WAVELET TRANSFORM
IN
IMAGE COMPRESSION
Presented By,
E . JEEVITHA
16MMAT05
M.Phil Mathematics
OVERVIEW
Introduction
Historical developments
Techniques
Methodology
Applications
Advantages
Conclusion
INTRODUCTION
Wavelets are mathematical functions that splits
up data into different frequency components, and
then study each component with a resolution
matched to its scale.
Wavelet transform decomposes a signal into a
set of basis functions. These basis functions are
called as “ wavelets ”.
HISTORICAL DEVELOPMENTS
1909 : Alfred Haar – Dissertation “On the orthogonal
function systems” for his doctoral degree. The first wavelet
related theory.
1910 : Alfred Haar : Development of a set of rectangular
basis functions.
1930 : Paul Levy investigated “ The brownian motion”.
Littlewood and Paley worked on localizing the
contributing energies of a functon.
1946 : Dennis Gabor : Used short time fourier transform.
1975 : George zweig - The first continuous wavelet
transform.
1985 : Meyer - Construction of orthogonal wavelet basis
functions with very good time and frequency localization.
1986 : Stephen Mallet – Developing the idea of Multiresolution analysis for DWT.
1988 : Daubechies and Mallet – The modern wavelet
theory.
1992 : Albert cohen and Daubechies constructed the
compactly supported biorthogonal wavelets.
TYPES
OF
WAVELET
TRANSFORM
WHY IMAGE COMPRESSION?
Digital images usually require a very
large number of bits, this causes critical
problem
for
digital
image
data
transmission and storage.
It is the art & science of reducing the
amount of data required to represent an
image.
It is one of the most useful and
commercially successful technologies in
the field of digital image processing.
WHY WAVELETS ?
Good approximation properties.
Efficient way to compress the
smooth data except in localized
region.
Easy to control wavelet properties.
( Example : Smoothness, better
accuracy near sharp gradients).
METHODS / STEPS
Digitize the source image to a signal s, which is a
string of numbers.
Decompose the signal into a sequence of wavelet
coefficients.
Use thresholding to modify the wavelet compression
from w to another sequence w’.
Use quantization to convert w’ to a sequence q.
Apply entropy coding to compress q into a sequence e.
STEP 1
A
B
C
D
A+B C+D A-B
L
C-D
H
STEP 2
A
C
B
D
L
C+D
A+B
LL
H
A-B
HL
C-D
LH
HH
LEVEL 1
LL1
LEVEL 2
HL1
LH1
HH1
LL3
HL3
LH3
HH3
LH2
HL2
LH2
HH2
LH1
HL1
HH1
HL2
HL1
LEVEL 3
HH2
LH1
LL2
HH1
ORIGINAL IMAGE
20
15
30
20
17
16
31
22
15
18
17
25
21
22
19
18
1st HORIZONTAL SEPERATION
1st VERTICAL SEPERATION
35
50
5
10
68
103
6
19
33
53
1
9
76
79
-4
-7
33
42
-3
-8
2
-3
4
1
43
37
-1
1
- 10
5
-2
-9
APPLICATION
LL2
HL2
HL
LH2
HH2
LH
LL3
HL3
LH3
HH3
HH
HL2
HL
LH2
HH2
LH
HH
OTHER APPLICATIONS
Wavelets are a powerful statistical tool which can
be used for a wide range of applications, namely
Signal processing.
Image processing.
Smoothing and image denoising.
Fingerprint verification.
Biology for cell membrane recognition, to
distinguish the normal from the pathological
membranes.
DNA analysis, protein analysis.
Blood-pressure, heart-rate and ECG analysis.
Finance (which is more surprising), for
detecting the properties of quick variation of values.
In Internet traffic description, for
designing the services size.
Speech recognition.
Computer graphics and multi-fractal
analysis.
ADVANTAGES
The advantage of wavelet compression is that, in contrast
to JPEG, wavelet algorithm does not divide image into
blocks, but analyze the whole image.
Wavelet transform is applied to sub images, so it produces
no blocking artifacts.
Wavelets have the great advantage of being able to
separate the fine details in a signal.
Wavelet allows getting best compression ratio, while
maintaining the quality of the images.
CONCLUSION
Image compression using wavelet transforms results
in an improved compression ratio as well as image quality.
Wavelet transform is the only method that provides both
spatial
and
properties
of
frequency
wavelet
domain
transform
information.
greatly
These
help
in
identification and selection of significant and nonsignificant coefficient. Wavelet transform techniques
currently provide the most promising approach to high
quality image compression, which is essential for many
real world applications.
THANK YOU