only on pixels belonging to texture
regions thereby reducing artifacts and improving performance of the algorithm. The
Peak Signal to Noise Ratio (PSNR) improvement on Barbara image is 6.66 dB and
SSIM index improvement is 0.355. Six different images are used to evaluate the
performance of this algorithm.
Mohideen et al. [118] in 2011 compared the wavelet and multi wavelet technique
to produce the best denoised mammographic images using efficient multi wavelet
algorithm. Mammographic images are denoised and enhanced using multi wavelet
with hard thresholding. Initially the images are pre-processed to improve its local
contrast and discrimination of faint details. Image suppression and edge enhancement
are performed. Edge enhancement is performed based on multi wavelet transform. At
each resolution, coefficients associated with the noise are modeled and generalized
by Laplacian random variables. The better denoising results depend on the degree
of noise; generally its energy is distributed over low frequency band while both its
noise and details are distributed over high frequency band. Also the applied hard
threshold in different scale of frequency sub bands limits the performance of image
denoising algorithms.
Ruikar and Doye [148] in 2011 proposed different approaches of wavelet based
image denoising methods. The main aim of authors was to modify the wavelet coefficients in the new basis, the noise could be removed from the data. They extended the
existing technique and providing a comprehensive evaluation of the proposed method
by using different noise, such as Gaussian, Poissons, Salt and Pepper, and Speckle.
A signal to noise ratio as a measure of the quality of denoising was preferred.
Liu et al. [104] in 2012 proposed a denoising method based on wavelet threshold and subband enhancement method for image de-noising. This method used soft
threshold method for the minimum scale wavelet coefficients, takes further decom-
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165
posing for other wavelet coefficient and takes effective enhancement and mixing
threshold processing for each subband after being decomposed. Thus making full
use of high frequency information of each of the multi-dimension could add image
details and got a better enhancement and de-noising effectively.
Kumar and Saini [94] in 2012 suggested some new thresholding method for image
denoising in the wavelet domain by keeping into consideration the shortcomings of
conventional methods and explored the optimal wavelet for image denoising. They
proposed a computationally more efficient thresholding scheme by incorporating the
neighbouring wavelet coefficients, with different threshold value for different sub
bands and it was based on generalized Gaussian Distribution (GGD) modeling of sub
band coefficients. In here proposed method, the choice of the threshold estimation was
carried out by analyzing the statistical parameters of the wavelet sub band coefficients
like standard deviation, arithmetic mean and geometrical mean. It was demonstrated
that their proposed method performs better than: VisuShrink, Normalshrink and
NeighShrink algorithms in terms of PSNR ratio. Further a comparative analysis had
been made between Daubechies, Haar, Symlet and Coiflet wavelets to explore the
optimum wavelet for image denoising with respect to Lena image. It had been found
that with Coiflet wavelet higher PSNR ratio was achieved than others.
Naik and Patel [122] in 2013 presented single image super resolution algorithm
based on both spatial and wavelet domain. Their algorithm was iterative and used
back projection to minimize reconstruction error. They also introduced wavelet based
denoising method to remove noise. PSNR ratio and visual quality of images were
also showed the effectiveness of algorithm.
Wavelets gave a superior performance in image denoising due to its properties
such as multi-resolution. Non-linear methods especially those based on wavelets have
become popular due to its advantages over linear methods. Abdullah Al Jumah [1] in
2013, applied non-linear thresholding techniques in wavelet domain such as hard and
soft thresholding, wavelet shrinkages such as Visu-shrink (nonadaptive) and SURE,
Bayes and Normal Shrink (adaptive), used Discrete Stationary Wavelet Transform
(DSWT) for different wavelets at different levels, denoised an image and determined
the best one out of them. Performance of denoising algorithm was measured using
quantitative performance measures such as Signal-to-Noise Ratio (SNR) and Mean
Square Error (MSE) for various thresholding techniques.
There are two disadvantages in variational regularization based image restoration
model. Firstly, the restored image is susceptible to noise because the diffusion coefficient depends on image gradient. Secondly, in the process of energy minimization,
the selection of Lagrange multiplier λ which is used to balance the regular term and
the fidelity term can directly affects the quality of the restored image.
Li et al. [102] in 2014 introduced multiresolution feature of multiscale wavelet
into the energy minimization model and proposed a wavelet based image restoration model to solve the above problems. They replaced Lagrange multiplier λ by an
adaptive weighting function λj in their proposed model, which is constructed by the
image wavelet transform coefficients. Experimental results and theoretical analysis
showed that the proposed model reduced iterations in the energy minimization process overcome the cartoon effects in the variational model and pseudo-Gibbs effect
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in traditional wavelet threshold methods and can well protect the detail features while
denoising.
Neha and Khera [124] in 2014 proposed new technique which is combination
of Enhanced Empirical Mode Decomposition (EEMD), has been presented along
with the standard wavelet thresholding techniques like hard thresholding to denoise
the image. Authors presented a comparative analysis of various image denoising
techniques using wavelet transforms and a lot of combinations had been applied in
order to find the best method that can be followed for denoising intensity images.
Varinderjit et al. [174] in 2014 defined a general mathematical and experimental
methodology to compare and classify classical image de-noising algorithms and
proposed a nonlocal means (NL-means) algorithm addressing the preservation of
structure in a digital image. The mathematical analysis was based on the analysis of
the method noise, defined as the difference between a digital image and its de-noised
version.
Chouksey et al. [37] in 2015 provided denoising scheme with a wavelet interscale
model based on Minimum mean square error (MMSE) and discussed the optimize
wavelet basis from family of wavelet. In their proposed method, the wavelet coefficients at the same spatial locations at two adjacent scales were represented as a
vector with orthogonal wavelet transform and the orthogonal wavelet based with
mmse was applied to the vector for enhancing the peak signal to noise ratio (PSNR).
The New algorithm filter also showed reliable and stable performance across a different range of noise densities varying from 10% to 90%. The performance of the
proposed method had been tested at low, medium and high noise densities on gray
scales and at high noise density levels the new proposed algorithm provided better
performance as compare with other existing denoising filters.
By considering the problem of generating a super-resolution (SR) image from a
single low resolution (LR) input image in the wavelet domain, Rakesh et al. [141]
in 2015 proposed an intermediate stage for estimating the high frequency (HF) sub
bands to achieve a sharper image. Experimental results indicated that the proposed
approach outperforms existing methods in terms of resolution.
Gadakh and Thorat [67] in 2015 presented a new and fast method for removal of
noise and blur from Magnetic Resonance Imaging (MRI) using wavelet transform.
They utilized a fact that wavelets can represent magnetic resonance images well,
with relatively few coefficients. They used this property to improve MRI restoration
with arbitrary k-space trajectories. They showed that their non-linear method was
performing fast than other regularization algorithms.
Wagadre and Singh [178] in 2016 described a method to remove the motion blur
present in the image taken from any cameras by which motion blurred. They restored
noisy image using Wiener and Lucy Richardson method then applied wavelet based
fusion Technique for restoration. The performance of the every stage was tabulated
for the parameters like SNR and RMSE of the restored images and it has been
observed that image fusion technique provided better results as compared to previous
techniques.
Sowmya et al. [160] in 2016 proposed a new image resolution enhancement
algorithm based on discrete wavelet transform (DWT), lifting wavelet transform
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(LWT) and sparse recovery of the input image. Firstly, a single low resolution (LR)
was decomposed into different subbands using two operators DWT and LWT. In
parallel, the LR image was subjected to a sparse representation interpolation. Finally,
The higher frequency sub-bands in addition to the sparse interpolated LR image
were combined to give a high resolution (HR) image using inverse discrete wavelet
transform (IDWT). The qualitative and quantitative analysis of our method showed
prominence over the conventional and various state-of-the art super resolution (SR)
techniques.
Rakheja and Vig [142] in 2016 combined neighborhood processing techniques
with wavelet Transform for image denoising and simulated results showed that the
combined algorithm performs better than both individually. They obtained simulated
results for Gaussian, Speckle and Salt & Pepper noise, for denoising median filter of
size 3X3, 5X5 and discrete wavelet Transform were used. Then results obtained were
evaluated on the basis of Peak signal to noise ratio which has improved remarkably.
Leena et al. [100] in 2016 presented a methodology to denoise an image based on
least square approach using wavelet filters. This work was the extension of the one
dimensional signal denoising approach based on least square (proposed by Selesnick)
to two dimensional image denoising. In their proposed work, the matrix constructed
using second order filter in the least square problem formulation was replaced with
the wavelet filters. The performance of the proposed algorithm was validated through
PSNR. From the results of PSNR values, it was evident that the proposed method performs equally well as the existing second order filter. The advantage of the proposed
method lies in the fact that it was simple and involves low mathematical complexity.
Thangadurai and Patrick [170] in 2017 proposed new set of blur invariant descriptors. These descriptors have been advanced in the wavelet domain 2D and 3D images
to be invariant to centrally symmetric blur. First, Image registration was done using
wavelet domain blur invariants. The method uses Daubuchies and B-spline wavelet
function which was used to construct blur invariants. The template image was chosen from the degraded image. The template images and the original images were
matched with its similarities. This wavelet domain blur invariants accurately register
an image compared to spatial domain blur invariants which might result in misfocus
registration of an image. Despite of the presence of harmful blurs, the image registration has been correctly performed. The experiments carried out by using SDBIs,
were failed in some of the image registration. Then regression based process is done
about to produce an image convolved with near diffraction limited PSF, which can
be shown as blur invariant. Eventually a blind deconvolution algorithm is carried
out to remove the diffraction limited blur from fused image the final output. Finally,
image was restored by using blind deconvolution algorithm and also PSNR values
are calculated. Hence the image quality was improved by using proposed method.
Sushil Kumar [97] in 2017 proposed a comparative study of image denoising
method using BlockShrink algorithm between the Wavelet transform (DWT) and the
Slantlet transform (SLT). Slantlet transform, which is also a wavelet-like transform
and a better candidate for signal compression compared to the DWT based scheme
and which can provide better time localization. BlockShrink was found to be a better
method than other conventional image denoising methods. It was found that DWT
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based BlockShrink thresholding was better option than BiShrink thresholding in
terms of PSNR for the Gaussian noise. The PSNR for the SLT based BlockShrink,
though found to be less than DWT based method, was better option as it provided
better time localization and better signal compression compared to the classical DWT.
Generally the Gaussian and salt Pepper noise occurred in images of different
quality due to random variation of pixel values. It is necessary to apply various
filtering techniques to denoise these images. There are lots of filtering methods proposed in literature which includes the haar, sym4, and db4 Wavelet Transform based
soft and hard thresholding approach to denoise such type of noisy images. Chaudhari and Mahajan [29] in 2017 analysed exiting literature on haar, db4 and sym4
Wavelet Transform for image denoising with variable size images from self generated grayscale database generated from various image sources such as satellite
images(NASA), Engineering Images and medical images. However this new proposed Denoising method showed satisfactory performances with respect to existing
literature on standard indices like Signal-to-Noise Ratio (SNR), Peak Signal to Noise
Ratio (PSNR) and Mean Square Error (MSE). Wavelet coefficient could be used to
improve quality of true image and from noise since wavelet transform represents
natural image better than any other transformations. The main aim of this work was
to eliminate the Gaussian and salt Pepper noise in wavelet transform domain. Subsequently a soft and hard threshold based denoising algorithm had been developed.
Finally, the denoised image was compared with original image using some quantifying statistical indices such as MSE, SNR and PSNR for different noise variance
which the experimental results demonstrate its effectiveness over previous method.
Wavelet transform is an effective method for removal of noise from image. But
traditional wavelet transform cannot improve the smooth effect and reserve images
precise details simultaneously, even false Gibbs phenomenon can be produced. Wang
et al. [179] in 2017 proposed a new image denoising method based on adaptive multiscale morphological edge detection beyond the above limitation. Firstly, the noisy
image was decomposed by using one wavelet base, then the image edge was detected
by using the adaptive multiscale morphological edge detection based on the wavelet
decomposition. On this basis, wavelet coefficients belonging to the edge position
were dealt with the improved wavelet domain wiener filtering and the others were
dealt with the improved Bayesian threshold and the improved threshold function.
Finally, wavelet coefficients were inversely processed to obtain the denoised image.
This method provided the better result from existing and this method can effectively
remove the image noise without blurring edges and highlight the characteristics of
image edge at the same time.
Ramadhan et al. [144] in 2017 proposed and tested a new method of image
de-noising based on using median filter (MF) in the wavelet domain. Various types of
wavelet transform filters were used in conjunction with median filter in experimenting with the proposed approach in order to obtain better results for image de-noising
process, and, consequently to select the best suited filter. Wavelet transform working
on the frequencies of sub-bands split from an image was a powerful method for
analysis of images. Experimental work showed that the proposed method provided
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better results than using only wavelet transform or median filter alone. The MSE and
PSNR values were used for measuring the improvement in de-noised images.
Neole et al. [125] in 2017 presented a novel approach to image denoising using
edge profile detection and edge preservation in spatial domain in presence of zero
mean additive Gaussian noise. A Noisy image was initially preprocessed using the
proposed local edge profile detection and subsequent edge preserving filtering in
spatial domain followed further by the modified threshold bivariate shrinkage algorithm. The proposed technique did not require any estimate of standard deviation of
noise present in the image. Performance of the proposed algorithm was presented
in terms of PSNR and SSIM on a variety of test images containing a wide range of
standard deviation starting from 15 to 100. The performance of the proposed algorithm was much better than NL means and Bivariate Shrinkage while its comparable
with BM3D.
5.9.2 Image Restoration Using Complex Wavelet Transform
The classical discrete wavelet transform (DWT) provides a means of implementing
a multiscale analysis, based on a critically sampled filter bank with perfect reconstruction. However, questions arise regarding the good qualities or properties of the
wavelets and the results obtained using these tools, the standard DWT suffers from
the following problems described as below:
1. Shift sensitivity: It has been observed that DWT is seriously disadvantaged by
the shift sensitivity that arises from down samples in the DWT implementation.
2. Poor directionality: An m-dimension transform (m > 1) suffers poor directionality when the transform coefficients reveal only a few feature in the spatial
domain.
3. Absence of phase information: Filtering the image with DWT increases its
size and adds phase distortions; human visual system is sensitive to phase distortion.
Such DWT implementations cannot provide the local phase information.
It is found that the above problems can be solved effectively by the complex
wavelet transform (CWT). The structure of the CWT is the same as in DWT, except
that the CWT filters have complex coefficients and generate complex output samples. However, a further problem arises here because perfect reconstruction becomes
difficult to achieve for complex wavelet decompositions beyond level 1, when the
input to each level becomes complex. For many applications it is important that the
transform must be perfectly invertible. A few authors, such as Lawton [98] and Belzer
et al. [15], have experimented with complex factorizations of the standard Daubechies
polynomials and obtained PR complex filters, but these do not give filters with good
frequency-selectivity properties. To provide shift invariance and directional selectivity, all of the complex filters should emphasize positive frequencies and reject
negative frequencies, or vice versa. Unfortunately, it is very difficult to design an
inverse transform, based on complex filters which has good frequency selectivity
and PR at all levels of the transform.
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To overcome this problem, in 1998, Kingsbury [86, 87] developed the dual-tree
complex wavelet transform (DTCWT), which added perfect reconstruction to the
other attractive properties of complex wavelets: shift invariance; good directional
selectivity; limited redundancy; and efficient order-N computation.
The dual-tree transform was developed by noting that approximate shift invariance
can be achieved with a real DWT by doubling the sampling rate at each level of the
tree. Now we are going to survey of image denoising techniques using dual tree
complex wavelet transforms.
Achim and Kuruoglu [2] in 2005, described a noval for removing noise from
digital images in the dual-tree complex wavelet transform framework. They designed
a bivariate maximum a posteriori estimator, which relies on the family of isotropic αstable distributions. Using this relatively new statistical model they were able to better
capture the heavy-tailed nature of the data as well as the interscale dependencies of
wavelet coefficients.
Chitchian et al. [33] in 2009, applied a locally adaptive denoising algorithm to
reduce speckle noise in time-domain optical coherence tomography (OCT) images
of the prostate. The algorithm was illustrated using DWT and DTCWT. Applying the
DTCWT provided improved results for speckle noise reduction in OCT images. The
cavernous nerve and prostate gland could be separated from discontinuities due to
noise, and image quality metrics improvements with a signal-to-noise ratio increase
of 14dB14dB were attained.
Xingming and Jing [189] in 2009 proposed a novel method based on HMT model
by the use of Fourier Wavelet Regulation Deconvolution (ForWaRD) algorithm
and compared with some conventional image restoration algorithms using complex wavelets. In the proposed method, they first applied the Wiener filter on the
blurring image in the Fourier domain, and then used the hidden Markov tree model
(HMT) to remove the unwanted noise in wavelet domain. Simulations for solving
the typical convolution and noised linear degraded model were made, in which the
performances based complex wavelets and real orthogonal wavelets were compared
in detail. Experimental results showed that the suggested method using complex
wavelets performed better in the view of visual effects and objective criterion than
the conventional methods.
Wang et al. [182] in 2010, proposed a technique based on the dual-tree complex
wavelet transform (DTCWT) to enhance the desired features related to some special
type of machine fault. Since noise inevitably exists in the measured signals, they
developed an enhanced vibration signals denoising algorithm incorporating DTCWT
with NeighCoeff shrinkage. Denoising results of vibration signals resulting from a
crack gear indicate the proposed denoising method can effectively remove noise and
retain the valuable information as much as possible compared to those DWT- and
SGWT-based NeighCoeff shrinkage denoising methods.
Sathesh and Manoharan [150] in 2010, proposed a image denoising technique
using Dual Tree Complex Wavelet Transform (DTCWT) along with soft thresholding.
In Medical diagnosis operations such as feature extraction and object recognition
will play the key role. These tasks will become difficult if the images are corrupted
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with noise. Raja and Venkateswarlub [139] in 2012 proposed the denoising method
which used dual tree complex wavelet transform to decompose the image and shrinkage operation to eliminate the noise from the noisy image. In the shrinkage step
They used semi-soft and stein thresholding operators along with traditional hard and
soft thresholding operators and verified the suitability of dual tree complex wavelet
transform for the denoising of medical images. Their results proved that the denoised
image using Dual Tree Complex Wavelet Transform (DTCWT) had a better balance
between smoothness and accuracy than the DWT and less redundant than Undecimated Wavelet Transform (UDWT). They used the SSIM along with PSNR to assess
the quality of denoised images.
Kongo et al. [90] in 2012 presented a new denoising method for ultrasound medical image in restoration domain. The approach was based on analysis in Dual-tree
wavelet Transform (DT-CWT). Various methods had been developed in the literature; most of them used only the standard wavelet transform (DWT). However,
the Discrete Wavelet Transform (DWT) had some disadvantages that undermine its
application in image processing. They investigated a performances complex wavelet
transform (DT-CWT) combined with Bivariate Shrinkage and Visu-shrinkage. The
proposed method was tested on a noisy ultrasound medical image, and the restored
images show a great effectiveness of DT-CWT method compared to the classical
DWT.
Vijay and Mathurakani [176] in 2014, proposed a image denoising technique
using Dual Tree Complex Wavelet Transform (DTCWT) along with Byes thresholding. Convolution based 2D processing was employed for simulation resulted in
improvement in PSNR.
Mitiche et al. [116] in 2013 proposed a denoising approach basing on dual tree
complex wavelet and shrinkage (where either hard and soft thresholding operators of
dual tree complex wavelet transform for the denoising of medical images are used).
The results proved that the denoised images using DTCWT (Dual Tree Complex
Wavelet Transform) have a better balance between smoothness and accuracy than
the DWT and are less redundant than SWT (Stationary Wavelet Transform). They
used the SSIM (Structural Similarity Index Measure) along with PSNR (Peak Signal
to Noise Ratio) and SSIM Map to assess the quality of denoised images.
Naimi et al. [123] in 2015, proposed a denoising approach basing on dual tree
complex wavelet and shrinkage with the Wiener filter technique (where either hard
or soft thresholding operators of dual tree complex wavelet transform for the denoising of medical images are used). The results proved that the denoised images using
DTCWT (Dual Tree Complex Wavelet Transform) with Wiener filter have a better
balance between smoothness and accuracy than the DWT and were less redundant
than SWT (Stationary Wavelet Transform). They used the SSIM (Structural Similarity Index Measure) along with PSNR (Peak Signal to Noise Ratio) and SSIM map
to assess the quality of denoised images.
Rao and Ramakrishna [145] in 2015 proposed a new algorithm for image denoisinig based on DTCWT. In this algorithm, the decomposed coefficients combined
with the bivariate shrinkage model for the estimation of coefficients in high frequency sub bands and Bayesian shrinkage method was applied in order to remove
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the noise in highest frequency sub-band coefficients. The experimental results were
compared with the existing shrinkage methods Visu and Bayes shrinkage methods in
terms of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM).
The experimental outcome showed that the proposed technique could eliminate the
noise efficiently and protected the edge information well, this algorithm provided
good denoising effect as well as peak signal to noise ratio and structural similarity
index was better than traditional denoising methods.
Yaseena et al. [195] in 2016 compared image denoising techniques based on
real and complex wavelet-transforms. Possibilities provided by the classical discrete
wavelet transform (DWT) with hard and soft thresholding were considered, and influences of the wavelet basis and image resizing were discussed. The quality of image
denoising for the standard 2-D DWT and the dual-tree complex wavelet transform
(DT-CWT) was studied. It was shown that DT-CWT outperforms 2-D DWT at the
appropriate selection of the threshold level.
Kumar and Reddy [96] in 2017 assumed the noisy image was to be complex
image and its real part and imaginary parts were separated. These were subjected to
Bi-shrink filter separately into different stages of decomposition depending upon the
severity of noise. The obtained de-noise image was compared with original image
using different parametric measures like Peak Signal to Noise Ratio, Structural similarity Index measure, Covariance and Root mean square Error whose values were
tabulated. The values of retrieved image obtained yields much better visual effect
and hence this method was said to be a better one when compared with de-noising
methods using Weiner Filter and various Local Adaptive Filters.
5.9.3 Image Restoration Using Quaternion Wavelet
Transform
As a mathematical tool, wavelet transform is a major breakthrough of the Fourier
transform and Fourier transform window known to many people since it has good
time-frequency features and multiple resolution. Wavelet analysis theory has become
one of the most useful tools in signal analysis, image processing, pattern recognition
and other fields. In image processing, the basic idea of the wavelet transform is
to decompose image multiresolution that is the original image is decomposed into
different space and different frequency sub-image, and then coefficients of sub-image
are processed. Mostly used wavelet transforms are real discrete wavelet transform
and complex wavelet transform and so on.
The discrete wavelet transform (DWT) and dual-tree complex wavelet transform
(DTCWT) however suffer from two major drawbacks. The first drawback is the real
discrete wavelet transform signal small shift will produce the energy of wavelet coefficient distribution change, making it difficult to extract or model signal information
from the coefficient values. Dual-tree complex wavelet although overcame the first
problem but it can generate signal phase ambiguity when represented two dimen-
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sional images features. While the quaternion wavelet transform is a new multiscale
analysis image processing tool, it is based on the Hilbert two-dimensional transform
theory, which has approximate shift invariance and can well overcome the above
drawbacks as a result the signal to noise ratio was improved to a greater extent as
compared with DWT.
Quaternion wavelet research is divided into two branches: first is based on quaternion numerical function multiresolution analysis theory of quaternion wavelet, using
a single tree structure. Mitrea [117] in 1994 gave quaternion wavelet form concept.
Traversoni [172] in 2001 used real wavelet transform and complex wavelet transform
by quaternion Haar kernel and proposed discrete quaternion wavelet transform and
gave some applications in image processing. He and Yu [74] in 2004, used matrix
value function multiresolution analysis structure for consecutive quaternion wavelet
transform and provided some properties. Bahri [12] constructed discrete quaternion
wavelet transform (DQWT) through complex matrix function and proved some basic
properties. Bahri et al. [11] in 2011 introduced through quaternion wavelet admissibility conditions, systematically extended the consecutive wavelet transform concept
to consecutive quaternion wavelet concept and provided the reconstruction theorem
and continuous quaternion wavelet basic properties. But these are mainly the concepts and properties of promotion, because its filters structure and implementation
are difficulties, it has not made any progress in application at present.
Another branch is based on Bulow quaternion analytic signal, by using real filter and dual-tree structure Bulow [19] in 1999 constructed the quaternion wavelet
transform. The filter has the advantages of simple structure, relatively easy, and there
was quaternion signal application background. Corrochano [41] in 2006 constructed
quaternion wavelet transform (QWT) through quaternion Gabor filter and discussed
the QWT properties and wavelet pyramid algorithm. He pointing out that the DWT
is without phase, CWT only has one phase, while QWT can provide three phases
and putting forward the image multiresolution disparity estimation method based
on the theory of QWT. Based on the dual-tree complex wavelet, Chan et al. [24]
in 2008 used the concepts and properties of Bulow quaternion analytic signal and
quaternion Fourier transform, constructed dual-tree quaternion wavelet transform,
and worked out the meaning of three phases, two of which represent the image of
local displacement information, another as image texture feature, which can be used
to estimate the image of the local geometric features. Xu et al. [191] in 2010 used
quaternion wavelet transforms amplitude and phase method and applied it to the face
recognition also obtained certain result. Soulard and Carre [159] in 2011 applied the
quaternion wavelet transform to image texture analysis and proved the feasibility
of this method. Now we are going to provide some applications quaternion wavelet
transform in the field of image denoising.
Yin et al. [197] in 2012, mainly studied some of the concepts and properties of
quaternion wavelet Transform and applied the quaternion wavelet in image denoising. They provided forward Bayesian denoising method based on quaternion wavelet
transform, considering wavelet coefficients correlation, and generalized Gaussian
distribution was used to model the probability distribution function of wavelet coefficients magnitude and the best range of the Bayesian thresholding parameter was
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found out. The experimental results showed that their method both in visual effect
and PSNR were better than many current denoising methods.
Tianfeng [171] in 2014, introduced the denoising performance evaluation standard and image wavelet threshold denoising method and then provided the quaternion
based on wavelet transform domain hidden markov tree model for image denoising
(Q-HMT), non-gaussian distribution model and hybrid statistical model of image
denoising algorithm. Good results had been achieved on the subjective and objective. On the basis of the introduction of additive model through quaternion wavelet
transform, author using improved coefficient classification criterion, the coefficients
were divided into two categories: important coefficient and the important factor, proposed to improve the Donoho threshold and the new threshold function, and it deals
with the important coefficient, estimate the excluding of quaternion wavelet transform coefficient, and the coherence of SAR image was obtained. Coherent speckle
noise suppression experiments of real SAR image, on the objective indicators and
visual effect, the proposed method was superior to the current many methods.
Fang-Fei et al. [62] in 2014, proposed an algorithm by combining the quaternion
wavelet transform model with the traditional HMT model. The new algorithm had
the advantage of good translation without deformation and the advantage of the
rich phase. The experimental results showed that this algorithm was superior to the
traditional algorithm of denoising in peak signal-to-noise ratio and image effect was
more superior to the traditional denoising algorithm.
Kadiri et al. [82] in 2014, studied the potential of the quaternion wavelet transform
for the analysis and processing of multispectral images with strong structural information. They showed an application of this transform in satellite image denoising
and proposed approach relies on the adaptation of thresholding procedures based
on the dependency between magnitude quaternionic coefficients in local neighborhoods and phase regularization. In addition, they introduced a non-marginal aspect
of multispectral representation. The results obtained indicate the potential of this
multispectral representation with magnitude thresholding and phase smoothing in
noise reduction and edge preservation compared with classical wavelet thresholding
methods that do not use phase or multiband information.
Malleswari and Madhu [112] in 2016, proposed an algorithm of image denoising
based on Quaternion Wavelet Transform model. The experimental results showed
that this algorithm was superior to the traditional algorithm of denoising in peak
signal to noise ratio and the visual appearance of the image was also better when
compared to the traditional denoising algorithm.
5.9.4 Image Restoration Using Ridgelet Transform
Wavelet is a useful technique to extract piecewise smooth information from a one
dimensional signal but does not give satisfactory performance in case on a two dimensional signal. Wavelet transform detect point singularities but it fails to capture line
singularities. An image contains singularities along line or curve which cannot be
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represented efficiently by wavelet based techniques. Therefore, new image processing schemes which are based on true two dimensional transform are expected to
improve the performance over the current wavelet based methods.
The higher dimension issues which cannot he handled by wavelets are put forth
by Candes and Donoho [20] in 1999. They discussed line like phenomena in dimension 2 and plane like phenomena in dimension 3. They gave the idea of analysis of
ridge functions ψ(u1 x1 + · · · + un xn ) whose ridge profiles ψ are used for performing
wavelet analysis in Radon domain. They discussed ridgelet frames, ridgelet orthonormal basis, ridges and described a new notion of smoothness naturally attached in the
ridgelet algorithm.
Do and Vetterli [48] in 2000 presented a finite implementation of the ridgelet transform. The transform is invertible, non-redundant and achieved via fast algorithms.
Furthermore they showed that this transform was orthogonal, hence it allowed one to
use linear approximations for representation of images. Finite ridgelet transform was
constructed using finite radon transform. The finite radon transform was redundant
and non-orthogonal. Radon transform was defined as a line integral of the image
intensity f (x, y), over the line that is at a distance s from the origin and perpendicular
to a line passing through origin, at an angle θ to the x-axis.
Do and Vetterli [47] in 2000, proposed a new finite orthonormal image transform
based on the Ridgelet. The Finite ridgelet image transform (FRIT) was shown to
represent effectively images with linear discontinuities. Authors observed that FRIT
has potential in restoring images that are smoth away from edges. Finite ridgelet
image transform was expected to work well also for images with smooth edges by
applying on suitable size block.
Cane and Andres [23] in 2004, proposed a new implementation of the ridgelet
transform based on discrete analytical 2D line, called discrete analytical ridgelet
transform (DART). Discrete radon transform was computed using Fourier strategy.
The innovative step given by Cane and Andre was the radial discrete analytical lines
in Fourier domain. These discrete analytical lines in Fourier domain were having a
parameter called arithmetical thickness, which could be used for specific application
such as denoising. Authors applied DART to each tile for denoising of digital images.
But the interesting approach laid by authors was its extendibility to higher dimension.
The thresholded image was regrouped and subjected to Wiener filtering to obtain
the final denoised image. Zhou et al. [204] in 2004 were used PSNR and SSIM indices
for denoising performance evaluation. For Lena and Barbara images corrupted by
noise of standard deviation equal to 10, the corresponding denoised images had
PSNR values of 35.11 dB and 33.99 dB. The corresponding SSIM values were 0.963
and 0.971.
A new denoising method, by integrating the dual-tree complex wavelets into
the ordinary ridgelet has been proposed by Chen et al. [30] in 2007. The dualtree complex wavelet has shift invariance property and ridgelet transform have high
directionality. Complex Ridgelet was obtained by applying 1-D dual-tree complex
wavelet transform on radon transform coefficients. Hard thresholding was used for
denoising application. PSNR improvement on Lena image was 7.13 dB using this
technique.
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Kang and Zhang [83] in 2012, proposed a method to remove the noise in QuickBird
images based on the ridgelet transform. Experimental results showed that ridgelet
transform performs effectively in removing the noise in QuickBird images compared
with other methods.
Shuyuan et al. [156] in 2013 proposed a ridgelet support vector based image
denoising algorithm. A multiscale ridgelet support vector filter and Geometric Multiscale Ridgelet Support Vector Transform (GMRSVT) are derived which have shift
invariant property. This is a dictionary based approach so the performance of algorithm is dependent on the extent of learning. Performance of this algorithm is validated on Lena image. Lena image corrupted by Gaussian noise of standard deviation
20 is denoised and the PSNR is obtained as 33.34 dB.
Krishnanaik et al. [93] in 2013 proposed a new image denoising technique by
integrating the dual-tree complex wavelets into the usual ridgelet transform. In this
procedure normal hard thresholding of the complex ridgelet coefficient was used
and results obtained showed better than VisuShrink, the ordinary ridgelet image
denoising, and wiener2 filter, and also Complex ridgelets applied to curvelet image
denoising.
Kaur and Mann [85] in 2013, proposed the ridgelet transform with M-band wavelet
transform, called M-band ridgelet transform, for medical image segmentation. The
performance of the proposed method was tested on ultrasound images under Gaussian
noise. The results of the proposed method were compared with the ridgelet and
curvelet transform in terms of PSNR, MSE. The results after being investigated
showed significant improvements compared to the ridgelet and curvelet denoising
algorithms.
Liu et al. [106] in 2014 proposed finite ridgelet transform based algorithm for
digital image denoising. Patch wise denoising was performed to obtain better PSNR
and superior visual quality. The noisy image was first grouped into patches followed
by finite ridgelet transform and hard thresholding.
Vetrivelan and Kandaswamy [175] in 2014 proposed ridgelet transform hard
thresholding algorithm for image denoising to preserve the details of the image.
Ridgelet transform was used as it is concentrated near the edges of the image and
it represented one-dimensional singularity in two-dimensional spaces. Wavelet was
good in representing point singularities. When wavelet was linked with ridgelet,
denoised image quality would be improved.
Due to the fusion of the properties of two transforms i.e. Wavelet transform and
Radon transform, the Ridgelet Transform possess improved denoising and edge preserving capabilities. Kumar and Bhurchandi [94] in 2015, proposed algorithm for
image denoising using Ridgelet transform and cycle spinning. The proposed algorithm yielded better PSNR for low and moderate magnitudes of zero mean Gaussian
noise. The algorithm showed remarkable denoising capability of Ridgelet transform
in while protecting edges compared to other contemporary algorithms.
Krishnanaik and Someswar [92] in 2016, introduced sliced ridgelet transform
for image de-noising, and to achieved the scalability and accuracy and in a reliable
manner of image processing. Sliced ridgelet transforms ridge function was segregated to multiple slices with constant length. Single dimension wavelet transforms
5.9 Application of Multiscale Transform in Image Restoration
177
were used to compute the angle values of each slice in sliced ridgelet transform.
Ridgelet co-efficient were obtained for the base threshold calculation to implement
the accurate de-noising. The proposed method was based on two operations: one was
the redundant directional wavelet transform based on the radon transform, and other
was threshold designing of the ridgelet coefficient. Authors compared the accuracy
and scalability of image de-noising with other popular approaches like wavelets,
curvlets and some other inter-relevant technologies. Experimental results were proving that the sliced ridgelet approach was having the better performance than the other
popular techniques.
Huang et al. [77] in 2016, proposed a new multiscale decomposition algorithm
called adaptive digital ridgelet (ADR) transform. This algorithm could adaptively
deal with line and curve information in an image by considering its underlying structure. As the key part of the adaptive analysis, the curve parts of an image were detected
accurately by a new curve part detection method. ADR transform was applied to
image denoising experiment and experimental results demonstrate its efficiency for
reducing noises as PSNR values could be improved maximally 5 dB compared with
other methods and MAE values were also considerably improved. A new comparison criterion was also proposed and using this criterion, it was shown that ADR
transform can provide a better performance in image denoising.
5.9.5 Image Restoration Using Curvelet Transform
With the introduction of wavelet transform in early 1980, several general and hybrid
denoising models have been proposed, in which either mathematical algorithms or
heuristic algorithms were used for specific applications. Wavelet has been explored
so much that it has more or less become a household word. Everyday life we come
across with electronic instruments which have wavelet applications. Not only engineers but also non-technical persons know about wavelets. Multiresolution capacity
which means representation should allow images to be successively approximated
from course to fine resolutions. The basis element should be able to localize frequency components in multiple directions with very less redundancy. Wavelet does
not perform well, when the singularities are in higher dimensions. These deficiencies
inspired the researchers to extend the wavelet transform. Some of these extensions
are done through geometry of the space and some by preserving properties of the
transform such as invariance, translation, rotation and singularities along certain
directions.
One of the generalizations of the wavelet transform is the directional wavelet
transform which in addition to scaling and translation, accounts for rotation. This
makes the directional wavelet transform better in detecting singularities of higher
dimensions.
In the last few years, there have been developments which discovered new multiscale transforms to overcome wavelets limitations. Those are ridgelets and curvelets.
Ridgelet and curvelet performed better even in the early stages of their development.
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Ridgelet transform which is an application of 1D wavelet transform is also based on
radon transforms. Radon transforms detect singularities along straight line, while the
Curvelet transform which may be viewed as directional parabolic dilation transform,
in which the parabolic dilation is applied in the frequency domain. It can detect singularities along curves better than the ridgelet and wavelet transform, in addition to
singularities along lines.
The problem of detecting singularities is closely related to image processing
because in two dimensional images, smooth regions are separated by edges and
smooth cures, which suggest that wavelet transform and ridgelet transform cant
be the best tool for some image processing applications. Till now we have gone
through wavelet, complex wavelet, quaternion wavelet, ridgelet and to overcome the
drawbacks of wavelet and ridgelet, now we will move on to curvelet transform. The
curvelet transform represents curves better than wavelet and ridgelet.
Candes and Donoho [21] in 2001 introduced the curvelet transform well-matched
for objects which are smooth and away from discontinuities across curves. Curvelet
transform has been proved to be effective at processing of images along curves instead
of radial directions. Curvelet can also capture structural information along multiple
scales, locations and orientations. Curvelet captures this structural information in
the frequency domain. Even if we subject complex texture structure of CT images,
curvelet will reasonability improve quality (texture) of the image.
Starck et al. [161] in 2002 used curvelet transform for representing astronomical
images. The noisy image was given as an input to the curvelet transform and the
resultant image was compared with the established method of thresholded coefficient
of wavelet transform. Curvelet transform enhance elongated features and better ring
and edges are seen in the astronomical images. The experiments results described
that curvelet reconstruction did not contain the quantity of disturbing artifacts along
edges that one see in wavelet reconstructions. PSNR of denoised Lenna image was
31.95 dB which was better than most of the algorithms.
Starck et al. [162, 163] in 2003, were designed the curvelet transform to represent
edges and other singularities along curves much more efficiently than traditional
transforms i.e. using many fewer coefficients for a given accuracy of reconstruction.
In the curvelet transform, the frame elements are indexed by scale, location and
direction. The curvelets elements obey special scaling law, where the length of the
support of a frame elements and the width of the support are linked by the relation
widthlength2 . All these properties are very attractive and have already led to a wide
range of applications such as tomography, astronomy etc.
Ma and Plonka [109] in 2007 presented an almost optimal non-adaptive sparse representation for curve like features and edges using curvelets. The authors described
a broad range of applications involving image processing, seismic data exploration,
image denoising etc. They proposed formula for soft, hard and continuous garrote and
total variation (TV) constraint curvelet shrinkage for denoising digital images. They
proved that total variation constraint curvelet shrinkage leads to a promising PSNR
at the expense of computation time. PSNR improvement of 4.77 dB was obtained for
Barbara image using this technique.
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Patil and Singhai [132] in 2010, proposed a soft thresholding multiresolution
analysis technique based on local variance estimation for image denoising. This
adaptive thresholding with local variance estimation effectively reduced image noise
and also preserved edges. In the proposed method, 2D fast discrete curvelet transform
(2D FDCT) out performed wavelet based image denoising and PSNR using 2D FDCT
was found approximately doubled.
Patil et al. [131] in 2010 proposed an approach of reconstruction of SR image using
a sub-pixel shift image registration and Curvelet Transform (CT) for interpolation.
The experimental results demonstrated that Curvelet Transform performed better as
compared to Stationary Wavelet Transform. Also, it was experimentally verified that
the computational complexity of the SR algorithm was also reduced by using CT for
interpolation.
Patil and Singhai [133] in 2011 proposed SR reconstruction using a sub-pixel
shift image registration and Fast Discrete Curvelet transform (FDCT) for image
interpolation. Experimentation results showed appropriate improvements in PSNR
and MSE and also it was experimentally verified that the computational complexity
of the SR algorithm was reduced.
Palakkal and Prabhu [129] in 2012 proposed a denoising algorithm for images
corrupted by Poisson noise. They had used fast discrete curvelet transform and waved
atom technique along with a variance stabilizing transform for denoising of images.
This algorithm was tested on standard images corrupted by poisson noise. For Barbara
image having maximum intensity of 120 and input PSNR value of 24.04 dB, the
denoised image PSNR was 29.45 dB. The other algorithms that were discussed focus
on denoising for Gaussian noisy image whereas this algorithm examines Poisson
noisy images.
A curvelet is a effective spectral transform, which allows sparse representations
of complex data. This spectral technique is based on directional basis functions that
represent objects having discontinuities along smooth curves. Oliveira et al. [127] in
2012 applied this technique to the removal of Ground Roll, which was an undesired
feature signal present in seismic data obtained by sounding the geological structures
of the Earth. They decomposed the original seismic data by curvelet transform in
scales and angular domains. For each scale the curvelet denoising technique allowed
a very efficient separation of the Ground Roll in angle sections. The precise identification of the Ground Roll pattern allowed an effective erasing of its coefficients. In
contrast to conventional denoising techniques they did not use any artificial attenuation factor to decrease the amplitude of the Ground Roll coefficients. They estimated
that, depending on the scale, around 75 percent of the energy of the strong undesired
signal is removed.
Kaur et al. [84] in 2012 applied curvelet transform denoising method to noisy
images comparison of images. The results were compared qualitatively (visually)
and quantitatively (using quality metrics) and it was proved that values of curvelet
methods for all quality metrics were better than the other methods.
Vaghela [173] in 2013 proposed a new method for image restoration using curvelet
transform. Experimental results showed that curvelet transform coefficient technique
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was conceptually simpler, faster and far less redundant than the existing technique
in case of Gaussian noise irrespective of type of images.
Parlewar and Bhurchandi [130] in 2013 developed a modified version of curvelet
transform called the 4 quadrant curvelet transform for image denoising. The authors
used curvelet transform by taking radon projections in four quadrants and then averaging the result. The 4 quadrant curvelet transform was applied only to patches which
consist of edges instead of applying to complete image. PSNR of denoised Barbara
image corrupted by noise of standard deviation 10 was obtained as 32.7 dB.
Denoising of the astronomical images is still a big challenge for astronomers and
people who process astronomical data. Anisimova et al. [7] in 2103 proposed an algorithm based on Curvelet and Starlet transform for astronomical image data denoising.
The proposed algorithm had been tested on image data from MAIA (Meteor Automatic Imager and Analyser) system. Their influence on important photometric data
like stellar magnitude and FWHM (Full Width at Half Maximum) had been studied
and compared with conventional denoising methods.
Wu et al. [187] in 2014 developed a curvelet transform based non-local means
algorithm for digital image denoising. The noisy image was subjected to curvelet
transform followed by inverse curvelet transform. The similarity between curvelet
transform reconstructed image pixels and the noisy image pixels was used for denoising the image by employing non-local means method.
Bains and Sandhu [13] in 2015, presented a comparative analysis of various image
denoising techniques using curvelet transforms. A lot of combinations had been
applied in order to find the best method that could be followed for denoising intensity
images. The experimental results demonstrated that curvelet transform outperforms
other transform for denoising all of the above mentioned images. Curvelet transform
denoised the images with more precision as compared to DWT because of its inborn
quality of keeping the data intact to a greater extent. PSNR showed an apprehensive
improvement, if the noisy images were denoised using curvelet transform.
Anjum and Bhyri [9] in 2015, used non-linear technique such as curvelet transform
and edge detection in image processing for removing of noise present in image. The
best results were obtained with denoising the test images corrupted by random noise,
spackel noise, Gaussian noise and salt and pepper noise in terms of PSNR and it
was noticed that the lowest PSNR gain was obtained for biomedical images when
compared to satellite images.
Raju et al. [140] in 2016 proposed the denoising of remotely sensed images
based on Fast Discrete Curvelet Transform (FDCT). The Fast Discrete Curvelet
Transform had been discussed via Wrapping (WRAP) and Unequally-Spaced Fast
Fourier Transform (USFFT). With its optimal image reconstruction capabilities, the
curvelet outperformed the wavelet technique in terms of both visual quality and Peak
Signal to Noise Ratio (PSNR). Mainly the author focused on the analysis of denoising
the Linear Imaging Self Scanning Sensor III (LISS III) images, Advanced Very High
Resolution Radiometer (AVHRR) images from National Oceanic and Atmospheric
Administration 19 (NOAA 19), METOP satellites for the Tirupati region, Andhra
Pradesh, India. Numerical illustrations demonstrated that this method was highly
effective for denoising the satellite images.
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Zanzad and Rawat [200] in 2016, described a comparison of the discriminating
power of the various multiresolution based thresholding techniques i.e. wavelet,
Curvelet for image denoising. The experimental results showed that the curvelet
transform gives better results/performance than wavelet transform method.
Talbi and Cherif [169] in 2017, proposed a new image denoising technique which
combines two denoising approaches. The first one was a curvelet transform based
denoising technique and the second one was a two-stage image denoising by principal
component analysis with local pixel grouping (LPG-PCA). This proposed technique
consisted at first step in applying the first approach to the noisy image in order to
obtain a first estimate of the clean image. The second step consisted in estimating
the level of noise corrupting the original image. The third step consisted in using this
first clean image estimation, the noisy image and this noise level estimate as inputs
of the second image denoising system (LPG-PCA based image denoising) in order
to obtain the final denoised image. The proposed image denoising technique was
applied on a number of noisy images and the obtained results from PSNR and SSIM
computations show its performance.
5.9.6 Image Restoration Using Contourlet Transform
Contourlet transform is designed to efficiently represent images made of smooth
region and curved boundaries. The contourlet transform has a fast implementation based on laplacian pyramid decomposition followed by directional filter banks
applied on each band pass subbands.
Donoho employed a discrete domain multiresolution and multi-direction expansion using non-separable filter banks to construct contourlet transform. Discrete contourlet transform used an iterative filter bank that requires N operations for N -pixel
image. The link was developed between the filter banks and associated continuous wavelet domain via a directional multiresolution analysis. Donoho [56] in 1998
proved that due to directional filter bank, contourlet was able to develop smooth
object boundaries. However, the major drawback was that its basis images were not
localized in frequency domain. Donoho analyzed and proposed a new contourlet that
had basis which were localized in frequency domain. This algorithm outperformed
other contemporary algorithms quantitatively as well as visually.
Do and Vertalli [49] in 2005, constructed a discrete-domain multiresolution and
multidirection expansion using non-separable filter banks, in much the same way
that wavelets were derived from filter banks. This construction resulted in a flexible multiresolution, local, and directional image expansion using contour segments,
called contourlet transform. Furthermore, they established a specific link between the
developed filter bank and the associated continuous domain contourlet expansion via
a directional multiresolution analysis framework and showed that contourlets, with
parabolic scaling and sufficient directional vanishing moments, achieved the optimal
approximation rate for piecewise smooth functions with discontinuities along twice
continuously differentiable curves. Finally, they provided some numerical experi-
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ments demonstrating the potential of contourlets in several image processing applications.
Matalon et al. [113] in 2005 proposed a novel denoising method based on the
Basis Pursuit Denoising (BPDN) method. Their method combined the image domain
error with the transform domain dependency structure, resulting in a general objective function, applicable for any wavelet like transform. They focused on the Contourlet Transform (CT), a relatively new transform designed to sparsely represent
images. The resulting algorithm proved superior to the classic Basis-Pursuit Denoising (BPDN), which did not account for these dependencies.
Eslami and Radha [60] in 2005 developed a procedure to obtain a TI version of
a general multi-channel multidimensional subsampled FB. They proposed a generalized algorithm a trous, and then applied the derived approach to the contourlet
transform. In addition to the proposed TI contourlets, they introduced semi-TI contourlets, which was less redundant than the TICT. Furthermore, they employed their
proposed schemes in conjunction with the TIWT to image denoising. Their simulation results indicated the potential of the TICT and STICT for image denoising,
where one can achieve better visual and PSNR performance at most cases when
compared with the TIWT.
The research of Eslami and Hayder was extended by Alparone et al. [5] in 2006 to
modify contourlet transform for image denoising using cycle spinning. In the classical denoising algorithms, many visual artifacts were produced due to the lack of
translation invariance. They used cycle spinning based technique to develop translation invariant contourlet denoising scheme. The results demonstrated enhancement
on the images corrupted with additive Gaussian noise.
Cunha et al. [43] in 2006, developed the nonsubsampled contourlet transform
(NSCT) and study its applications. They constructed NSCT based on a nonsubsampled pyramid structure and nonsubsampled directional filter banks. This construction
provided a flexible multiscale, multidirection, and shift-invariant image decomposition that could be efficiently implemented via the trous algorithm. They applied
NSCT in image denoising and enhancement applications and found the better results
to other existing methods in the literature.
Po and Do [135] in 2006 provided detailed study on the statistics of the contourlet
coefficients of natural images: using histograms to estimate the marginal and joint
distributions, and mutual information to measure the dependencies between coefficients. They provided a model for contourlet coefficients using a hidden Markov tree
(HMT) model with Gaussian mixtures that can capture all inter-scale, inter-direction,
and inter-location dependencies. In addition, They presented experimental results
using this model in image denoising and texture retrieval applications. In denoising,
the contourlet HMT outperformed other wavelet methods in terms of visual quality,
especially around edges.
Sun et al. [168] in 2008 adopted multiscale geometry method, distilled the principal component from the image after Contourlet transform, lowered the dimension
of the high frequency subdomains, eliminated the noise by minimum variance cost
function. The entire arithmetic without estimate noise, compared to Contourlet hard
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threshold denoising and wavelet hard threshold denoising, PSNR increased 1 dB, the
denoising effect was better than other methods in the experiment.
Sivakumar et al. [158] in 2009, used the new algorithm based on the Contourlet
Transform. This algorithm was more efficient than the wavelet algorithm in Image
Denoising particularly for the removal of speckle noise. The parameters considered
for comparing the wavelet and Contourlet Transforms were SNR and IEF. The results
showed that the proposed algorithm outperformed the wavelet in terms of SNR, IEF
values and visual perspective as well.
Guo et al. [71] in 2011, introduced the characteristics of multi-resolution and
multi-direction decomposition about Contourlet transform, by comparing with the
wavelet transform, the theoretical basis and advantages of Contourlet transform.
Authors strained on analyzing the principle of threshold denoising, and laid forward
a new kind method of multi-threshold image denoising. Experimental results showed
that the effect of image denoising method better than wavelet transform image and
individually threshold denoising effect was good, and this method was simple on
calculation with fast speed.
Liu et al. [104] in 2012, presented an image denoising algorithm based on nonsubsampled contourlet transform (NSCT). A second order random walk with restart
kernel was employed to describe the geometric features like edges and texture. NSCT
was then used to capture these features. This algorithm used an iterative approach for
image denoising. The iterative process continues till the current RMSE is smaller than
the preceding RMSE. Denoising of Barbara image corrupted by noise of variance
0.01, yields RMSE value slightly less than 6.5.
Zhou and Wang [205] in 2012, proposed an image denoising algorithm based on
nonsubsampled contourlet transform. It used the Symmetric Normal Inverse Gaussian (SNIG) model and models the pixel values as random variables. First the image
was decomposed using NSCT and the noise was estimated. SNIG model was then
applied to the noisy image. The algorithm was validated on standard image corrupted by noise of different strengths. For Barbara and Lena images corrupted by
white noise of standard deviation equal to 10, the corresponding denoised images had
PSNR values of 34.08 and 30.77 dB. This algorithm was validated only on images
corrupted by Gaussian noise.
Borate and Nalbalwar [17] in 2012, proposed a technique to recover the superresolved image from a single observation using contourlet based learning and useful
when multiple observations of a scene were not available so one must make the best
use of a single observation to improve its resolution. Experimental results showed
appropriate improvements over conventional interpolation techniques.
Shah et al. [153] in 2013 discussed a novel approach of getting high resolution
image from a single low resolution image. In their method, the Non Sub-sampled
Contourlet Transform (NSCT) based learning was used to learn the NSCT coefficients at the finer scale of the unknown high-resolution image from a dataset of high
resolution images. The cost function consisting of a data fitting term and a Gabor
prior term was optimized using an Iterative Back Projection (IBP). By making use
of directional decomposition property of the NSCT and the Gabor filter bank with
various orientations, the proposed method was capable to reconstruct an image with
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less edge artifacts. The experimental results showed the better performance of the
proposed method in terms of RMS measures and PSNR measures.
Shen et al. [155] in 2013, proposed an adaptive window based contourlet transform
domain method for image denoising. An elliptical shaped window was used instead of
a square window so as to estimate the signal variance. The improvement in denoising
performance by using elliptical window was represented graphically. For Lena image,
denoising by this algorithm improved the PSNR from 22.10 to 31.25 dB i.e. an
improvement of 9.15 dB.
Wang et al. [181] in 2013, presented a SVM classification based non-subsampled
contourlet transform (NSCT) domain technique for denoising of images. To distinguish between noisy pixels and edge pixels, the authors have used SVM to classify
NSCT coefficients into smooth regions and texture regions. A significant increase
in PSNR value of Lena image from 22.11 to 31.64 dB was obtained using this technique. This algorithm yielded the highest PSNR improvement of 9.53 dB in case of
Lena image.
Yin et al. [198] in 2013, proposed a modified version of non-subsampled contourlet
transform based image denoising algorithm. The derived non-subsampled dual-tree
complex contourlet transform (NSDTCT) was obtained from the merger of dual-tree
complex wavelet transform and the non-subsampled directional filter banks. The
NSDTCT was followed by non-local means filtering to obtain the denoised image.
PSNR of denoised Lena image corrupted by noise of standard deviation 10 was
obtained as 35.98 dB.
Padmagireeshan et al. [128] in 2013, proposed a medical image denoising algorithm using contourlet transform with directional filter banks and Laplacian pyramid.
The performance of the proposed method was analysed with the existing methods of
denoising using wavelet transform and block DCT. Simulation results showed that
contourlet transform had better denoising capabilities compared to existing methods.
Sakthivel [149] in 2014, proposed contourlet based image denoising algorithm
which can restore the original image corrupted by salt and pepper noise, Gaussian
noise, Speckle noise and the Poisson noise. The noisy image was decomposed into
sub bands by applying contourlet transform, and then a new thresholding function
was used to identify and filter the noisy coefficient and take inverse transform to
reconstruct the original image. The simulation result of the proposed method was
compared with other simulation results which used the various thresholding functions
namely Bayes Shrink and Visu Shrink. It was observed that the proposed algorithm
can remove Poisson and speckle noises effectively.
Wei et al. [184] in 2015, proposed the image denoising method based on an
improved Contourlet to remove the noise of the traction machine and the steel wire
rope effectively and protect the details of the image better. The Experimental results
showed that the proposed algorithm was improved in the denoising performance and
the visual effects. Meanwhile, the image details were protected better.
Image denoising is a very important step in cryo-transmission electron microscopy
(cryo-TEM) and the energy filtering TEM images before the 3D tomography reconstruction, as it addresses the problem of high noise in these images, which leads
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to a loss of the contained information. High noise levels contribute in particular to
difficulties in the alignment required for 3D tomography reconstruction.
Ahmed et al. [4] in 2015 investigated the denoising of TEM images that were
acquired with a very low exposure time, with the primary objectives of enhancing
the quality of these low-exposure time TEM images and improving the alignment
process. They proposed denoising structures to combine multiple noisy copies of the
TEM images and the structures were based on Bayesian estimation in the transform
domains instead of the spatial domain to build a novel feature preserving image
denoising structures via wavelet domain, the contourlet transform domain and the
contourlet transform with sharp frequency localization. Numerical image denoising
experiments demonstrated the performance of the Bayesian approach in the contourlet transform domain in terms of improving the signal to noise ratio (SNR) and
recovering fine details that may be hidden in the data. The SNR and the visual quality
of the denoised images were considerably enhanced using these denoising structures
that combine multiple noisy copies.
Li et al. [101] in 2015 proposed an algorithm for image denoising based on
Nonsubsampled Contourlet Transform (NSCT) and bilateral filtering in the spatial domain is proposed. The noisy image was first decomposed into multi-scale
and multi-directional subbands by NSCT, and direction subbands of each high-pass
component was processed by the new threshold function which was obtained by the
Bayes threshold that based on stratified noise estimation. During the reconstruction,
the low-pass subband constructed image was further denoised by the bilateral filtering in the spatial domain. Experimental results demonstrated that the proposed
method improved de-noising performance.
Divya and Sasikumar [46] in 2015, proposed a technique of noise removal from
digital images. In this process, the image was first transformed to the nonsubsampled
contourlet transform (NSCT) domain and then support vector machine (SVM) was
used for classifying noisy pixels from the edge related ones. The proposed method
had the advantage of achieving a good visual quality with very less quantity of
disturbing artifacts.
Jannath et al. [78] in 2016 compared the performance of DWT and Contourlet
transform for image denoising. They found that Contourlet transform Speckle noise
was better removal of Block Shrink and Poisson noise was work well for the Bayes
Shrink.
Jannath et al. [79] in 2016, denoised Gaussian noises and Speckle noises in MR
images undergo a contourlet domain for decomposition of input images. After decomposition some threshold methods were applied such as Bayes Shrink, Neigh Shrink,
and Block Shirnk. These Threshold methods were used to unfasten the noises. Finally
they analysed the performance of denoised image to find the better result. Performance of medical image denoising was reckoning by Peak signal to noise ratio
(PSNR), structural similarity index (SSIM), image quality index (IQI) and normalized cross correlation (NCC).
Kourav and Chandrawat [91] in 2017, proposed non-subsampled contourlet transform (NSCT) for image denoising and compared the result with discrete wavelet
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transform (DWT). They observed that the performance of NSCT was better than the
DWT on the basis of PSNR, RMSE and SSIM.
Oad and Bhavana [126] in 2017 proved to the pros of the contourlet transform over
wavelet transform. The proposed methodology integrating the wiener and median
filtering to enhance the performance of the denoising over wavelet transform. The
simulation was performed on three images Lena, Peppers and Barbara with three
parameters peak signal to noise ratio (PSNR), root mean square error (RMSE) and
elapsed time found that the proposed contourlet transform was better than the wavelet
transform based denoising.
5.9.7 Image Restoration Using Shearlet Transform
Removing or reducing noises from image is very important task in image processing.
Image Denoising is used to improve and preserve the fine details that may be hidden
in the data. In Image processing, noise is not easily eliminated as well as preserving
edges is also difficult. The shearlet representation has emerged in recent years as
one of the most effective frameworks for the analysis and processing of multidimensional data. Shearlet is the greatest method for preserving the edges. Shearlet
Transform combines multiscale and multi-directional representation and is very efficient to capture intrinsic geometry of the multidimensional image and is optimally
sparse in representing image containing edges, which enable them to capture intrinsic geometric features of image. It can work well in both natural images and medical
images for identifying the Anistropic features and preserved smooth edges. Shearlet
is best because it has retained the accurate information. During these advantages It
motivates and justified to do work in Shearlet transform.
Unlike wavelets, shearlets form a pyramid of well-localized functions defined
not only over a range of scales and locations, but also over a range of orientations
and with highly anisotropic supports. As a result, shearlets are much more effective
than traditional wavelets in handling the geometry of multidimensional data, and
this was exploited in a wide range of applications from image and signal processing.
However, despite their desirable properties, the wider applicability of shearlets is
limited by the computational complexity of current software implementations. For
example, denoising a single 512 512 image using a current implementation of the
shearlet-based shrinkage algorithm can take between 10 s and 2 min, depending on
the number of CPU cores, and much longer processing times are required for video
denoising. On the other hand, due to the parallel nature of the shearlet transform, it
is possible to use graphics processing units (GPU) to accelerate its implementation.
Easley et al. [57] in 2008 introduced a new discrete multiscale directional representation called the discrete shearlet transform. This approach combined the power
of multiscale methods with a unique ability to capture the geometry of multidimensional data and was optimally efficient in representing images containing edges.
They described two different methods of implementing the shearlet transform. The
numerical experiments presented in this paper demonstrated that the discrete shearlet
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transform was very competitive in denoising applications both in terms of performance and computational efficiency.
Chen et al. [31] in 2009, proposed a new local adaptive shrinkage threshold denoising method based on the shearlet transform by incorporating neighbouring shearlet
coefficients of image and noise level. Experimental results showed that the method
outperformed the several methods.
Li et al. [103] in 2011, presented a new image denoising scheme by combining
the shearlet shrinkage and improved total variation (TV). According to the artifacts
that appear in the result image after applying shearlet denoising approach, the image
was further denoised by a TV model, which was improved on the fidelity term.
Experiment results showed that the proposed scheme could remove image noise and
preserved the edge texture, removed Gibbs-like artifacts effectively and had lower
computational complexity.
Deng et al. [44] in 2012, proposed an efficient algorithm for removing noise
from corrupted image by incorporating a shearlet-based adaptive shrinkage filter
with a non-local means filter. Firstly, an adaptive Bayesian maximum a posteriori
estimator, where the normal inverse Gaussian distribution was used as the prior
model of shearlet coefficients, was introduced for removing the Gaussian noise from
corrupted image. Secondly, the nonlocal means filter was used to suppress unwanted
nonsmooth artifacts caused by the shearlet transform and shrinkage. Experimental
results demonstrated that the proposed method can effectively preserve the image
features while suppressing noise and unwanted nonsmooth artifacts. It achieved stateof-the-art performance in terms of SSIM and PSNR.
Fan et al. [61] in 2013 proposed an filter algorithm which comprehensive utilize
Multi-Objective Genetic Algorithm (MOGA) and Shearlet transform based on a
Multi-scale Geometric Analysis (MGA) theory. Experimental results showed that
their algorithm was more effective in removing Rician noise, and giving better Peak
Signal Noise Ratio (PSNR) gains, without manual intervention in comparison with
other traditional filters.
Nonsubsampled shearlet transform (NSST) is an effective multi-scale and multidirection analysis method, it not only can exactly compute the shearlet coefficients
based on a multiresolution analysis, but also can provide nearly optimal approximation for a piecewise smooth function. Yang et al. [194] in 2014, proposed a
new edge/texture-preserving image denoising using twin support vector machines
(TSVMs) Based on NSST. In this proposed method, firstly, the noisy image was
decomposed into different subbands of frequency and orientation responses using
the NSST, then, the feature vector for a pixel in a noisy image was formed by the
spatial geometric regularity in NSST domain, and the TSVMs model was obtained
by training. Next, the NSST detail coefficients were divided into information related
coefficients and noise-related ones by TSVMs training model. Finally, the detail subbands of NSST coefficients were denoised by using the adaptive threshold. Experimental results demonstrated that their method could obtain better performances in
terms of both subjective and objective evaluations than those state-of-the-art denoising techniques. Especially, the proposed method could preserve edges and textures
very well while removing noise.
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Gilbert et al. [69] in 2014 presented an open source stand-alone implementation
of the 2D discrete shearlet transform using CUDA C++ as well as GPU-accelerated
MATLAB implementations of the 2D and 3D shearlet transforms. They had instrumented the code so that they could analyze the running time of each kernel under
different GPU hardware. In addition to denoising, authors described a novel application of shearlets for detecting anomalies in textured images. In that application,
computation times could be reduced by a factor of 50 or more, compared to multicore
CPU implementations.
Image normally has both dots-like and curve structures. But the traditional wavelet
or multidirectional wave (ridgelet, contourlet, curvelet, etc.) could only restore one
of these structures efficiently so that the restoration results for complex images are
unsatisfactory. Ding and Zhao [45] in 2015 proposed a combined sparsity regularization method for image restoration based on the respective advantages of shearlet and
wavelet in the sparsity regularization method. This method could efficiently restore
the dots-like and curve structures in images, generating higher SNR of restored
images. They did not use the traditional soft and hard-threshold algorithm to improve
convergence rate, but adopted semismooth Newton method with super linear convergence rate. Numerical results showed the analysis results of Lena image restoration
from various perspectives and demonstrated that the combined sparsity regularization method could restore more accurately and efficiently in the way of SNR, residual
errors and local dots or curve structures of images.
To remove noise and preserve detail of image as much as possible, Hu et al. [76] in
2015, proposed image filter algorithm which combined the merits of Shearlet transformation and particle swarm optimization (PSO) algorithm. Experimental results
had shown that proposed algorithm eliminates noise effectively and yields good peak
signal noise ratio (PSNR).
Sharma and Chugh [154] in 2015 presented a proficient approach intended for
image denoising based on Shearlet transform and the Bayesian Network. The projected technique used the geometric dependencies in the shearlet domain in the
direction of the Bayesian Network which was next used for predict the noise probability. The Shearlet transform provided improved approximation particularly in
different scales, and directional discontinuities which make it preferable designed
used within support of processing the pixel around the edge. The later result proved
that the future technique better wavelet base method visually and mathematical in
conditions of PSNR (peak signal-to-noise ratio).
Satellite images have become universal standard in almost all applications of
image processing. However, satellite images are mostly degraded due to the inaccuracy or limitations of the transmission and storage devices. Development of a denoising algorithm in satellite images is still a challenging task for many researchers. Most
of the state of the art denoising schemes employ wavelet transform but the main limitation of wavelet transform is it can capture only limited information along different
directions. Hence edges in an image get distorted. Shearlet transformation is a sparse,
multiscale and multidimensional alternative to wavelet transform.
Anju and Raj [8] in 2016 presented a novel image denoising algorithm utilizing
shearlet transform and Otsu thresholding for denoising the satellite images and it was
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found to exhibit superior performance among other state of the art image denoising
algorithms in terms of peak signal to noise ratio (PSNR) and visual quality.
Ehsaeyan [58] in 2016 presented a new approach for image denoising based on
shearlet transform, Wiener filter and NeighShrink SURE model. In this method, the
low frequency sub-band coefficients were denoised by applying the adaptive Wiener
filter. As for the high frequency sub-band coefficients, they were refined according
to the NeighShrink rule. The visual effect image and detailed measurements showed
that his method was more effective, which was not only better in reducing noise,
but also had an advantage in preserving the information of edges. Measured results
revealed that his scheme had the best PSNRs in most cases.
Mugambi et al. [121] in 2016 proposed an algorithm for image denoising based
on Shearlet Transform and PCA (Principle Component Analysis). The combined
method gave better results both byhuman visual and by PSNR values.
Fathima et al. [63] in 2017 proposed the noise removal shearlet transform by hard
threshold for denoising. The multiscale and multidirectional aspects of the shearlet
transform provided a better estimation capability for images exhibiting piecewise
smooth edges, Quantitative performance measure such as MSE, RMS, PSNR were
used to evaluated the denoised image effect. The Shearlet Transform with hard threshold was an efficient technique for improving the quality of the image.
Bharath et al. [16] in 2017 proposed a technique by integrating Wavelet and
Shearlet transform which effectively removes the noise to the maximum extent and
restored the image by edge detection which can be identified. The simulation was
done on synthetic image and showed improvement with existing methods.
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Chapter 6
Image Enhancement
The purpose of image enhancement and image restoration techniques is to perk up a
quality and feature of an image that result in improved image than the original one.
Unlike the image restoration, image enhancement is the modification of an image to
alter impact on the viewer. Generally enhancement distorts the original digital values;
therefore enhancement is not done until the restoration processes are completed.
In image enhancement the image features are extracted instead of restoration of
degraded image. Image enhancement is the process in which the degraded image is
handled and the appearance of the image by visual is improved. It is a subjective
process and increases contrast of image but image restoration is a more objective
process than image enhancement. Performance of image restoration can be measured
very precisely, whereas enhancement process is difficult to represent in mathematical
form.
6.1 Overview
Image enhancement is one of the measurement issues in high quality pictures such as
digital camera and HDTV. Since clarity of image is very easily affected by weather,
lighting, wrong camera exposure or aperture settings, high dynamic range of scene,
etc. These conditions make an image suffer from loss of information. Image enhancement techniques have been widely used in many applications of image processing
where the subjective quality of images is important for human interpretation. Image
enhancement aims to improve the visual appearance of an image, without affecting the original attributes (i.e.,) image contrast is adjusted and noise is removed to
produce better quality image. Image enhancement improves the interpretability or
perception of information in images. Contrast is an important factor in any subjective evaluation of image quality. Contrast is created by the difference in luminance
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7_6
199
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6 Image Enhancement
reflected from two adjacent surfaces. In other words, contrast is the difference in
visual properties that makes an object distinguishable from other objects and the
background. Due to low contrast image enhancement becomes challenging and also
objects cannot be extracted clearly from dark background. Since color images provide more and richer information for visual perception than that of the gray images,
color image enhancement plays an important role in digital image processing. Many
algorithms for accomplishing contrast enhancement have been developed and applied
to problems in image processing.
In the literature there exist many image enhancement techniques that can enhance
a digital image without spoiling its important features. The image enhancement
methods can be divided into the following two categories:
1. Spatial-domain processing and
2. Frequency-domain processing.
Spatial-domain image enhancement acts on pixels directly. The pixel values are
manipulated to achieve desired enhancement. Spatial domain techniques enhance
the entire image uniformly which at times results in undesirable effects.
Frequency-domain image enhancement is a term used to describe the analysis
of mathematical functions or signals with respect to frequency and operate directly
on the image transform coefficients. The image is first transformed from spatial to
frequency domain, and the transformed image is then manipulated. It is, in general,
not easy to enhance both low- and high-frequency components at the same time using
the frequency-domain technique.
These traditional techniques thus do not provide simultaneous spatial and spectral resolution. Wavelet Transform is capable of providing both frequency and spatial
resolution. Wavelet Transform is based upon small waves with varying frequency
and limited duration called wavelets. Since higher frequencies are better resolved
in time and lower frequencies are better resolved in frequency, the use of wavelets
therefore ensure good spatial resolution at higher frequencies and good frequency resolution at lower frequencies. Hence wavelet-based techniques can solve drawbacks
of frequency-domain techniques by providing flexibility in analyzing the signal over
the entire time range.
Newly developed wavelet-based multiscale transforms include ridgelet, curvelet,
contourlet and shearlet.
6.2 Spatial Domain Image Enhancement Techniques
In the spatial domain image enhancement technique, transformations are directly
applied on the pixels. The pixel values are manipulated to achieve desired enhancement. Spatial domain processes can be expressed as
g(x, y) = T [ f (x, y)],
(6.2.1)
6.2 Spatial Domain Image Enhancement Techniques
201
where f (x, y) is the input image, g(x, y) is the processed image, and T is an operator
on f, defined over the neighborhood of (x, y). Spatial domain techniques like the
logarithmic transforms, power law transforms, histogram equalization are based in
the direct manipulation of the pixels in image. Spatial domain methods can again be
divided into two categories; (1) point processing and (2) spatial filtering operations.
Now we briefly summarize various spatial domain techniques.
6.2.1 Gray Level Transformation
Gray level transformation is the simplest image enhancement techniques. The values
of pixels, before and after processing, will be denoted by r and s, respectively, these
values are related by an expression of the form s = T (r ), where T is a transformation
that maps a pixel value r into a pixel value s. Gray level transformations are applied to
improve the contrast of the image. This transformation can be achieved by adjusting
the gray level and dynamic range of the image, which is the deviation between
minimum and maximum pixel value.
In gray level transformations, three basic types of functions are used frequently
for image enhancement: linear, logarithmic and power-low.
1. Image Negative
In this method, reverses the pixel value i.e. each pixel is subtracted from L, where,
L is the maximum pixel value of the image. This can be expressed as
s = L − 1 − r,
(6.2.2)
where s is the negative image or output image, L − 1 is the maximum pixel value and
r is the input image. The pixel range for both the input image and negative image is
in the range (0, L − 1). This type of processing is particularly used for enhancing
white or gray detail embedded in dark regions of an image, especially when the black
areas are dominant in size.
2. Log Transformation
The general form of the log transformation is given by
s = c log(1 + r ),
(6.2.3)
where c is a constant and r ≥ 0. This transformation maps a narrow range of low
gray-level values in the input image into a wider range of output levels. Hossain and
Alsharif [27] is used Log transformation to expand the dark pixels and compress the
202
6 Image Enhancement
brighter pixel. This compressed the dynamic range of the image with large variations
in pixel values.
3. Power-Law Transformation
The general form of the power-law transformation is given by
s = c rγ,
(6.2.4)
where c and γ are positive constants. Sometimes the Eq. (6.2.4) can be written as
s = c (r + )γ
(6.2.5)
to account for a measurable output when the input is zero. A variety of devices
used for image capture, printing and display respond according to power law. This
process is also called gamma correction. Gamma correction has become increasingly
important in the past few years, as use of digital images for commercial purposes
over the internet as increased. In addition, gamma correction are useful for general
purpose contrast manipulation.
6.2.2 Piecewise-Linear Transformation Functions
A practical implementation of some important transformations can be formulated
only as piecewise linear functions.
Contrast Stretching
Low-contrast images can result of poor illumination, lack of dynamic range in the
imaging sensor or even wrong camera. One of the simplest piecewise linear functions
is a contrast-stretching transformation. The basic idea behind contrast stretching is
to increase the dynamic range of the gray levels in the image being processed. In
contrast stretching, upper and lower threshold are fixed and the contrast is stretched
between these thresholds. It is contrast enhancement method based on the intensity
value as shown
(6.2.6)
I0 (x, y) = f (I (x, y))
where, the original image is I (x, y), the output image is I0 (x, y) after contrast
enhancement. The transformation function T is given by
s = T (r ),
(6.2.7)
6.2 Spatial Domain Image Enhancement Techniques
203
where s is given as
⎧
if 0 ≤ r < a
⎨ l · r,
s = m · (r − a) + v, if 0 ≤ r < b
⎩
n · (r − b) + w, if 0 ≤ r < l − 1
where l, m, n are the slopes of the three regions, the s is the modified gray level
and r is the original gray level. a and b are the limits of lower and upper threshold.
Contrast stretching tend to make the bright region brighter and vice versa.
6.2.3 Histogram Processing
Intensity transformation functions based on information extracted from image intensity histograms play a basic role in image processing.
Histogram Equalization
With L total possible intensity levels in the range [0, G], the histogram of a digital
image is defined as the discrete function
h(rk ) = n k ,
(6.2.8)
where rk is the kth intensity level in the interval [0, G] and n k is the number of pixels in
the image whose intensity level is rk . it is useful to work with normalized histograms,
which is obtained by dividing all elements of h(rk ) by the total number of pixels in
nk
h(rk )
= , for k = 1, 2, . . . , L .
the image, which we denote by n, i.e. p(rk ) =
n
n
Let Pr (r ) denote the probability density function (PDF) of the intensity levels in
a given image and assume that intensity levels are continuous quantities normalized
to the range [0, 1], where the subscript is used for differentiating between the PDFs
of the input and output images. Then
s = T (r ) =
r
Pr (w)dw.
(6.2.9)
0
Gonzalez and Woods [23] in 2002, showed that the probability density function of
the output levels is uniform; that is,
Ps (s) =
l if 0 ≤ s ≤ 1,
0 otherwise.
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6 Image Enhancement
In other words, the preceding transformation generates an image whose intensity
levels are equally likely, and cover the entire range [0, 1]. The result of this intensitylevel equalization process is an image with increased dynamic range, which will tend
to have higher contrast. It is clear that the transformation function is the cumulative
distribution function (CDF). When dealing with discrete quantities we work with
histograms and call the preceding technique histogram equalization. In general, the
histogram of the processed image will not be uniform, due to the discrete nature of
the variables. Let Pr (r j ), j = 1, 2, . . . , L , denote the histogram associated with the
intensity levels of a given image and the values in a normalized histogram are approximations to the probability of occurrence of each intensity level in the image. For
discrete quantities, we work with summations, and the equalization transformation
becomes
k
k
nj
Pr (r j ) =
sk = T (rk ) =
n
j=1
j=1
for k = 1, 2, . . . , L , where sk is the intensity value in the output (processed) image
corresponding to value rk in the input image.
Histogram equalization produces a transformation function that is adaptive, in
the sense that it is based on the histogram of a given image. However, once the
transformation function for an image has been computed, it does not change unless
the histogram of the image changes.
Histogram equalization is used for contrast adjustment using the image histogram When ROI is represented by close contrast values, this histogram equalization
enhances the image by increasing the global contrast. As a result, the intensities are
well scattered on the histogram and low contrast region is converted to region with
higher contrast. This is achieved by considering more frequently occurring intensity
value and spreading it along the histogram. Histogram equalization plays a major
role in images having both ROI and other region as either darker or brighter. Its
advantage is, it goes good with images having high color depth. For example images
like 16-bit gray-scale images or continuous data. This technique is widely used in
images that are over-exposed or under-exposed, scientific images like X-Ray images
in medical diagnosis, remote sensing, and thermal images. Same way this technique
has its own defects, like unrealistic illusions in photographs and undesirable effect
in low color depth images.
6.2.4 Spatial Filtering
Spatial filtering is used to remove noise whose detail was introduced in this chapter.
6.3 Frequency Domain Image Enhancement Techniques
205
6.3 Frequency Domain Image Enhancement Techniques
Image enhancement in the frequency domain is straightforward. We simply compute
the Fourier transform of the image to be enhanced, multiply the result by a filter (low
pass filter, high pass filter and homomorphic filter) rather than convolve in the spatial
domain, and take the inverse transform to produce the enhanced image. The idea
of blurring an image by reducing its high frequency components or sharpening an
image by increasing the magnitude of its high frequency components is intuitively
easy to understand.
6.3.1 Smoothing Filters
The noises, edges and other sharp transitions in the gray level contribute significantly
to the high frequency. Hence smoothing or blurring is achieved by attenuating a
specified range of high frequency components in the transform of a given image,
which can be done using a lowpass filter.
A filter that attenuates high frequencies and retains low frequencies unchanged
is called lowpass filter. Since high frequencies are blocked, this results a smoothing
filter in the spatial domain. Three are three types of lowpass filters: Ideal lowpass
filter, Gaussian lowpass filter and Butterworth lowpass filter.
1. Ideal LowPass Filter
The most simple lowpass filter is the ideal lowpass filter (ILPF). It suppresses all
frequencies higher than the cut-off frequency r0 and leaves smaller frequencies
unchanged:
l if D(u, v) ≤ r0 ,
H (u, v) =
(6.3.1)
0 if D(u, v) > r0 ,
where r0 is called the cutoff frequency (nonnegative quantity), and D(u, v) is the
distance from point (u, v) to the frequency rectangle. If the image is of size M × N ,
then
M
N
u−
+ v−
D(u, v) =
2
2
The lowpass filters considered here are radially symmetric about the origin. The
drawback of the IDLF function is a ringing effect that occurs along the edges of
the filtered image. In fact, ringing behavior is a characteristic of ILPF. As we know
that multiplication in the Fourier domain corresponds to a convolution in the spatial
domain. Due to the multiple peaks of the ideal filter in the spatial domain, the filtered image produces ringing along intensity edges in the spatial domain. The cutoff
206
6 Image Enhancement
frequency r0 of the ILPF determines the amount of frequency components passed
by the filter. Smaller the value of r0 , more the number of image components eliminated by the filter. In general, the value of r0 is chosen such that most components
of interest are passed through, while most components not of interest are eliminated.
Hence, it is clear that ILPF is not very practical.
2. Butterworth LowPass Filter
A commonly used discrete approximation to the Gaussian is the Butterworth filter.
Applying this filter in the frequency domain shows a similar result to the Gaussian
smoothing in the spatial domain. The transfer function of a Butterworth lowpass
filter (BLPF) of order n, and with cut-off frequency at a distance r0 from the origin,
is defined as
1
.
(6.3.2)
H (u, v) =
2n
D(u,v)
1+
r0
Is can be easily seen that, frequency response of the BLPF does not have a sharp
transition as in the ideal LPF and as the filter order increases, the transition from the
pass band to the stop band gets steeper. This means as the order of BLPF increases,
it will exhibit the characteristics of the ILPF. The difference can be clearly seen
between two images with different orders but the same cutoff frequency.
3. Gaussian LowPass Filter
Gaussian filters are important in many signal processing, image processing and communication applications. These filters are characterized by narrow bandwidths, sharp
cutoffs, and low overshoots. A key feature of Gaussian filters is that the Fourier transform of a Gaussian is also a Gaussian, so the filter has the same response shape in
both the spatial and frequency domains. The form of a Gaussian lowpass filter in
two-dimensions is given by
H (u, v) = e−D
2
(u,v)/2σ 2
.
(6.3.3)
The parameter σ measures the spread or dispersion of the Gaussian curve. Larger
the value of σ, larger the cutoff frequency and milder the filtering is. Let σ = r0 . the
Eq. (6.3.3) becomes
2
2
H (u, v) = e−D (u,v)/2r0 .
(6.3.4)
When D(u, v) = r0 , the filter is down to 0.607 of its maximum value of 1.
The Gaussian has the same shape in the spatial and Fourier domains and therefore
does not incur the ringing effect in the spatial domain of the filtered image. This
is an advantage over ILPF and BLPF, especially in some situations where any type
of artifact is not acceptable, such as medical image. In the case where tight control
6.3 Frequency Domain Image Enhancement Techniques
207
over transition between low and high frequency needed, Butterworth lowpass filter
provides better choice over Gaussian lowpass filter; however, tradeoff is ringing
effect.
The Butterworth filter is a commonly used discrete approximation to the Gaussian.
Applying this filter in the frequency domain shows a similar result to the Gaussian
smoothing in the spatial domain. But the difference is that the computational cost
of the spatial filter increases with the standard deviation, whereas the costs for a
frequency filter are independent of the filter function. Hence, the Butterworth filter
is a better implementation for wide lowpass filters, while the spatial Gaussian filter
is more appropriate for narrow lowpass filters.
6.3.2 Sharpening Filters
Sharpening filters emphasize the edges, or the differences between adjacent light and
dark sample points in an image. A highpass filter yields edge enhancement or edge
detection in the spatial domain, because edges contain many high frequencies. Areas
of rather constant gray level consist of mainly low frequencies and are therefore
suppressed. A highpass filter function is obtained by inverting the corresponding
lowpass filter. An ideal highpass filter blocks all frequencies smaller than r0 and
leaves the others unchanged. The transfer function of lowpass filter and highpass
filter can be related as follows:
Hhp (u, v) = 1 − Hlp (u, v),
(6.3.5)
where Hhp (u, v) and Hlp (u, v) are the transfer function of highpass and lowpass
filter respectively.
1. Ideal HighPass Filter
The transfer function of an ideal highpass filter with the cutoff frequency r0 is:
H (u, v) =
0 if D(u, v) ≤ r0 ,
1 if D(u, v) > r0 .
(6.3.6)
2. Butterworth High Pass Filter
The transfer function of Butterworth highpass filter (BHPF) of order n and with
cutoff frequency r0 is given by:
208
6 Image Enhancement
H (u, v) =
1
1+
r0
D(u,v)
2n
.
(6.3.7)
The frequency response does not have a sharp transition as in the IHPF. It can be seen
that BHPF behaves smoother and has less distortion than IHPF. Therefore, BHPF is
more appropriate for image sharpening than the IHPF. Also less ringing is introduced
with small value of the order n of BHPF.
3. Gaussian HighPass Filter
The transfer function of a Gaussian high pass filter (GHPF) with the cutoff frequency
r0 is given by:
2
2
H (u, v) = 1 − e−D (u,v)/2r0 .
(6.3.8)
The parameter σ, measures the spread or dispersion of the Gaussian curve. Larger
the value of σ, larger the cutoff frequency and milder the filtering is.
6.3.3 Homomorphic Filtering
An image can be expressed as the product of illumination and reflectance components:
f (x, y) = i(x, y)r (x, y),
(6.3.9)
where i(x, y) and r (x, y) are illumination and reflectance components respectively.
f (x, y) must be non zero and finite, the reflectance is bounded by 0 (total absorption)
and 1 (total reflectance) and nature of illumination is determined by the illumination
source, i.e., 0 < i(x, y) < ∞. Since Fourier transform is not distributive over
multiplication, first take natural log both side, we have
z(x, y) = ln f (x, y) = ln i(x, y) + ln r (x, y),
(6.3.10)
and then apply Fourier transform:
F{z(x, y)} = F{ln f (x, y)} = F{ln i(x, y)} + F{ln r (x, y)},
or
Z (u, v) = Fi (u, v) + Fr (u, v),
(6.3.11)
where Fi (u, v) and Fr (u, v) are the Fourier transform of ln i(x, y) and ln r (x, y),
respectively. If we process Z (u, v) by means of a filter function H (u, v) then,
6.3 Frequency Domain Image Enhancement Techniques
209
Fig. 6.1 Block diagram of homomorphic filtering
S(u, v) = H (u, v)Z (u, v) = H (u, v)Fi (u, v) + H (u, v)Fr (u, v)
(6.3.12)
where S(u, v) is the Fourier transform of the result. In the spatial domain,
s(x, y) = F −1 {S(u, v)} = F −1 {H (u, v)Fi (u, v)} + F −1 {H (u, v)Fr (u, v)}
(6.3.13)
Finally, as z(x, y) was formed by taking the logarithm of the original image f (x, y),
the inverse(exponential) operation yields the desired enhanced image, denoted by
g(x, y), i.e.,
g(x, y) = es(x,y) = e F
−1
{H (u,v)Fi (u,v)} F −1 {H (u,v)Fr (u,v)}
e
−1
= i 0 (x, y)r0 (x, y), (6.3.14)
−1
where, i 0 (x, y) = e F {H (u,v)Fi (u,v)} and r0 (x, y) = e F {H (u,v)Fr (u,v)} are the illumination and reflectance components of the output image. The enhancement approach
using the homomorphic filtering approach is described in Fig. 6.1.
The illumination component of an image generally is characterized by slow spatial
variations, while the reflectance component tends to vary abruptly, particularly at
the junctions of dissimilar objects. These characteristics lead to associating the low
frequencies of the Fourier transform of the logarithm of an image with illumination
and the high frequencies with reflectance. Although these associations are rough
approximations, they can be used as an advantage in image enhancement.
6.4 Colour Image Enhancement
Color images provide more and richer information for visual perception than that
of the gray images. Color image enhancement plays an important role in Digital
Image Processing. The purpose of image enhancement is to get finer details of an
image and highlight the useful information. During poor illumination conditions,
the images appear darker or with low contrast. Such low contrast images needs to
be enhanced. In the literature many image enhancement techniques such as gamma
correction, contrast stretching, histogram equalization, and Contrast-limited adaptive
histogram equalization (CLAHE) have been discussed. These are all old techniques
which will not provide exact enhanced images and gives poor performance in terms
of Root Mean Square Error (RMSE), Peak Signal to Noise Ratio (PSNR) and Mean
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Absolute Error (MAE). Use of the old enhancement technique will not recover exact
true color of the images. Recently, Retinex, Homomorphic and Wavelet MultiScale
techniques have been popular for enhancing images. These methods are shown to
perform much better than those listed earlier.
6.5 Application of Multiscale Transforms in Image
Enhancement
Image enhancement is one of the measure issues in high-quality pictures from digital
cameras and in high definition television (HDTV). Since clarity of the image is
easily affected by weather, lighting, wrong camera exposures or aperture settings, a
high dynamic range in the scene, etc., these conditions lead to an image that may
suffer from loss of information. Many techniques have been developed to recover
information in an image.
Color image enhancement plays an important role in digital image processing
since color images provide more and richer information for visual perception than
gray images. The main purpose of image enhancement is to obtain finer details of
an image and to highlight useful information. The images appear darker or with
low contrast under poor illumination conditions. Such low-contrast images need
to be enhanced. Image enhancement is basically improving the interpretability or
perception of information in images for human viewers, and providing better input
for other automated image processing techniques.
Because some features in an image are hardly detectable by eye, we often transform images before display. Histogram equalization is one of the most well-known
methods for contrast enhancement. Such an approach is generally useful for images
with poor intensity distribution. Histogram equalization (HE) is one of the common
methods used for improving contrast in digital images. However, this technique is
not very well suited to be implemented in consumer electronics, such as television,
because the method tends to introduce unnecessary visual deterioration such as the
saturation effect. One of the solutions to overcome this weakness is by preserving
the mean brightness of the input image inside the output image.
In most cases, Brightness preserving histogram equalization (BPDHE) successfully enhances the image without severe side effects, and at the same time, maintains
the mean input brightness. BPDHE preserves the intensity of the input image, it
is disadvantageous to highlight the details in areas of low intensity. BPDHE is not
suitable for non-uniform illumination images.
Local histogram equalization (LHE) method tries to eliminate the above problem. It makes use of the local information remarkably. However, LHE (Kim et al.,
[39]) demands high computational cost and sometimes causes over enhancement
in some portion of the image. Nonetheless, these methods produce an undesirable
checkerboard effects on the enhanced images. It makes use of the local information
remarkably. However, LHE demands high computational cost and sometimes causes
over enhancement in some portion of the image. Nonetheless, these methods produce
an undesirable checkerboard effects on the enhanced images.
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211
The brightness preserving Bi-HE (BBHE) (Kim, [41]) method decomposes the
original image into two sub-images, by using the image mean gray level, and then
applies the HE method on each of the sub images independently. At some extent
BBHE preserves brightness of image; however generated image might not have a
natural appearance. Dualistic sub-image histogram equalization (DSIHE) is similar
to BBHE but DSIHE uses median value as separation intensity to divide the histogram into two sub-histogram. The algorithm enhances only the image visual information effectively, but does not preserve the details and naturalness. The essence
of the named brightness preserving histogram equalization with maximum entropy
(BPHEME) [13] tried to find the mean brightness was fixed, then transformed the
original histogram to that target one using histogram specification. In the consumer
electronics such as TV, the preservation of brightness is highly demanded.
Minimum mean brightness error bi-histogram equalization (MMBEBHE) was
proposed to preserve the brightness optimally. MMBEBHE is an extension of the
BBHE method and in this the separation intensity is minimum mean brightness
error between input image and output image. The RMSHE method proposed for
performing image decomposition recursively, up to a scalar r , generating 2r subimage.
Multi histogram equalization(MHE) techniques had been proposed to further
improve the mean image brightness preserving capability. MHE proposed a technique for image enhancement based on curvelet transform and perception network.
Since edges play a fundamental role in image understanding, one good way to
enhance the contrast is to enhance the edges. Multiscale-edge enhancement (Toet
[77]) can be seen as a generalization of this approach, taking all resolution levels
into account.
In color images, objects can exhibit variations in color saturation with little or
no correspondence in luminance variation. Several methods have been proposed
in the past for color image enhancement. Many image enhancement techniques,
such as gamma correction, contrast stretching, histogram equalization, and contrast
limited adaptive histogram equalization (CLAHE) have been discussed. These are
all old techniques that will not provide exact, enhanced images, and that give poor
performance in terms of root mean square error (RMSE), peak signal-to-noise ratio
(PSNR) and mean absolute error (MAE). Use of the old enhancement technique will
not recover an exact true color in the image. Recently, retinex, single and multiscale
retinex, and homomorphic and wavelet multiscale techniques have become popular
for enhancing images. These methods are shown to perform much better than those
listed earlier by Hanumantharaju et al. [24].
The retinex concept was introduced by Land in 1986 as a model for human color
constancy. The single scale Retinex (SSR) method [31] consists of applying the
following transform to each band of the color image:
Ri (x, y) = log(Ii (x, y)) − log(F(x, y) ∗ Ii (x, y))
where Ri (x, y) is the retinex output, Ii (x, y) is the image distribution in the i th
spectral band, F is a Gaussian function, and ∗ is convolution. The retinex method is
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efficient for dynamic range compression, but does not provide good tonal rendition
[62]. Multiscale Retinex (MSR) combines several SSR outputs to produce a single
output image that has both good dynamic range compression and color constancy
(color constancy may be defined as the independence of the perceived color from
the color of the light source [49, 53]), and good tonal rendition [31]. Multiscale
retinex leads the concept of multiresolution for contrast enhancement. It implements
dynamic range compression and can be used for different image processing goals.
Barnard and Funt [10] in 1999, presented improvements to the algorithm, leading
to better color fidelity. MSR softens the strongest edges and leaves the faint edges
almost untouched.
Using the wavelet transform, the opposite approach was proposed by Velde [81]
in 1999 for enhancing the faintest edges and keeping the strongest untouched. The
strategies are different, but both methods allow the user to see details that were hardly
distinguishable in the original image, by reducing the ratio of strong features to faint
features.
The review concludes that histogram equalization cannot preserve the brightness
and color of the original image, and a homomorphic filtering technique has a problem
with bleaching of the image. Modern technique retinex (SSR and MSR) performs
much better than those listed above, because it is based on the color constancy theory,
but it still suffers from color violation and the unnatural color rendition problem, as
the wavelet transform is a very good technique for image enhancement and denoising,
and input images always face noise during image processing.
Wavelet analysis [40] has proven to be a powerful image processing tool in recent
years. When images are to be viewed or processed at multiple resolutions, the wavelet
transform is the mathematical tool of choice. In addition to being an efficient, highly
intuitive framework for the representation and storage of multiresolution images, the
WT provides powerful insight into an images spatial and frequency characteristics.
The image detail parts are stored in the high-frequency parts of the image transformed
by the wavelet, and the imagery constant part is stored in the low-frequency part.
Because the imagery constant part determines the dynamic range of the image, the
low frequency part determines the dynamic range of the image. We attenuate the
low-frequency part in order to compress the dynamic range. But details must be
lost when the low frequency part is attenuated (Xiao et al., [85]). As some details
are stored in the high-frequency parts very well, the image reconstructed by inverse
wavelet transform has more detail. The wavelet framework was selected instead of
the Fourier, because wavelets provide an intrinsically local frequency description of
a signal directly related to local contrast, while the Fourier transform provides only
global frequency information.
6.5.1 Image Enhancement Using Fourier Transform
The Fourier Transform is used in a wide range of applications in image processing,
such as image analysis, image filtering, image reconstruction, image enhancement
and image compression. Fast Fourier Transform (FFT) is an efficient implementation
6.5 Application of Multiscale Transforms in Image Enhancement
213
of DFT and it is an important tool used in image processing. The main disadvantages
of using DFT, the speed is very slow when compared with FFT. The number of
computations for a DFT is on the order of N Squared.
Fast Fourier Transform is applied to convert an image from the image (spatial)
domain to the frequency domain. Applying filters to images in frequency domain is
computationally faster than to do the same in the image domain. An inverse transform
is then applied in the frequency domain to get the result of the convolution. The
interesting property of the FFT is that the transform of N points can be rewritten
as the sum of two N /2 transforms (divide and conquer). This is important because
some of the computations can be reused thus eliminating expensive operations. The
FFT-based convolution method is most often used for large inputs. For small inputs
it is generally faster to use im-filter.
A discrete cosine transform (DCT) expresses a finite sequence of data points in
terms of a sum of cosine function oscillating at different frequencies. The use of cosine
rather than sine functions is critical in those applications, for compression it turns
out that cosine functions are much more efficient, whereas for differential equations
the cosines express a particular choice of boundary conditions. The Discrete Cosine
Transform (DCT) is used in many applications by the scientific, engineering and
research communities and in data compression in particular.
Kanagalakshmi and Chandra [33] in 2012, proposed and implemented a Frequency domain enhancement algorithm based on Log-Gabor and FFT. They found
that the maximum variations between original and enhanced images; and also the
increased number of terminations and decreased number of bifurcations due to the
un-smoothing and noisiness. The results proved that the proposed algorithm can be
a better one for the frequency domain enhancement.
Tarar and Kumar [74] in 2013, designed a fingerprint enhancement algorithm
which can increase the degree of clarity of ridges and valleys. Since high quality fingerprint image acquired by using an adaptive fingerprint image enhancement
method was critical to the quality of any fingerprint identification system, a fingerprint enhancement method based on iterative Fast Fourier Transformation had been
designed by tarar et al. and comparative analysis with the existing method, i.e., Fast
Fourier Transformation had been shown with the help of graph. The performance of
the algorithm was evaluated using the goodness index of minutia extraction process.
Experiments on a public domain fingerprint database (i.e., FVC 2006) demonstrates
that the use of minutia descriptors leads to an order of magnitude reduction in the
false accept rate without significantly affecting the genuine accept rate. Based on
the observation of good quality rolled images, the ridge and valleys intervals of each
image are considered in order to select the Region of Interest (ROI) for effective
enhancement. Experimental results showed that our algorithm improved the goodness index as well as matching performance of the FIS. Also their algorithm dealt with
the broken ridges/false minutia problem and removed them from further processing.
Steganography is used to send the data secretly in the carrier. While sending
this information, noise may get added and it will distort the message which is sent.
For the removal of noise we require the features of image enhancement. Hence
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Singh et al. [69] worked on these two important fields and were combined together
so that the receiver could get the image which was noise free and the message had
been delivered.
Li [46] in 2013 proposed a novel method of image enhancement with respect
to the fractional Fourier transform. This method filtered image in the fractional
Fourier domain instead of the Fourier domain which was usually applied to the
classical image enhancement. The fractional Fourier transform had a rotation angle a,
characters of image thus change in different transform domain. In a proper fractional
Fourier domain with angle a, ideal low-pass filter made image smoother and ideal
high-pass filter loses less information of image than in the traditional Fourier domain,
which provided an alternative way to enhance image with proper filter designing.
Umamageswari et al. [79] in 2014, proposed a naturalness preserved enhancement
method for non-uniform illumination images using transformation techniques. Their
proposed method did not only enhances the details of the image but also preserved
the naturalness.
With the advancement of technologies, the images are created with more and more
enhancement. Yeleshetty et al. [86] in 2015, proposed a generalized equalization
model for image enhancement and further improved the same using Fast Fourier
Transform and Bilog Transformation. Here, we analysed the relationship between
image histogram and contrast enhancement and white balancing. In the proposed
system, they enhance not only images, but also videos both live and recorded. The
original image was stored in RAW format which was too big for normal displays.
Arunachalam et al. [6] in 2015 implemented two-dimensional Fast Fourier Transform (FFT) and Vedic algorithm based on Urdhva Tiryakbhyam sutra. The algorithm
was presented using MATLAB program. The input image was divided into blocks
and two-dimensional FFT was applied to enhance or filter the image. The proposed
two-dimensional FFT design was based on using Urdhva Tiryakbhyam sutra. FFT
computations using Vedic multiplication sutra gave a significant performance as
compared to the conventional FFT.
Ramiz and Quazi [63] in 2017 proposed a hybrid method which was very effective
in enhancing the images. Initially, frequency domain analysis was done followed by
spatial domain procedures. The performance of the proposed method was assessed
on the basis of two parameters i.e. Mean Square Error (MSE) and Peak Signal to
noise ratio (PSNR). Their proposed algorithm provided better PSNR and MSE.
6.5.2 Image Enhancement Using Wavelet Transform
Image enhancement is applied in every field where images are ought to be understood
and analyzed. It offers a wide variety of approaches for modifying images to achieve
visually acceptable images. The choice of techniques is a function of the specific task,
image content, observer characteristics, and viewing conditions. Wavelet transform is
capable of providing the time and frequency information of a signal simultaneously.
But sometimes we cannot know what spectral component exists at any given time
6.5 Application of Multiscale Transforms in Image Enhancement
215
instant. The best we can do is to investigate that what spectral components exist at
any given interval of time. DWT is a linear transformation that operates on a data
vector whose length is an integer power of two, transforming it into a numerically
different vector of the same length.
An advancement of wavelet theory has taken the interest of researchers in its
application to image enhancement. Discrete wavelet transform (DWT) is one of
the latest wavelet transform. It is simple mathematical tool for image processing.
DWT composes filter bank (High pass filter and low pass filter). DWT decomposes
image into four sub-bands. Image resolution enhancement in the wavelet domain is a
relatively new research topic and recently many new algorithms have been proposed.
Another recent wavelet transform, named stationary wavelet transform (SWT), has
been used in several image processing applications. SWT is similar to DWT but it
does not use down-sampling, hence the sub-bands will have the same size as the input
image. The SWT is an inherently redundant scheme as the output of each level of
SWT contains the same number of samples as the input. So for a decomposition of N
levels there is a redundancy of N in the wavelet coefficients. Translation invariance
property of DWT is overcome in SWT.
In (Mallat and Hwang [48]) the enhancement was done by noise removing and
edge enhancement. The algorithm relied on a multiscale edge representation where
the noise is connected to the singularities.
Donoho [16] in 1993, proposed the nonlinear wavelet shrinkage algorithm which
reduced wavelet coefficients toward zero, based on a level-dependent threshold. He
provided a detailed mathematical analysis of a directional wavelet transform and
revealed its connection with the edge enhancement. In addition, he discussed a single
level edge sharpening, followed by its refinement to a multiscale sharpening.
Directional wavelet transform decomposes an image into four-dimensional space
which augments the image by the scale and directional information. Heric and Potocnik [26], in 2006, proposed a novel image enhancement technique by using directional wavelet transform. They showed that the directional information significantly
improved image enhancement in noisy images in comparison with the classical techniques. Image enhancement was based on the multiscale singularity detection with
an adaptive threshold whose value was calculated via maximum entropy measure.
The proposed technique was tested on synthetic images at different signal-to-noise
ratios and clinical images and showed that proposed image enhancement technique
was robust, accurate, and effective in noisy images too.
Al-Samaraie et al. [4] in 2011, proposed a new method to enhance the satellite
image which using intelligent aspect of filtering and describe multi-threshold technique with an additional step in order to obtain the perceived image. In the proposed
scheme, the edge detected guided smoothing filters succeeded in enhancing low
satellite images. This was done by accurately detecting the positions of the edges
through threshold decomposition and the detected edges were then sharpened by
applying smoothing filter. By utilizing the detected edges, the scheme was capable to effectively sharpening fine details whilst retaining image integrity. The visual
examples shown have demonstrated that the proposed method was significantly bet-
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ter than many other well-known sharpener type filters in respect of edge and fine
detail restoration.
Kakkar et al. [32] in 2011, proposed a enhancement algorithm using gabor filter
in wavelet domain. Present algorithm effectively improved the quality of fingerprint
image. A refined gabor filter was applied in fingerprint image processing, then a good
quality of fingerprint image was achieved, and the performance of the fingerprint
recognition system had been improved.
Prasad et al. [59] in 2012 developed a method to enhance the quality of the given
image. The enhancement is done both with respect to resolution and contrast. The
proposed technique uses DWT and SVD. To increase the resolution, the proposed
method uses DWT and SWT. These transforms decompose the given image into four
sub-bands, out of which one is of low frequency and the rest are of high frequency.
The HF components are interpolated using conventional interpolation techniques.
Then we use IDWT to combine the interpolated high frequency and low frequency
components. To increase the contrast, we use SVD and DWT. The experimental
results show that proposed technique gives good results over conventional methods.
Neena et al. [55] proposed a image enhancement technique with respect to resolution and contrast based on bi cubic interpolation, stationary wavelet transform,
discrete wavelet transform and singular value decomposition. They tested proposed
technique on different satellite images and experimental results showed the proposed
method provided good results over conventional methods.
Narayana and Nirmala [54] in 2012 proposed an image resolution enhancement
technique based on the interpolation of the high frequency subbands obtained by
DWT and SWT. The proposed technique had been tested on well-known benchmark
images, where their PSNR, Mean Square Error and Entropy results showed the superiority of proposed technique over the conventional and state-of-art image resolution
enhancement techniques.
Saravanan et al. [65] in 2013, proposed that a new image enhancement scheme
using wavelet transform, smooth and sharp approximation of a piecewise nonlinear
filter technique after converting the RGB values of each pixel of the original image to
HSV. The method had effectively achieved a successful enhancement of color images.
The experimental result vividly displays the proposed algorithm was efficient enough
to remove the noise resulting good enhancement.
Karunakar et al. [34] in 2013 proposed a new resolution enhancement technique
based on the interpolation of the high-frequency sub band images obtained by DWT
and the input image. The proposed technique had been tested on well-known benchmark images, where their PSNR and RMSE and visual results show the superiority
of the proposed technique over the conventional and state-of-art image resolution
enhancement techniques. The PSNR improvement of the proposed technique was
up to 7.19 dB compared with the standard bicubic interpolation.
Bagawade and Patil [9] in 2013 used SWT (Stationary Wavelet Transform) and
DWT (Discrete Wavelet Transform) to enhance image resolution and then intermediate subbands of image produced by SWT and DWT were interpolated by using
Lanczos interpolation. Finally they combined all subbands by using IDWT (Inverse
6.5 Application of Multiscale Transforms in Image Enhancement
217
Discrete Wavelet Transform). This approach provided better result in comparison to
other method. For Baboon image they got PSNR value 27.0758dB.
Miaindargi V.C. and Mane A.P. [52] in 2013 proposed an image resolution
enhancement technique based on interpolation of the high frequency subband images
obtained by discrete wavelet transform (DWT) and the input image. The image edges
were enhanced by introducing an intermediate stage by using stationary wavelet
transform (SWT). DWT was applied in order to decompose an input image into different subbands. The quantitative and visual results showed the superiority proposed
decimated resolution technique over the conventional system and state-of-art image
resolution enhancement techniques.
Panwar and Kulkarni [58] in 2014 provided a technique of image resolution
enhancement based on SWT and DWT. The proposed technique was compared with
conventional and state-of-art image resolution enhancement techniques. They have
also provided subjective and objective comparison of resultant images and PSNR
table showed the superiority of the proposed method over conventional methods.
Provenzi and Caselles [61] in 2014, proposed a variational model of perceptually
inspired color correction based on the wavelet representation of a digital image.
Qualitative and quantitative tests about the wavelet algorithm had shown that it was
able to enhance both under and over exposed images and to remove color cast.
Moreover, in the quantitative test of color normalization property, i.e. the ability
to reduce the difference between images of the same scene taken under diverse
illumination conditions, the wavelet algorithm had performed even better than that
of existing methods.
Pai et al. [56] in 2015, provided a method to obtain the sharpened image mainly
for medical image enhancement by using the wavelet transforms using Haar wavelet
followed by the Laplacian operator. First, a medical image was decomposed with
wavelet transform. Secondly, all highfrequency sub-images were decomposed with
Haar transform. The contrast of the image was adjusted by linear contrast enhancement approaches. Filters were applied to identify the edges. Finally, the enhanced
image was obtained by subtracting resulting image from the original image. Experiments showed that this method can not only enhance an images details but can also
preserve its edge features effectively.
Khatkara and Kumar [38] in 2015 presented a method to enhance the biomedical
images using combination of wavelets, as image enhancement is the main issue for
biomedical image diagnosis. The results of the proposed method had been compared
with other wavelets on the basis of different metrics like PSNR (Peak signal to noise
ratio) and Beta coefficient and it had been found that the proposed method provides
better results than the other methods.
Brindha [12] in 2015, proposed a satellite image contrast enhancement technique
based on DWT and SVD. The experimental result showed that the better performance
and high accuracy when compared with other methods.
Thorat and Katariya [76] in 2015, proposed an image resolution enhancement
technique based on the interpolation of the high frequency subbands obtained by
DWT, correcting the high frequency subband estimation by using SWT high frequency subbands and the input image. The proposed technique had been tested on
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well-known benchmark images, where their PSNR and visual results showed the
superiority of proposed technique over the conventional and state-of-art image resolution enhancement techniques.
Kole and Patil [42] in 2016 proposed a new resolution enhancement technique
based on the interpolation of the high-frequency sub band images obtained by DWT.
The proposed technique had been tested on well-known benchmark images, where
their PSNR and RMSE and visual results show the superiority of the proposed technique over the conventional and state-of-art image resolution enhancement techniques. This procedure was successful in obtaining the enhanced images to obtain
even the minute details when related to satellite images. The PSNR improvement of
this technique was up to 7.19 dB compared with the standard bicubic interpolation.
Sumathi and Murthi [73] in 2016, proposed a new satellite image resolution
enhancement technique based on the interpolation of the high frequency sub bands
obtained by discrete wavelet transform (DWT) and the input image. The proposed
technique had been tested on satellite benchmark images. The quantitative and visual
results showed the superiority of the proposed technique over the conventional and
state of art image resolution enhancement techniques.
Arya and Sreeletha [7] in 2016 provided image resolution enhancement methods using multi-wavelet and interpolation in wavelet domain. They discussed about
improvement in the resolution of satellite images based on the multi-wavelet transform using interpolation techniques. The quantitative metrics (PSNR, MSE) of the
image calculated showed the superiority of DWT-SWT technique.
Shanida et al. [66] in 2016, proposed a denoising and resolution enhancement
technique for dental radiography images using wavelet decomposition and reconstruction. Salt and pepper noise present in image was removed by windowing the
noisy image with a median filter before performing the enhancement process. The
better performance was achieved using the proposed technique than the conventional
techniques. Quantitative assessment of the image quality was performed by means
of peak signal to noise (PSNR) calculation.
Light scattering and color change are two major sources of distortion for underwater photography. Light scattering is caused by light incident on objects reflected and
deflected multiple times by particles present in the water before reaching the camera.
This in turn lowers the visibility and contrast of the image captured. Color change
corresponds to the varying degrees of attenuation encountered by light traveling in
the water with different wavelengths, rendering ambient underwater environments
dominated by a bluish tone. Any underwater image will have one or more combinations of the Inadequate range visibility, Non uniform illumination, Poor contrast,
Haziness, Inherent noise, Dull color, Motion blur effect due to turbulence in the
flow of water, Scattering of light from different particles of various sizes, Diminished intensity and change in color level due to poor visibility conditions, Suspended
moving particles and so on.
Badgujar and Singh [8] in 2017 proposed an efficient systematic approach
to enhance underwater images using generalized histogram equalization, discrete
wavelet transform and KL transform. The proposed system provided properly
6.5 Application of Multiscale Transforms in Image Enhancement
219
enhanced underwater image output and the quality of the image was up to the mark
regarding contrast and resolution. The PSNR value of the image was higher than
other methods like DWT-KLT, DWT-SVD and GHE.
6.5.3 Image Enhancement Using Complex Wavelet
Transform
Image-resolution enhancement in the wavelet domain is a relatively new research
topic, and, recently, many new algorithms have been proposed. Complex wavelet
transform based approach of image enhancement is one of the recent approaches
used in image processing and also an improvement technique of discrete wavelet
transform. A one-level CWT of an image produces two complex-valued lowfrequency subband images and six complex-valued high-frequency subband images.
The high-frequency subband images are the result of direction-selective filters.
They show peak magnitude responses in the presence of image features oriented
at +75, +45, +15, 15, 45, and 75. Resolution enhancement is achieved by using
directional selectivity provided by CWT. The high frequency subband in 6 different
directions contributes to the sharpness of high frequency details such as edges. The
DT-CWT has good directional selectivity and has the advantage over discrete wavelet
transform (DWT). It also has limited redundancy. The DT-CWT is approximately
shift invariant, unlike the critically sampled DWT. The redundancy and shift invariance of the DT-CWT mean that DT-CWT coefficients are inherently interpolable.
Demirel and Anbarjafari [15] in 2010 proposed a satellite image resolution
enhancement technique based on the interpolation of the high-frequency subband
images obtained by DT-CWT and the input image. The proposed technique used
DT-CWT to decompose an image into different subband images, and then the highfrequency subband images were interpolated. An original image was interpolated
with half of the interpolation factor used for interpolation of the high-frequency subband images. Afterward, all these images were combined using IDT-CWT to generate
a super-resolved image. The proposed technique had been tested on several satellite
images, where their PSNR and visual results show the superiority of the proposed
technique over the conventional and state-of-the-art image resolution enhancement
techniques.
Tamaz et al. [75] in 2012 proposed a satellite image enhancement system consisting of image denoising and illumination enhancement technique based on dual
tree complex wavelet transform (DT-CWT). The technique firstly decomposed the
noisy input image into different frequency subbands by using DT-CWT and denoised
these subbands by using local adaptive bivariate shrinkage function (LA-BSF) which
assumed the dependency of subband detail coefficients. In LA-BSF, model parameters were estimated in a local neighborhood which results in improved denoising
performance. Further, the denoised image once more was decomposed into the different frequency subbands by using DT-CWT. The highest singular value of the low
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frequency subbands were used in order to enhance the illumination of the denoised
image. Finally the image was reconstructed by applying the inverse DT-CWT (IDTCWT). The experimental results showed the superiority of the proposed method over
the conventional and the state-of-art techniques.
Iqbal et al. [28] in 2013 proposed wavelet-domain approach for RE of the satellite images based on dual-tree complex wavelet transform (DT-CWT) and nonlocal
means (NLM). Objective and subjective analyses revealed superiority of the proposed
technique over the conventional and state-of-the-art RE techniques.
Bhakiya and Sasikala [11] in 2014 proposed a new satellite image resolution
enhancement technique based on the interpolation of the high-frequency sub bands
obtained by dual tree complex wavelet (DTCWT) transform and the input image.
The proposed technique had been tested on satellite benchmark images. The quantitative peak signal-to-noise ratio, root mean square error and visual results showed the
superiority of the proposed technique over the conventional and state-of-art image
resolution enhancement techniques. The PSNR improvement of the proposed technique was up to 19.79 dB.
Mahesh et al. [47] in 2014 proposed a wavelet-domain approach based on double
density dual-tree complex wavelet transform (DDDT-CWT) for RE of the satellite
images and compared with dual-tree complex wavelet transform(DT-CWT). Firstly, a
satellite image was decomposed by DDDT-CWT to obtain highfrequency subbands.
Then the high frequency subbands and the low-resolution (LR) input image were
interpolated and the high frequency subbands were passed through a low pass filter.
Finally, the filtered high-frequency subbands and the LR input image were combined
using inverse DT-CWT to obtain a resolution-enhanced image. Objective and subjective analyses revealed superiority of the proposed technique over the conventional
and state-of-the-art RE techniques.
Kumar and Kumar [44] in 2015 presented Multi scale decomposition for SRE of
the satellite images based on dual-tree complex wavelet transform and edge preservation. In their proposed method, a satellite input image was decomposed by DTCWT to obtain high-frequency sub bands. The high-frequency sub band and the low
resolution (LR) input image were interpolated using the bi-cubic interpolator. The
simulated results showed that technique used in this process provided better accuracy
rather than prior methods.
Sharma and Chopra [68] in 2015 proposed A method based on dual tree complex wavelet transform (DTCWT), contrast limited adaptive histogram equalization
(CLAHE) and Wiener filter for enhancing the visual quality of the X-Ray images.
Quantitative analysis of proposed algorithm was done by evaluating MSE, PSNR,
SNR and Contrast Ratio (CR). Their proposed algorithm showed that it outperformed
other conventional method for improving visual quality of the X-Ray image. Wiener
filter took less time as compared to NLM filter, which was the advantage in emergency
situations.
Sharma and Mishra [67] in 2015 presented multi scale decomposition based on
dual-tree complex wavelet transform and edge preservation for SRE of the satellite
images. DTCWT decomposed the low resolution input image into high frequency
subbands and low frequency subbands, since DTCWT is nearly shift invariant. High
6.5 Application of Multiscale Transforms in Image Enhancement
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frequency subbands were interpolated with factor 2(α) and input image is interpolated with factor 1(α/2). Nonlocal means filter diminished the remaining artifacts. The PSNR and the RMSE values had been calculated and compared with conventional methods which showed the effectiveness and the superiority of proposed
method.
Arulmozhi and Keerthikumar [5] in 2015 presented that noise reduction of
enhanced images using Dual tree complex wavelet transform and Bivariate shrinkage
filter. In their method, initially noisy image pixels were decorrelated to obtain coarser
and finer components and more noise details were contaminated in high frequency
subbands. In order to reduce the spatial distortion during filtering, bivariate shrinkage
function were used in the DTCWT domain. Experimental results showed that the
resultant algorithms produced images with better visual quality and evaluation was
carried out in terms of various parameters such as Peak Signal to Noise Ratio, mean
Structural Similarity and Coefficient of Correlation.
Firake and Mahajan [21] in 2016 proposed based on dual-tree complex wavelet
transform (DT-CWT) and nonlocal means (NLM) for satellite images of RE. Simulation results showed the superior performance of proposed techniques.
Deepak and Jain [14] in 2016 improved the medical image such as CT image
quality which was degraded through the Gaussian noise using dual-tree complex
wavelet transform. To improve the image quality, noise reduction techniques were
used over lower dose images and noise was reduced with preserving all clinically
relevant structures. The proposed scheme was tested on various test images and the
obtained results showed the effectiveness of the proposed scheme.
Kaur and Vashist [35] in 2017 proposed a hybrid approach algorithm using
DTCWT, NLM filter and SVD for Medical Image Enhancement and had been tested
on a set of medical images. In their method, firstly, The medical input image was
decomposed using DTCWT. Less artifacts were generated with the help of DTCWT
compared to that of DWT because of nearly shift invariance characteristic of DTCWT.
Further image quality was improved using NLM filtering approach and SVD was used
for to get originality of image and obtain a better quality image both quantitatively
and qualitatively. Simulation results showed that proposed technique outperforms
other conventional techniques for improving visual quality of medical images for
proper manual interpretation and computer based diagnosis.
HemaLatha and Vardarajan [25] in 2017 presented a image resolution enhancement of LR image using the dualtree complex wavelet transform. In their method,
dual tree complex wavelet transform was applied to low resolution (LR) satellite
image. Further, the high resolution (HR) image was reconstructed from the low resolution image, together with a set of wavelet coefficients, using the inverse DT-CWT.
Finally, the inverse dual tree complex transform was taken. Output was high resolution image and the DT-CWT had better performance in terms of PSNR, RMSE, CC
and SSIM compared to DWT technique.
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6.5.4 Image Enhancement Using Curvelet Transform
Wavelet bases present some limitations, because they are not well adapted to the
detection of highly anisotropic elements, such as alignments in an image, or sheets
in a cube. Recently, other multiscale systems have been developed, which include
in particular ridgelets and curvelets and these are very different from wavelet-like
systems. Curvelets and ridgelets take the form of basis elements which exhibit very
high directional sensitivity and are highly anisotropic. The curvelet transform uses
the ridgelet transform in its digital implementation. The Curvelet Transform is based,
on decomposing the image into different scales, then partitioning into squares whose
sizes based on the corresponding scale.
Since edges play a fundamental role in image understanding, one good way to
enhance the contrast is to enhance the edges. The curvelet transform represents
edges better than wavelets, and is therefore well-suited for multiscale edge enhancement. Enhancement of clinical image research based on Curvelet has been developed
rapidly in recent years.
Starck et al. [72] in 2003 provided a new method for contrast enhancement based
on the curvelet transform and compared this approach with enhancement based on
the wavelet transform, and the Multiscale Retinex. They found that curvelet based
enhancement out-performed other enhancement methods on noisy images, but on
noiseless or near noiseless images curvelet based enhancement was not remarkably
better than wavelet based enhancement.
Ren and Yang [64] in 2012 proposed a new method for color microscopic image
enhancement based on curvelet transform via soft shrinkage and the saturation adjustment. They presented a new curvelet soft thresholding method called modulus square
function, which modifies the high frequency curvelet coefficients of the V component. The experimental results showed that the method consistently produce the
satisfactory result for the V component degraded by Random noise, Gaussian noise,
Speckle noise, and Poisson noise, and the S component was adjusted to render the
microscopic image colorfulness by adaptive histogram equalization. Hence, the proposed method was an efficient and reliable method for hue preserving and color
microscopic image enhancement.
Kumar [43] in 2015 proposed a new method to enhance the colour image based
on Discrete Curvelet Transform (DCT) and multi structure decomposition. Experimental results showed that this method provided better qualitative and quantitative
results.
Abdullah et al. [1] in 2016 proposed an efficient method to enhance low contrast
in gray image based on fast discrete curvelet transform via unequally spaced fast
Fourier transform (FDCT-USFFT). Results showed that the proposed technique was
computationally efficient, with the same level of the contrast enhancement performance and proposed technique was better than histogram equalization and wavelet
transform in image quality.
Abdullah et al. [2] in 2017 proposed a new method for contrast enhancement gray
images based on Fast Discrete Curvelet Transform via Wrapping (FDCT-Wrap).
6.5 Application of Multiscale Transforms in Image Enhancement
223
Experimental results showed that the proposed technique gave very good results
in comparison to the histogram equalization and wavelet transform based contrast
enhancement method.
Farzam and Rastgarpour [18] in 2017 presented a method for image contrast
enhancement for cone beam CT (CBCT) images based on fast discrete curvelet
transforms (FDCT) that work through Unequally Spaced Fast Fourier Transform
(USFFT). Their proposed method first used a two-dimensional mathematical transform, namely the FDCT through unequal-space fast Fourier transform on input image
and then applied thresholding on coefficients of Curvelet to enhance the CBCT
images. Consequently, applying unequal-space fast Fourier Transform lead to an
accurate reconstruction of the image with high resolution. The experimental results
indicated the performance of the proposed method was superior to the existing ones in
terms of Peak Signal to Noise Ratio (PSNR) and Effective Measure of Enhancement
(EME).
6.5.5 Image Enhancement Using Contourlet Transform
The wavelet transform may not be the best choice for the contrast enhancement
of natural images. This observation is based on the fact that wavelets are blind to
the smoothness along the edges commonly found in images. Thus, there must be
a new multiresolution approach which is more flexible and efficient in capturing
the smoothness over the edges of the images should be used in image enhancement
applications.
The enhancement approach which is proposed by Starck et al. [72] in 2003, based
on the curvelet transform domain, is a modified version of the Veldes [81] algorithm.
This approach successfully removes the noise, as the noise are not parts of structural
information of the image, and the curvelet transform will not generate coefficients for
the noise. But the curvelet transform is defined in the polar coordinate which makes
it difficult to translate it back to the Cartesian coordinate. Analyzing these problems
Do and Vetterli in 2005 proposed another method called the contourlet transform.
The contourlet framework provides an opportunity to achieve the tasks of capturing important features of the image and is defined in Cartesian coordinates. It
provides multiple resolution representations of an image, each of which highlights
scale-specific image features. Since features in those contourlet transformed components remain localized in space, many spatial domain image enhancement techniques
can be adopted for the contourlet domain. For high dynamic range and low contrast
images, there is a large improvement by using contourlet transform-based image
enhancement since it can detect the contours and edges quite adequately.
Literature dictates that contourlet transform has better performance in representing
the image salient features such as edges, lines, curves, and contours than wavelets for
its anisotropy and directionality and is therefore well suited for multiscale edge-based
image enhancement.
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The evaluation of retinal images is widely used to help doctors diagnose many
diseases, such as diabetes or hypertension. Due to the acquisition process, retinal
images often have low grey level contrast and dynamic range. This problem may
seriously affect the diagnostic procedure and its results.
Feng et al. [20] in 2007 presented a new multi-scale method for retinal image contrast enhancement based on the Contourlet transform. Firstly they modify the Contourlet coefficients in corresponding subbands via a nonlinear function and took the
noise into account for more precise reconstruction and better visualization. Authors
compared this approach with enhancement based on the Wavelet transform, Histogram Equalization, Local Normalization and Linear Unsharp Masking and found
that the proposed approach outperformed other enhancement methods on low contrast and dynamic range images, with an encouraging improvement, and might be
helpful for vessel segmentation.
Melkamu et al. [50] in 2010 provided a new algorithm for image enhancement
by fusion based on Contourlet Transform. The experimental results showed that the
fusion algorithm gives encouraging results for both multi modal and multi focal
images.
AlSaif and Abdullah [3] in 2013 proposed a new approach for enhancing contrast
of color image based on contourlet transform and saturation components. In their
method, the color image was converted to HSI (hue, saturation, intensity) values.
The S, which represented the Saturation value of color image, decomposed to its
coefficients by non-sampling contourlet transform, then applying grey-level contrast
enhancement technique on some of the coefficients. Then, inverse contourlet transform was performed to reconstruct the enhanced S component. The I component
was enhanced by histogram equalization while the H component did not changed to
avoid degradation color balance between the HSI components. Finally the enhanced
S and I together with H were converted back to its original color system. The algorithm effectively enhanced the Contrast images especially the fuzzy ones with low
Contrast. At the same time, this method was easy one and a new approach to achieve
the later transformation on contrast enhancement.
Song [71] in 2013 proposed a useful image enhancement scheme based on nonsubsampled contourlet transform. Experimental results showed that the proposed
enhancement scheme was able to enhance the detail and increase the contrast of the
enhanced image at the same time.
Kaur and Singh [37] in 2014 proposed a new algorithm based on the firefly algorithm and the contourlet transform for sharpening of ultra sound images. To improve
the results of contourlet transform, authors implemented a new approach based on
firefly and it was providing very high percentage of image quality for ultra sound
images. The results demonstrated the improvement in the quality of the ultra sound
images to find the optimal solution and parameters were calculated which showed
that the proposed approach was performing better than the existing solutions.
Melkamu et al. [51] in 2015 presented a new image enhancement algorithm using
the important features of the contourlet transform. The results obtained are compared
with other enhancement algorithms based on wavelet transform, curvelet transform,
bandlet transform, histogram equalization (HE), and contrast limited adaptive his-
6.5 Application of Multiscale Transforms in Image Enhancement
225
togram equalization. The performance of the enhancement based on the contourlet
transform method was superior than the other methods.
Kaur and Aashima [36] in 2015 proposed a algorithm based on Contourlet transformation and BFOA for Medical image enhancement. Firstly, the contourlet utilized
LP and FB decomposition filter to divide the image into different segments and these
segments had been enhanced independently using different contourlet transformation
equations. After the implementation of contourlet transformation the BFO algorithm
was implemented for optimization process. This is a nature inspired approach that
have been used or optimization on the basis of bacterias. After these steps various
parameters had been analyzed for validation of purposed work and different parameters were introduced and on the basis of these parameters their system provided them
better results.
6.5.6 Image Enhancement Using Shearlet Transform
About 20 years ago, the emergence of wavelets represented a milestone in the development of efficient encoding of piecewise regular signals. The wavelet basis function
is the optimal basis which represents a point singularity objective function, and has
good time-frequency locality, multi-resolution and sparse representation for a point
singularity piecewise smooth function. In the two-dimensional image, high dimensional singular curves such as edges, contours and textures, contain most of the
image information. Nonetheless, a two-dimensional wavelet frame spanned by two
one dimensional wavelets can just describe the location of singular points in the
image. In the support region, the wavelet basis function only has horizontal, vertical
and diagonal direction, and its shape is isotropic square. In fact, the wavelet frame is
optimal for approximating data with point-wise singularities only and cannot equally
well handle distributed singularities such as singularities along curves. Therefore,
the wavelet frame, lacking of direction and anisotropy, is hard to sparsely represent
high dimensional singular characteristics like edges and textures.
In order to overcome the classical wavelet frame defects, scholars propose a variety
of multi-scale geometric analysis methods, in light of the characteristics of visual
cortex receiving outside scene information. Notable methods include the curvelet
and the contourlet. The curvelet is the first and so far the only construction providing
an essentially optimal approximation property for 2D piecewise smooth functions
with discontinuities along C 2 curves. However, the curvelet is not generated from the
action of a finite family of operators on a single function, as is the case with wavelets.
This means its construction is not associated with a multiresolution analysis. This and
other issues make the discrete implementation of the curvelet very challenging, as is
evident by the fact that two different implementations of it have been suggested by the
originators. In an attempt to provide a better discrete implementation of the curvelet,
the contourlet representation has been recently introduced. This is a discrete timedomain construction, which is designed to achieve essentially the same frequency
tiling as the curvelet representation. But the directional sub-bands of contourlet exists
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spectrum aliasing. This leads to high similarity between the high frequency sub-band
coefficients. However, both of them do not exhibit the same advantages the wavelet
has, namely a unified treatment of the continuum and digital situation, a simple
structure such as certain operators applied to very few generating functions, and a
theory for compactly supported systems to guarantee high spatial localization.
The shearlet frame was described in 2005 by Labate, Lim, Kutnyiok and Weiss
with the goal of constructing systems of basis-functions nicely suited for representing
anisotropic features (e.g. curvilinear singularities). Many scholars put forward image
enhancement algorithms based on the shearlet frame for different types of images,
such as medical, cultural relic, remote sensing, infrared and ultrasound images. The
common idea is decomposing the original image into low frequency coefficients and
high frequency sub-band coefficients of various scales and directions in the shearlet
domain, and then enlarging or reducing these two components according to the aim
of enhancement. These methods enlarge high frequency coefficients include hard
thresholding and soft thresholding. Meanwhile, there is fuzzy contrast enhancement
to deal with low frequency coefficients.
The advantage of the Shearlet frame, in particular, is to provide a unique ability
to control the geometric information associated with multidimensional data. Thus,
the Shearlet appears to be particularly promising as a tool, enhancing the component
of the 2D data associated with the weak edges.
Wang and Zhu [83] in 2014 proposed a single image dehazing algorithms to
improve the contrast of the foggy images based on shearlet Transform. Firstly, algorithm executed shearlet transforms for foggy images, got low frequency coefficients
and high frequency of in all directions and scale factor, then executed fuzzy contrast
enhancement for Low-frequency coefficients, then executed fuzzy enhancement for
high-frequency coefficients of different scales in different directions. Finally, applied
shearlet inverse transform for low-frequency coefficients and high frequency coefficients of treated. Experimental results showed that this algorithm can effectively
improve the visual effect of the foggy images, and enhance the contrast of the foggy
images.
Premkumar and Parthasarathi [60] in 2014 proposed an efficient approach for
colour image enhancement using discrete shearlet transform. They proposed a novel
method for image contrast enhancement based on Discrete Shearlet Transform (DST)
for colour images. In order to obtain high contrast enhancement image, the RGB
image was first converted into HSV (Hue, Saturation and Value) color space. The
converted hue color channel was only taken into the account for DST decomposition.
After that higher sub bands of hue component were eradicated and lower sub bands
were only considered for reconstruction. Finally, high contrast image was obtained
by using reconstructed Hue for HSV color space and then it was converted to RGB
color space.
Wubuli et al. [84] in 2014 proposed a novel enhancement algorithm for medical image processing based on Shearlet transform and unsharp masking. In their
method, Firstly, histogram equalization was applied to the medical image, then, the
medical image was decomposed into low frequency component and high frequency
component using shearlet transform. Next, the adaptive threshold denoising and
6.5 Application of Multiscale Transforms in Image Enhancement
227
linear enhancement was applied to the high frequency components while the low
frequency components were not processed. Finally, the coefficients were increased
through unsharp masking algorithm behind the Shearlet inverse transform process.
The benchmark results for this algorithms showed that the proposed method could
significantly enhance the medical images and thus improve the image qualities.
Sivasankari et al. [70] in 2014 introduced the effective speckle reduction of SAR
images based on a new approach of Discrete Shearlet Transform with Bayes Shrinkage Thresholding. The combined effect of soft thresholding in shearlet transform
worked better when compared to the other spatial domain filter and transforms and
it also performed better in the curvilinear features of SAR images.
Wang et al. [82] in 2015 proposed an image enhancement algorithm based on
Shearlet transform. In their method, the image was decomposed into low frequency
components and high frequency components by Shearlet transform. Firstly, MultiScale Retinex (MSR) was used to enhance the low frequency components of Shearlet decomposition to remove the effect of illumination on image then the threshold
denoising was used to suppress noise at high frequency coefficients of each scale.
Finally, the fuzzy contrast enhancement method was used to the reconstruction image
to improve the overall contrast of image. The experimental results showed that proposed algorithm provided significantly improvement in the image visual effect, and
it had more image texture details and anti-noise capabilities.
Pawade and Gaikwad [57] in 2016 implemented a novel method for image contrast
enhancement which includes enhancement in both Intensity/value using discrete
cosine transform and hue components using discrete Shearlet transform of HSV
colour image simultaneously. The results showed that the perceptibility of an image
was increased.
Fan et al. [17] in 2016 proposed a novel infrared image enhancement algorithm
based on the shearlet transform domain to improve the image contrast and adaptively
enhance image structures, such as edges and details. Experimental results showed that
the proposed algorithm could enhance the infrared image details well and produced
few noise regions, which was very helpful for target detection and recognition.
Tong and Chen [78] in 2017 presented a multi-scale image adaptive enhancement
algorithm for image sensors in wireless sensor networks based on non-subsampled
shearlet transform. The performance of the proposed algorithm was evaluated both
objectively and subjectively and the results showed that the visibility of the images
was enhanced significantly.
Favorskayaa and Savchinaa [19] in 2017 investigated a process of dental image
watermarking based on discrete shearlet transform. The proposed watermarking technique was tested on 40 dental gray scale images with various resolution. The experiments showed the highest robustness to the rotations and proportional scaling and
the medium robustness to the translations and JPEG compression. The SSIM estimators were found high for the rotation and scaling distortions that showed good HVS
properties.
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Appendix A
Real and Complex Number System
The set of natural numbers {1, 2, 3, 4, . . .} will be denoted by N, the set of integers
{. . . , −3, −2, −1, 0, 1, 2, 3, . . .} by Z, the set of rational numbers { xy ; y = 0} by
Q, the set of irrational numbers by Q , and union of rational & irrational numbers
√
shall be denoted by R. The set of complex numbers {a + ib, a, b ∈ R, i = −1}
will be denoted by C.
Field. A pair (F, +, ·), where F is a set with operations + (addition) and · (multiplication), is called a field if it satisfies the following properties:
For Addition:
1. (Closure) x + y ∈ F, for all x, y ∈ F.
2. (Commutativity) x + y = y + x, for all x, y ∈ F.
3. (Associativity) x + (y + z) = (x + y) + z, for all x, y, z ∈ F.
4. (Additive identity) There exists an element 0, in F such that x + 0 =
x, for all x ∈ F.
5. (Additive Inverse) There exists an element −x, in F such that x + (−x) =
0, for all x ∈ F.
For Multiplication:
1. (Closure) x · y ∈ F, for all x, y ∈ F.
2. (Commutativity) x · y = y · x, for all x, y ∈ F.
3. (Associativity) x · (y · z) = (x · y) · z, for all x, y, z ∈ F.
4. (Multiplicative identity) There exists an element 1, in F such that x · 1 =
x, for all x ∈ F.
5. (Multiplicative Inverse) There exists an element x1 ; x = 0 , in F such that
x · x1 = 1, for all x ∈ F.
Distributive Law:
x · (y + z) = (x · y) + (x · z) for all x, y, z ∈ F.
Clearly, (R, +, ·) and (C, +, ·), form a field.
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234
Appendix A: Real and Complex Number System
Ordered Field.
An ordered field is a field F with a relation < satisfying the following properties:
1. (Comparison) If x, y ∈ F, then the one of the following holds: x < y, y <
x, x = y.
2. (Transitivity) If x, y, z ∈ F, with x < y and y < z then x < z.
3. If x, y, z ∈ F, and y < z, then x + y < x + z.
4. If x, y ∈ F and x, y > 0, then x · y > 0.
Clearly Q and R with the usual relation < form an ordered field.
Triangle inequality in R. If x, y ∈ R, then
|x + y| ≤ |x| + |y|.
We interpret |x − y| as the distance between the point x and y in R.
Convergence in R. Let M ∈ Z and x ∈ R. A sequence {xn }∞
n=M of real numbers
converges to x ∈ R, if for all > 0, there exists N ∈ N such that |xn − x| < for
all n > N .
Cauchy Sequence in R. A sequence {xn }∞
n=M of real numbers is a Cauchy sequence
if, if for all > 0, there exists N ∈ N such that |xn − xm | < for all n, m > N .
Cauchy Criterion for convergence of a Sequence in R. Every Cauchy sequence
of real numbers converges.
The converse of Cauchy criterion is also true in R. Hence R with the usual addition and multiplication forms a complete ordered field.
The set of complex numbers C also form a complete field but not an ordered field.
We denote the elements of C in the usual way,
z = x + i y, where x, y ∈ R.
We call x the real part of z, denoted by Re(z) and y the imaginary part of z by I m(z),
respectively.
Define the complex conjugate z of z by
z = x − i y.
The modulus squared of z by
|z|2 = zz = (x + i y)(x − i y) = x 2 + y 2
Appendix A: Real and Complex Number System
235
and the modulus or magnitude |z| of z by
|z| =
|z|2 = x 2 + y 2
Properties of Complex Numbers. Suppose z, w ∈ Z. Then
(i). z = z.
, I m(z) = z−z
.
(ii). Re(z) = z+z
2
2i
(iii). z + w = z + w z · w = z · w.
(iv). |z| = |z|, |zw| = |z||w|.
(v). |Re(z)| ≤ |z|, |I m(z)| ≤ |z|.
(vi). |z + w| ≤ |z| + |w|. (Triangle inequality in C)
(vii). wz = wz , w = 0.
|z|
(viii). wz = |w|
.
Convergence in C. Let M ∈ Z and z ∈ C. A sequence {z n }∞
n=M of complex numbers
converges to z ∈ C, if for all > 0, there exists N ∈ N such that |z n − z| < for all
n > N.
Here |z − w| denotes the distance between two points z and w in the complex
plane. For example, if z = x1 + i y1 and w = x2 + i y2 , then
|z − w| = |(x1 − x2 ) + i(y1 − y2 )| =
(x1 − x2 )2 + (y1 − y2 )2 ,
which is the same as the usual distance in R2 between two points (x1 , y1 ) and (x2 , y2 ).
Cauchy Sequence in C. A sequence {z n }∞
n=M of complex numbers is a Cauchy
sequence if, if for all > 0, there exists N ∈ N such that |z n − z m | < for all
n, m > N .
This leads to the Cauchy criterion for the convergence of a sequence of complex
numbers.
Completeness of C. A sequence of complex numbers converges if and only if it is
a Cauchy sequence.
Appendix B
Vector Space
Vector Space Let F be a field. A vector space V over a field F is a set with operations
vector addition and scalar multiplication satisfying the following properties:
A. (i). u + v ∈ V for all u, v ∈ V.
(ii). u + v = v + u for all u, v ∈ V.
(iii). u + (v + w) = (u + v) + w for all u, v, w ∈ V.
(iv). There exist an element in V, denoted by 0, such that u + 0 = u for all u ∈ V.
(v). For each u ∈ V, there exist an element in V, denoted as −u, such that
u + (−u) = 0.
B. (i). α · u ∈ V, for all α ∈ F and u ∈ V.
(ii). 1 · u = u for all u ∈ V and 1 is the multiplicative identity of F.
(iii). α · (β · u) = (α · β) · u, for all α, β ∈ F and u ∈ V.
(iv). (a) α · (u + v) = α · u + α · v for all α ∈ F and u, v ∈ V.
(b) (α + β) · u = α · u + β · u for all α, β ∈ F and u ∈ V.
Linear Combination. Let V be a vector space over a field F, let n ∈ N, and let
v1 , v2 , . . . vn ∈ V. A linear combination of vectors v1 , v2 , . . . vn is any vector of the
form
n
αi vi = α1 v1 + α2 v2 + · · · + αn vn ,
i=1
where α1 , α2 , . . . , αn ∈ F.
Let V be a vector space over a field F, suppose U ⊆ V. The span of U is the set
of all linear combination of elements of U and is denoted by spanU.
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A. Vyas et al., Multiscale Transforms with Application to Image
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Appendix B: Vector Space
If U is a finite set, say, U = {u 1 , u 2 , . . . , u n }, then
spanU =
n
αi u i : αi ∈ F for all i = 1, 2, . . . , n .
i=1
Linear Dependence and Independence. Let V be a vector space over a field F and
let {v1 , v2 , . . . , vn } is linearly dependent if there exists α1 , α2 , . . . , αn ∈ F that are
not all zero, such that,
α1 v1 + α2 v2 + · · · + αn vn = 0.
We say that the set {v1 , v2 , . . . , vn } is linearly independent if
α1 v1 + α2 v2 + · · · + αn vn = 0,
holds only when αi = 0 for all i = 1, 2, . . . , n.
If U is an infinite subset of V, we say U is linearly independent if every finite
subset of U is linearly independent and we say U is linearly dependent if U has a
finite subset that is linearly dependent.
Basis. Let V be a vector space over a field F. A subset U of V is a basis for V if U
is a linearly independent set such that spanU = V.
Euclidean Basis for Rn or Cn .
Define E = {e1 , e2 , . . . , en } by
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎛ ⎞
1
0
0
0
⎜0⎟
⎜1⎟
⎜0⎟
⎜0⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜0⎟
⎜0⎟
⎜1⎟
⎜0⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎟
⎟
⎟
⎟
⎜
⎜
⎜
e1 = ⎜ · ⎟ , e2 = ⎜ · ⎟ , e3 = ⎜ · ⎟ , · · ·, en = ⎜
⎜ · ⎟.
⎜·⎟
⎜·⎟
⎜·⎟
⎜·⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝·⎠
⎝·⎠
⎝·⎠
⎝·⎠
0
0
0
1
These vectors can be regarded as elements of Rn or Cn .
If a vector space V has a basis consisting of finitely many elements, we say that
V is finite dimensional vector space. In this case, any two basis for V have the same
number of elements.
Dimension of Vector Space V . Suppose V is a finite dimensional vector space. The
number of elements in a basis for V, denoted as, dimV . If dimV = n, we say that
V is n-dimensional vector space.
Appendix C
Linear Transformation, Matrices
Linear Transformation. Let U and V be two vector space over the same field F. A
linear transformation T is a function T : U → V having the following properties:
(i). T (u + v) = T (u) + T (v) for all u, v ∈ U.
(ii). T (αu) = αT (x) for all α ∈ F and u ∈ U.
Any linear transformation T : X → Y can be represented in basis for X and Y
by matrix multiplication.
Matrix. For m, n ∈ N, an m × n matrix A over a field F is a rectangular array of the
form
⎞
⎛
a11 a12 · · · a1n
⎜ a21 a22 · · · a2n ⎟
⎟
⎜
⎜ a31 a32 · · · a3n ⎟
⎟
⎜
⎟,
· · · · ··
A=⎜
⎟
⎜
⎟
⎜
·
·
·
·
·
·
⎟
⎜
⎠
⎝
· · · · ··
am1 am2 · · · amn
where ai j ∈ F for all i = 1, 2, . . . , m and j = 1, 2, . . . , n. We call ai j the (i, j)th
entry of A. We also denote A by [ai j ].
Note that an n×1 matrix is a vector with n-components i. e. an element of Rn of Cn .
Transpose of a Matrix. let A = [ai j ] be an m × n matrix over C. The transpose of
matrix A, denoted by At , is the n × m matrix B = [bi j ] defined by bi j = a ji for all
i, j.
or the transpose At is obtained by interchanging the rows and columns of matrix A.
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A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
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240
Appendix C: Linear Transformation, Matrices
Conjugate Transpose of a Matrix. let A = [ai j ] be an m × n matrix over C. The
conjugate transpose A∗ of matrix A is the n ×m matrix C = [ci j ] defined by ci j = a ji
for all i, j.
or the conjugate transpose A∗ is obtained by taking the complex conjugate of all the
entries of At .
Let A be an m × n matrix over C. Then
Az, w = z, A∗ w ,
for every z ∈ Cn and w ∈ Cm . Furthermore, A∗ is the only matrix with this property.
Unitary Matrix. Let A be an n × n matrix over C. Then A is called unitary matrix
if A is invertible and A−1 = A∗ .
For a matrix over the real numbers, the conjugate transpose is the same as the
transpose. So a real unitary matrix A is one that satisfies A−1 = A∗ ; such a matrix
is called orthogonal.
Let A be an n × n matrix over C. Then the following statements are equivalent:
(i). A is unitary.
(ii). The columns of A form an orthonormal basis for Cn .
(iii). The rows of A form an orthonormal basis for Cn .
(iv). Matrix A preserves inner products, that is, Az, Aw = z, w for all z,
w ∈ Cn .
(v). ||Az|| = ||z||, for all z ∈ Cn .
Appendix D
Inner Product Space and Orthonormal Basis
An inner product is a generalization of the dot product for vectors in Rn .
Inner Product. Let V be a vector space over C. A (complex) inner product is a map
·, · : V × V → C with the following properties:
(i). (Additivity) u + v, w = u, w + v, w for all u, v, w ∈ V.
(ii). (Scalar Homogeneity) αu, v = α u, v for all α ∈ C and u, v ∈ V.
(iii). (Conjugate Symmetry) u, v = v, u for all u, v ∈ V.
(iv). (Positive Definiteness) u, u ≥ 0 for all u ∈ V, and u, u = 0 if and only
if u = 0.
A vector space V with complex inner product is called a complex inner product
space.
Conditions (i) and (iii) imply that
u, v + w = u, v + u, w for all u, v, w ∈ V.
Conditions (ii) and (iii) imply that
u, αv = α u, v for all α ∈ C and for all u, v ∈ V.
An inner product always yields a norm in the following way:
Norm. Let V be a vector space over C with a complex inner product ·, · . For u ∈ V,
define
||u|| = u, u ,
is called norm of u and denoted by || · ||.
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
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242
Appendix D: Inner Product Space and Orthonormal Basis
Cauchy-Schwarz Inequality. Let V be a complex inner product space. Then for
any u, v ∈ V,
| u, u | ≤ ||u||||v||.
Triangle Inequality. Let V be a complex inner product space. Then for any u, v ∈ V,
||u + v|| ≤ ||u|| + ||v||.
Orthogonality. Suppose V is a complex inner product space. For u, v ∈ V, we say
that u and v are orthogonal, if u, v = 0. It is denoted by u ⊥ v.
Note that if u ⊥ v, then we have
||u + v||2 = ||u||2 + u, v + v, u + ||v||2 = ||u||2 + ||v||2 .
Orthogonal Set. Suppose V is a complex inner product space. Let B be a collection
of vectors in V. B is called orthogonal set if any two different elements of B are
orthogonal. B is called an orthonormal set if B is an orthogonal set and ||v|| = 1 for
all v ∈ B.
Orthogonal sets of nonzero vectors are linearly independent.
Suppose V is a complex inner product space. Suppose B is an orthogonal set of
vectors in V and 0 ∈
/ B. Then Bis a linearly independent set.
Suppose V is a complex inner product space and B = {u 1 , u 2 , . . . , u n } is an
orthogonal set in V with u j = 0 for all j. If v ∈ span(B), then
v=
n
v, u j
u j.
||u j ||2
j=1
If B is orthonormal set, then
v=
n
j=1
v, u j u j .
Appendix D: Inner Product Space and Orthonormal Basis
243
Orthogonal Projection. Suppose V is a complex inner product space and B =
{u 1 , u 2 , . . . , u n } is an orthogonal set in V with u j = 0 for all j. Let S = span(B).
For v ∈ V, define the orthogonal projection PS (v) of v on S by
PS (v) =
n
v, u j
u j.
||u j ||2
j=1
The orthogonal projection operator PS has the following properties:
(i). PS is a linear transformation.
(ii). For every v ∈ V, PS (v) ∈ S.
(iii). If s ∈ S, then PS (S) = S.
(iv). (Orthogonality Property) For v ∈ V and s ∈ S, (v − PS (v)) ⊥ S.
(v). (Best approximation property) For any v ∈ V and s ∈ S, ||v − PS (v)|| ≤
||v − s||, with equality if and only if S = PS (v).
Gram-Schmidt Procedure. Suppose V is a complex inner product space and
{u 1 , u 2 , . . . , u n } is a linearly independent set in V. Then there exists an orthonormal
set {v1 , v2 , . . . , vn } with the same span.
Suppose V is a complex inner product space. An orthonormal basis for V is an
orthonormal set in V that is also a basis.
Let V be a complex inner product space with finite orthonormal basis R =
{u 1 , u 2 , . . . , u n }.
(i). For any v ∈ V, v = nj=1 v, u j u j .
(ii). (Parseval’s Relation) For any v, w ∈ V, v, w = nj=1 v, u j w, u j .
(iii). (Plancherel’s Formula) For any v ∈ V, ||v||2 = nj=1 | v, u j |2 .
Appendix E
Functions and Convergence
E.1 Functions
Bounded (L ∞ ) Functions. A piecewise continuous function f (x) defined on an
interval I is bounded (or L ∞ ) on I if there is a number M > 0 such that | f (x)| ≤ M
for all x ∈ I.
The L ∞ -norm of a function f (x) is defined by
|| f ||∞ = sup{| f (x)| : x ∈ I }.
If I is a closed and finite interval, then any continuous function f (x) ∈ I is also
in L ∞ (I ).
Integrable (L 1 ) Functions. A piecewise continuous function f (x) defined on an
interval I is integrable (or L 1 ) on I if the integral
| f (x)|d x
I
is finite.
The L 1 -norm of a function f (x) is defined by
|| f ||1 =
| f (x)|d x.
I
If f (x) is L ∞ (I ), then f (x) is L 1 (I ). Any continuous function on a finite closed
interval I is L 1 (I ), since such a function must be on L ∞ (I ). Any piecewise continuous function with only jump discontinuous on a finite closed interval I is on L 1 (I ).
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
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246
Appendix E: Functions and Convergence
First Approximation Theorem. Let f (x) ∈ L 1 (R), and let > 0 be given. Then
there exists a number R such that if
g(x) = f (x)χ[−R,R] (x) =
Then
∞
−∞
f (x) i f x ∈ [−R, R],
0
if x ∈
/ [−R, R].
| f (x) − g(x)|d x = || f − g||1 < .
This theorem says that any function in L 1 (R) can be approximated arbitrarily
closely in the sense of the L 1 -norm by a function with compact support.
Square Integrable (L 2 ) Functions. A piecewise continuous function f (x) defined
on an interval I is square integrable (or L 2 ) on I if the integral
| f (x)|2 d x
I
is finite.
The L 2 -norm of a function f (x) is defined by
1/2
|| f ||2 =
| f (x)|2 d x
.
I
Any bounded function f (x) on a finite interval I is also in L 2 (I ). This includes
continuous functions on closed intervals and functions piecewise continuous on
closed intervals with only jump discontinuous. Any functions that is L ∞ (I ) and
L 1 (I ), (I may be finite or infinite) is also in L 2 (I ).
Cauchy-Schwarz Inequality. Let f (x) and g(x) be two functions on L 2 (I ). Then
1/2 1/2
2
2
f (x)g(x) ≤
| f (x)| d x
|g(x)| d x
.
I
I
I
Let I be a finite interval. Cauchy-Schwarz inequality implies that any function
f (x) ∈ L 2 (I ) is also in L 1 (I ). This is not true in case I is infinite interval. For
example, let
1/x i f x ≥ 1,
f (x) =
0 i f x < 1.
Appendix E: Functions and Convergence
247
Clearly, f (x) is in L 2 (R) but not in L 1 (R). Converse of the theorem is false for both
finite and infinite intervals.
Minkowski’s Inequality. Let f (x) and g(x) be two functions on L 2 (I ). Then
1/2
| f (x) + g(x)|2 d x
I
1/2
≤
| f (x)|2 d x
I
1/2
+
|g(x)|2 d x
.
I
It is also known as the triangle inequality for L 2 functions; that is, it says that
|| f + g||2 ≤ || f ||2 + ||g||2 .
Approximation Theorem for L 2 (R). Let f (x) ∈ L 2 (R), and let > 0 be given.
Then there exists a number R such that if
g(x) = f (x)χ[−R,R] (x) =
Then
∞
−∞
f (x) i f x ∈ [−R, R],
0
if x ∈
/ [−R, R].
| f (x) − g(x)|2 d x = || f − g||22 < .
Differentiable C n Functions. Given n ∈ N, we say that a function f (x) defined on
an interval I is C n on I if it is n-times continuously differentiable on I. Function
f (x) is C 0 on I means that f (x) is continuous on I. Function f (x) is C ∞ on I if it
is C n on I for every n ∈ N.
We say that a function f (x) defined on an interval I is Ccn on I if it is C n on I
and compactly supported, Cc0 on I means it is C 0 on I and compactly supported, and
Cc∞ on I if it is C ∞ on I and compactly supported.
E.2 Convergence of Functions
Numerical Convergence. The sequence {an }n∈N converges to the number a if for
every > 0, there is an N > 0 such that if n ≤ N , then |an − a| < .
A series ∞
n converges to a number S if the sequence of partial sums {S N } N ∈N ,
n=1 a
N
defined by, S N = n=1
an converges to S. In this case, we write ∞
n=1 an = S.
A series
∞
n=1
an converges absolutely if
∞
n=1
|an | converges.
248
Appendix E: Functions and Convergence
Pointwise Convergence. A sequence of functions { f n (x)}n∈N defined on an interval
I converges pointwise to a function f (x) if for each x0 ∈ I, the sequence { f n (x0 )}n∈N
converges to f (x0 ).
∞
The
∞series n=1 f n (x) = f (x) pointwise on an interval I if for each x0 ∈
I,
n=1 f n (x 0 ) = f (x 0 ).
Uniform (or L ∞ ) Convergence. A sequence of functions { f n (x)}n∈N defined on an
interval I converges uniformly (or L ∞ ) to the function f (x) if for each > 0, there
is an N > 0 such that if n ≥ N , then | f n (x) − f (x)| < for all x ∈ I.
(x) = f (x) uniformly on an interval I if the sequence of
The series ∞
n=1 f n
N
partial sums S N (x) = n=1
f n (x) converges uniformly to f (x) on I.
Theorem If f n (x) converges to f (x) uniformly (or L ∞ ) on an interval I, then f n (x)
converges to f (x) pointwise on I.
Theorem If f n (x) converges to f (x) uniformly (or L ∞ ) on an interval I, and if
each f n (x) is continuous on I, then f (x) is continuous on I.
Mean (or L 1 ) convergence. The sequence { f n (x)}n∈N defined on an interval I
converges in mean to the function f (x) on I if
| f n (x) − f (x)|d x = 0 or limn→∞ || f n − f ||1 = 0.
limn→∞
I
∞
The series
∞ n=1 f n (x) = f (x) in mean on I if the sequence of partial sums
S N (x) = n=1 f n (x) converges in mean to f (x) on I.
Theorem If f n (x) converges to f (x) uniformly (or L ∞ ) on a finite interval I, then
f n (x) converges to f (x) in mean or L 1 on I.
Mean-square (or L 2 ) convergence. The sequence { f n (x)}n∈N defined on an interval
I converges in mean-square to the function f (x) on I if
| f n (x) − f (x)|2 d x = 0 or limn→∞ || f n − f ||1 = 0.
limn→∞
I
The series ∞
n=1 f n (x) = f (x) in mean-square on I if the sequence of partial
sums S N (x) = ∞
n=1 f n (x) converges in mean-square to f (x) on I.
Theorem If f n (x) converges to f (x) uniformly (or L ∞ ) on a finite interval I, then
f n (x) converges to f (x) in mean-square or L 2 on I.
Appendix E: Functions and Convergence
249
Theorem If f n (x) converges to f (x) in mean-square (or L 2 ) on a finite interval I,
then f n (x) converges to f (x) in mean or L 1 on I.
The following theorem gives several conditions under which interchanging the
limit and the integral is permitted.
Theorem
a. If f n (x) converges to f(x) on L 1 (I ), then
f n (x)d x =
limn→∞
f (x)d x.
I
I
b. If f n (x) converges to f(x) on L ∞ (I ), where I is the finite interval, then
f n (x)d x =
limn→∞
f (x)d x.
I
I
c. If f n (x) converges to f(x) on L 2 (I ), where I is the finite interval, then
f n (x)d x =
limn→∞
f (x)d x.
I
I
If I is an infinite interval, then the conclusions of Theorem (b) and (c) are false.
However, in case of infinite intervals, we have the following theorem by making an
additional assumption on the sequence { f n (x)}n∈N :
Theorem Suppose the for every R > 0, f n (x) converges to f (x) in L ∞ or in L 2 on
[−R, R]. That is for each R > 0,
R
limn→∞
−R
| f n (x)− f (x)|2 d x = 0 or limn→∞ sup{| f n (x)− f (x)| : x ∈ [−R, R]} = 0.
If f (x) is L 1 (I ) and if there is a function g(x) on L 1 (I ) such that for all x ∈ I
and for all n ∈ N, | f n (x)| ≤ g(x), then
f n (x)d x =
limn→∞
I
f (x)d x.
I
Index
A
Adaptive filters, 146
Additive noise, 137
Admissible shearlet, 114, 115
Alpha-trimmed mean filter, 146
Amplifier noise, 137
Amplitude, 17
Angular frequency, 16, 17
Application of multiscale transforms, 154,
210
Approximation adjoint, 60, 74
Approximation coefficient, 54, 63
Approximation operator, 48, 60, 71, 74
Arithmetic mean filter, 143
Auto white balance, 5
B
Band-limited wavelet, 48, 97
Basis, 70
Bayer pattern, 3
Biorthogonal, 70–72
Biorthogonal QMF, 76
Biorthogonal wavelet, 70
Bivariate ridgelet, 94
Butterworth LowPass filter, 206
C
Classical shearlet, 115, 116
Color Filter Array (CFA), 3
Colour image enhancement, 209
Compactly supported shearlets, 121
Compactly supported wavelet, 48, 96, 97
Complex Fourier series, 19
Complex-valued scaling function, 78
Complex-valued wavelet, 78
Complex wavelet transform, 77, 78
Cone-adapted continuous shearlet transform, 117
Cone-adapted discrete shearlet transform,
118, 119, 121
Conjugate symmetry, 24, 33
Constraint least-square filter, 151
Continuous Ridgelet transform, 94, 96, 98,
101
Continuous shearlet, 114
Continuous shearlet system, 113, 117
Continuous shearlet transform, 114
Continuous wavelet transform, 68, 69, 114
Contourlet, 106
Contourlet filter bank, 109, 111
Contourlet transform, 106, 107, 111
Contra-harmonic mean filter, 145
Contrast enhancement, 200, 202
Contrast stretching, 202, 203
Convolution, 28
Convolution theorem, 24, 32
CurveletG1, 102, 103
CurveletG1 transform, 102
CurveletG2, 102–105
Curvelets, 101
Curvelet transform, 102, 106, 107
D
Defocus blur, 141
Demosaicing, 5
Detail adjoint, 60, 74
Detail coefficient, 54, 63
© Springer Nature Singapore Pte Ltd. 2018
A. Vyas et al., Multiscale Transforms with Application to Image
Processing, Signals and Communication Technology,
https://doi.org/10.1007/978-981-10-7272-7
251
252
Detail operator, 48, 60, 71, 74
Differentiation property, 24
Digital image, 3
Digital image processing, 3
Dilation operator, 112
Direct inverse filtering, 151
Directional decomposition, 108
Discrete complex wavelet transform, 79
Discrete contourlet transform, 106, 109–111
Discrete cosine transform, 38, 39
Discrete curveletG1, 102
Discrete curveletG2, 102
Discrete Fourier transform, 25, 26, 29–31,
38, 97
Discrete quaternion wavelet transform, 88
Discrete Ridgelet transform, 97, 98
Discrete shearlet system, 115
Discrete shearlet transform, 116
Discrete sine transform, 38
Discrete wavelet transform, 54, 55, 57, 62,
63, 65, 67
Downsampling operator, 60
Dual, 71
Dual basis, 70
Duality, 24
Dual MRA, 71, 72
Dual-tree complex wavelet transform, 79–82
Dual wavelet, 72
Dyadic wavelet, 47
E
Efficiency, 65
Euclidean basis, 25
Euler identities, 19
Exponential noise, 139
F
Fast Fourier transform, 28, 33, 34, 36, 37, 64
Fast Slant Stack Ridgelet Transform, 100
Fast wavelet transform, 62
First-generation curvelet transform, 102
Formals adjoints, 60
Fourier coefficients, 18, 20
Fourier integral, 21
Fourier inversion formula, 27, 29
Fourier series, 18–21, 53
Fourier transform, 20–23, 38, 40, 41, 46, 47,
49, 51, 77, 78, 83, 97, 99, 100, 106
Frame, 112
Frequency domain filtering, 146
Index
G
Gaussian blur, 140
Gaussian distribution, 137
Gaussian highpass filter, 208
Gaussian lowpass filter, 206
Geometric mean filter, 143
Gridding in tomography, 96
H
Harmonic analysis, 18
Harmonic mean filter, 144
Heisenberg uncertainty principle, 40
Hilbert transform, 78, 80, 84, 86
Histogram equalization, 203
Histogram processing, 203
Homomorphic filtering, 208
I
Ideal highpass filter, 207
Ideal lowpass filter, 205
Image deblurring, 140
Image degradation, 133
Image denoising, 136
Image enhancement, 199, 200
Image negative, 201
Image restoration, 133
Image restoration algorithms, 142
Image Signal Processing (ISP), 3
Impulse train, 5
Impulsive noise, 138
Inverse continuous wavelet transform, 68
Inverse discrete cosine transform, 38
Inverse discrete Fourier transform, 29
Inverse discrete wavelet transform, 62, 68
Inverse dual-tree complex wavelet transform, 80
Inverse fast wavelet transform, 64
Inverse Fourier transform, 23, 47, 96–98
Inverse Radon transform, 97
Inversion formula, 22
Iterative blind deconvolution, 152
L
Laplacian pyramid, 107
Linear filters, 147
Linearity, 23, 32
Localization, 65
Local radon transform, 109
Local Ridgelet transform, 101, 102
Log transformation, 201
LPA-ICI algorithm, 153
Index
LP decomposition, 107
Lucy-Richardson algorithm, 154
M
Max and min filters, 145
Mean filters, 142
Median filter, 145
Midpoint filter, 145
Motion blur, 141
MRA wavelet, 49
Multiplicative noise, 137
Multiresolution analysis, 48, 49, 55, 62
Multiscale decomposition, 107
Multiscale transforms, 150
N
NAS-RIF, 152
Neural network approach, 154
Noise models, 136
O
Order-statistic filter, 145
Orthonormal basis, 26, 47–49, 52, 98
Orthonormal finite ridgelet transform, 100
P
Parseval frame, 112
Parseval’s relation, 24, 27, 95
Perfect reconstruction, 60
Period, 16, 17, 20, 29
Periodic function, 16, 18, 20, 29, 50
Phase, 17
Plancherel theorem, 27, 99
Power-Law transformation, 202
Pyramidal directional filter bank, 109
Q
Quadrature mirror filter, 57, 61
Quantization, 5
Quaternion algebra, 84, 85
Quaternion Fourier transform, 84
Quaternion multiresolution analysis, 88
Quaternion wavelet, 83, 84, 86–88
Quaternion wavelet scale function, 87
Quaternion wavelet transform, 83, 88
R
Radon domain, 95
253
Radonlets, 109
Radon transform, 95, 96, 100
Rayleigh noise, 138
Rectangular blur, 141
RectoPolar Ridgelet transform, 96
Refinement equation, 49, 55, 62
Restoration process, 133
Reversal, 24
Ridgelet, 95
Ridgelet transform, 94, 96, 97, 100, 101, 103
S
Sampling, 5
Scaling, 23
Scaling coefficient, 54
Scaling filter, 49, 57, 61, 72, 74
Scaling function, 48, 55, 71, 72, 74, 97
Scaling relation, 49
Second-generation curvelet transform, 102,
103
Separability, 31, 32
Sharpening filters, 207
Shearing matrix, 113
Shearing operator, 113
Shearlet, 111
Shearlet generators, 120
Shearlet group, 113, 115
Shearlet scaling function, 120
Shearlet transform, 114
Shifting, 23, 33
Shift operator, 60
Short-time Fourier transform, 40
Signal-to-noise ratio, 5
Similarity, 31
Sinusoidal function, 17
Smoothing filters, 205
Sparse representation, 101, 103, 105, 123
Sparsity, 65
Spatial filtering, 142, 204
Spatial-frequency filtering, 147
Speckle noise, 139
Stationary wavelet transform, 69, 70
Structure extraction, 64
Superresolution, 141
Superresolution restoration algorithm, 152
T
Threshold filters, 147
Tight frame, 112
Translation operator, 112
254
U
Undecimated wavelet transform, 69
Uniform noise, 138
Upsampling operator, 60
W
Wavelet, 46–49, 51, 53–55, 65, 69, 94, 95
Wavelet coefficient, 54
Index
Wavelet coefficient model, 149
Wavelet domain, 147
Wavelet filter, 53, 57, 61, 74
Wavelet series, 53, 54
Wavelet transform, 53, 64, 68, 83, 96, 97,
106, 111
Wiener filter, 146
Windowed Fourier transform, 40, 69
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