International Journal of Engineering Trends and Technology (IJETT) – Volume 22 Number 10 - April 2015
Image De-noising using Double Density Discrete
Wavelet Transform& Median Filtering
NARAYAN DEV GUPTA1, DEVANAND BHONSLE2
1
ME Student, Department of ET&T, SSCET
Bhilai, India
2
Senior Assistant Professor, Department of EEE, SSCET
Bhilai, India
Abstract— Noise affects images during transmission or
processing. The quality of image is affected by noise so to
retain the quality of image an operation is performed that
is called de-noising. De-noising basically deals with the
removal of noise from noise contaminated version of image.
There are different method of image de-noising depend
upon the noise that is present in the image. This paper
presents image de-noising method that is based on double
density discrete wavelet transform and median filtering. In
this paper we presents different types of noise that can
affect the image .When image is corrupted with Gaussian
noise then double density discrete wavelet transform
outperform the best and when image is corrupted with
salt and paper noise then median filtering approach gives
better result. In this proposed algorithm a new hybrid
method is used to remove mixed noise and proposed
method gives better result in terms of MSE, PSNR and
SSIM.
functions which are not localized in time or space the edge
information spread across frequencies. The FFT based denoising results in smearing of the edges but localized nature of
wavelet transform in time and space gives us de-noising with
edge preservation.
Keywords— Image De-noising, Image Noise, Median
Filter, Wavelet Domain De-noising, Image Decomposition,
Wavelet Threshold
Where, si(x, y) is the original image intensity and no(x, y)
denotes the noise introduced to produce the corrupted signal
z(x, y) at (x, y) pixel location. Noise signal gets multiplied to
the original signal. The multiplicative noise model follows the
following rule:
I. INTRODUCTION
The critical issue in image processing is restoration of
image in noisy environment. When noise introduced in the
image then pixel value get altered from its actual value that do
not demonstrate true intensity of real scene. The process that
deals with the manipulation of image data to produce high
class image is called image de-noising. The image de-noising
method can be classified in two categories that is spatial
filtering method and transform domain method .In spatial
filtering method we use a low pass filter on group of pixel in
the image with pre-emption that noise affects higher
frequency band of the spectrum. The spatial domain filtering
method removes noise from the image but it also blur the
image due to which edges became invisible. While the highpass filters can make edges even sharper and enhance the
spatial resolution but will also amplify.
There is a trade-off between the signal to noise ratio and
spatial resolution of signal/image processed when we used
Fourier transform domain filtering. When we used fast fourier
transform for de-noising then edges in the de-noised image is
not as sharp as in the original image. Because of the FFT basis
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II DIFFERENT TYPES OF NOISE IN IMAGE
Any deterioration in the image signal due to any external
disturbance is noise. Noise in the image occurs from different
sources. It can affect image spatial resolution, details of the
image and can produce distortion of important image features.
Noise present in image can be modeled either as an additive
noise or as a multiplicative noise. The noise signal that get
added to the image is called additive noise and can be
represented as
z(x, y) = si(x, y) + no(x, y)
z(x, y) = si(x, y) × no(x, y)
The different types of Noise are following

Additive White Gaussian noise

Salt-and-pepper noise

Speckle Noise
A. Additive White Gaussian noise
Most of the time during transmission or acquisition when an
image is corrupted with the channel noise generally we
consider it as additive white Gaussian noise. Gaussian noise
is equally distributed over the signal. This implies that each
pixel in the corrupted image is the sum of the actual pixel
value and a random Gaussian distributed noise value. This
noise follows a Gaussian distribution, which has a bell shaped
probability distribution function given by
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International Journal of Engineering Trends and Technology (IJETT) – Volume 22 Number 10 - April 2015
A. Median Filter
Where g shows the gray level, m is the mean or average of the
function and σ is the standard deviation of the noise.
B. Salt-and-pepper noise
Salt and pepper noise is also known as impulse type of
noise. This is originating generally because of errors in data
transmission. It has only two probable values a and b. The
expectation of occurring of each is typically less than0.1. The
altered pixels are placed alternatively to the minimum or to
the maximum value, giving the image a presence of “salt and
pepper” noise. Unaltered pixels remain constant. If we
consider an 8-bit image, the normal value for pepper noise is 0
and for salt noise 255. The salt and pepper noise is normally
produced due to defect or fault in functioning of pixel
elements in the camera sensors, faulty memory locations, or
timing errors in the different digitization method used.
The median filter comes under the class of nonlinear filter
generally used to eliminate noise. Such noise elimination is a
normal pre-processing step to enhance the results of
subsequent processing (for example, edge detection on an
image).As median filter preserves edges while removing noise
from images it is generally used in digital image processing. It
is also known as a rank filter, this spatial filter suppresses
isolated noise by replacing each pixel‟s intensity by the
median of the intensities of the pixels in its neighbourhood.
Median filter is used extensively in de-noising and image
smoothing applications. Due to the edge-keeping feature of
median filter (cf. linear methods such as average filtering
tends to blur edges), which is very advantageous for many
image processing application as edges carry significant
information for labelling, segmenting and keeping details in
image. This filter may be represented by Eq. (1)
........1
C. Speckle Noise
Speckle noise comes under the category of multiplicative
noise unlike the Gaussian or Salt and Pepper type noise.
Speckle noise is granular noise that naturally be present in and
reduces the feature of the active radar and synthetic aperture
radar (SAR) images. Speckle noise in traditional radar occurs
from random fluctuations in the return signal from an object
that is no larger than a single image-processing segment. It
extended the mean grey level of a local area. Speckle noise is
generated by signals from fundamental scatterers, the gravitycapillary ripples, and demonstrates as a pedestal image,
underneath the image of the sea waves. All coherent imaging
system suffers from this noise such as laser, acoustics and
SAR (Synthetic Aperture Radar) imagery. The origin of this
noise is made from random interference between the coherent
returns. Speckle noise pursues a gamma distribution and can
be given as [11].
Where variance is
andg is the gray level
III.DENOISING OF IMAGES BY FILTERING
When the images are processed through channels, they
are corrupted with impulse noise due to noisy channels. This
impulse noise comprise of large positive and negative spikes.
The positive spikes have values much larger than the
background and thus they appear as bright spots, while the
negative spikes have values smaller than the background and
they appear as darker spots. Both the spots for the Positive and
negative spikes are observable to the human eye. Gaussian
noise also affects the image. Thus, filters are required for
removing noises before processing.
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Where
Filter window with pixel ( ,) as its middle.
IV.WAVELET BASED DENOISING
The term „wavelet‟ indicates an oscillatory vanishing
wave with time-limited extend, which has the capability to
illustrate the time-frequency plane, with atoms of divergent
time supports. The wavelet transform provides a multi
resolution representation using a set of analyzing functions
that are dilations and translations of a few functions. It is
preferred to work in Wavelet domain because the Discrete
Wavelet Transform (DWT) make the signal energy
concentrate in a small number of coefficients, hence, the
DWT of the noisy image contains a small number of
coefficients having high Signal to Noise Ratio (SNR) while
comparatively huge number of coefficients is having low SNR.
After discarding the noisy sub band having low SNR the
image is reconstructed by using inverse DWT. Due to which
noise is throw out from observation[12]. The basic principle
of image de-noising using discrete wavelet transform consists
of following three steps [15].
1. Apply wavelet transform to the noisy image to produce the
noisy wavelet coefficients.
2. Select appropriate threshold limit at each level by using
threshold method to remove the noises.
3. Inverse wavelet transform is applied to threshold wavelet
coefficients to obtain a de-noised image.
V.DOUBLE DENSITY DISCRETE WAVELET
TRANSFORM
The Undecimated DWT is exactly shift-invariant and for
de-noising, it performs substantially better than the critically
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sampled DWT. The Double-Density discrete wavelet
transform is based on a single scaling function and two
distinct wavelets where the two wavelets are designed to be
offset from one another by one half, the integer translates of
one wavelet fall midway between the integer translates of the
other wavelet. In this way, the Double-Density DWT
approximates the continuous wavelet transform (having more
wavelets than necessary gives a closer spacing between
adjacent wavelets within the same scale). The Double-Density
DWT is two-times expansive regardless of the number of
scales implemented-potentially much less than the Undecimated DWT. The Double-Density DWT has twice as
many coefficients as the critically sampled DWT. The double
density DWT is an improvement upon the critically sampled
DWT with important additional properties.
1. In double density discrete wavelet transform we use two
wavelet and one scaling function which are offset from one
another.
2. The double density DWT is over completing by a factor of
two.
3. Shift–invariant property exists in double density discrete
wavelet transform.
There is a difference between un-decimated discrete
wavelet transform and oversampled discrete wavelet
transforms. Shift-invariant property exists in un-decimated
discrete wavelet transforms and it has an expansion factor of
logN. It expands an N-sample data vector to N logN samples.
VI. IMAGE DECOMPOSITION
By using discrete wavelet transform an image can be
decomposed into sequence of different spatial resolution
images. If we consider N level decomposition of the image (2
D image) then it results in 3N+1 different frequency bands
denoted as, LL, LH, HL and HH.When we apply discrete
wavelet transform on the image due to the decomposition of
an image the original image is decomposed into four pieces
which is normally denoted as LL1, LH1, HL1 and HH1 as
shown in the fig.1a. When we again decomposed LL1 band
then it gives us four sub bands denoted as LL2, LH2, HL2 and
HH2 as shown in the fig 1 (b).
Fig. 1 Image decomposition by using DWT
VII.WAVELET THRESHOLD
VIII.METHODOLOGY
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For de-noising of the image, there are different
thresholding methods that can be used. The selection of the
thresholding technique is one of the important steps which
affect the quality of the image during de-noising of image.
Hard and soft thresholding is used in discrete wavelet
transform. The thresholding technique which is based on keep
and kill rule is called hard thresholding and thresholding
technique which is based on shrink or kill rule is called soft
thresholding. As soft thresholding gives more pleasant image
and also reduces abrupt sharp changes in the image as
compared to hard thresholding, soft thresholding is preferred
over hard thresholding. By default, hard thresholding is used
for compression and soft thresholding is used for de-noising in
MATLAB [13].
A. Threshold Selection Rules
During the selection of the threshold value in de-noising we
should select threshold value in such a manner that it gives us
maximum peak signal to noise ratio. If we select a small
threshold then it will pass noisy sub band of the image so we
get noisy image and if we select a large threshold it will make
more coefficients to zero, which results in blurring of the
image and we may loss some pixel value. Selection of
threshold can be divided into two parts that is non adaptive
threshold and adaptive threshold.
B. Non Adaptive Threshold
A thresholding method that depends only on a number of
data points is called visu shrink and it comes under the
category of non adaptive threshold. When number of pixel
reaches infinity, it gives us best performance in terms of mean
square error. As it depends on large number of pixel its
threshold value is quite large. It is suitable for only additive
noise because it cannot remove multiplicative noise. The
formula that is used for calculating threshold is given by
T=  2 log M
Where σ is the noise level and M is the length of the noisy
signal [1].
C. Adaptive Threshold
There are two types of adaptive threshold i.e. Sure Shrink
and Bayes Shrink. Sure Shrink derived from minimizing
Stein‟s Unbiased Risk Estimator, an estimate of MSE risk. It
is a combination of universal threshold and SURE threshold.
It is used for suppression of noise by thresholding the
empirical wavelet coefficient. The aim of Sure Shrink is to
minimize the mean square error. Sure shrink suppresses the
noise by thresholding the empirical wavelet coefficient [1].
The Bayes Shrink technique has been attracting attention
latterly as an algorithm for setting different thresholds for
every sub band. Here sub bands are frequency bands that
differ from each other in level and direction [14].
In our implementation we are trying to get original image
from noise contaminated version of image. The images taken
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are corrupted with two different types of noises. We will take
a portable network graphic format image available in matlab.
Gray scale conversion of the image is performed. Gray scale
conversion is done in order to make a single page matrix for
processing of data through digital filter structure and then the
intensity values in a gray scale image are equalized through
Histogram Equalization. Now noise is added to the image.
Here two different types of noise are added to the image that
is Gaussian noise and salt and pepper noise. When addition of
noise is completed then we will apply double density discrete
wavelet transform and then median filtering and try to get denoised image. In this proposed method we will use single
level decomposition of the image. The flow chart of proposed
method is shown in fig .2.
.
Input Image
used for image quality measurement. By subtracting the test
signal from the reference, and then computing the average
energy of the error signal. Let f = {fi |i = 1, 2… M} and g = {gi|
i=1, 2… M} be the original and the de-noised images
respectively.MSE may be defined by
MSE =
Where M is the number of elements in the image.
PSNR: This can be calculated by comparing two images one
is original image and other is distorted image. The PSNR has
been computed using the following formula
PSNR = 10 log10 (R2/MSE)
R is the maximum fluctuation in the input image data type.
RGB to gray scale conversion
X. EXPERIMENTAL RESULTS
The proposed image de-noising algorithm has been applied on
natural grayscale Lena images (512×512) that are
contaminated by additive Gaussian noise and salt and pepper
noise. The implementation of this work is performed in
MATLAB 7.8 software. Performance measure is shown in the
Table 1.The PSNR, MSE, SSIM value is calculated for
different value of noise variance and accordingly de-noised
image is shown in fig.3.
Histogram equalization
Set variance of noise
distribution
Table1. Performance Measure
Add noise
Compute forward Double density
DWT
Filter structure for cut-off
thresholding
Noise
variance
PSNR(db)
MSE
Correlation
0.01
31.30
48.15
0.87
0.79
0.05
31.18
49.55
0.80
0.72
0.10
31.08
50.66
0.74
0.76
Noisy Image
(Gaussian and
Salt and Pepper)
Compute Inverse Double Density
DWT
SSIM
De-noised Image using proposed method
Noise Variance=0.01
Median Filter
De-noised image
Fig.2. flow chart of proposed method
IX. PERFORMANCE PARAMETER
Noise Variance=0.05
Performance of proposed algorithm is evaluated by using
MSE (Mean Square error) and PSNR (Peak signal to noise
ratio). Mean square error is Simplest, and the most widely
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International Journal of Engineering Trends and Technology (IJETT) – Volume 22 Number 10 - April 2015
[6]. Mrs. C. Mythili and Dr. V. Kavitha “Efficient Technique for Color Image
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Noise Variance=0.10
[9]. S.Arivazhagan, S.Deivalakshmi, K.Kannan, “Performance Analysis of
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[10]. Mukesh C. Motwani, Mukesh C. Gadiya, Rakhi C. Motwani “survey of
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Fig.3 various de-noised Lena image using proposed method
XI.CONCLUSIONS
In this paper a new method is proposed by hybridizing
double density discrete wavelet transform and median filter to
deal with the mixed noise (Gaussian and salt and pepper
noise). The performance of proposed algorithm is tested at
different noise level and evaluated in terms of PSNR, MSE
and SSIM. The proposed algorithm gives us better noise
removal of mixed noise than individual algorithm on
particular noise.
ACKNOWLEDGMENT
[13]. Akhilesh Bijalwan, Aditya Goyal, Nidhi Sethi, “Wavelet Transform
Based Image De-noise Using Threshold Approaches”, International Journal of
Engineering and Advanced Technology (IJEAT), June 2012, Vol.1, Issue-5.
[14]. E.Jebamalar, Leavline, S.Sutha, D.Asir Antony Gnana Singh, “Wavelet
Domain Shrinkage Methods for Noise Removal in Images: A Compendium”,
International Journal of Computer Applications, November 2011, Vol. 33,
No. 10.
[15]. Sudipta Roy, Nidul Sinha & Asoke K. Sen,” A New Hybrid image denoising method,” International Journal of Information Technology and
Knowledge Management, Vol. 2, No. 2, P. 491-497,2010.
The authors are greatly indebted to the Department of
Electronics & Telecom, SSCET Bhilai, CSVTU, India for
providing necessary support and facilities that made this work
possible.
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