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ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
126
Compression and Denoising - Comparative Analysis
from still Images using Wavelet Techniques
B. B. S. Kumar1 & P. S. Satyanarayana
2
1Dept. of ECE, Rajarajeswari College of Engineering, Bangalore. 2Dept. Of ECE, Cambridge Institute of Technology, Bangalore.
E-mail : bbskumarindia@yahoo.com1, pssvittala@gmail.com
2
Abstract – With the growth of the multimedia technology
over the past decades, the demand for digital information
has increased dramatically. This enormous demand poses
difficulties for the current technology to handle. One
approach to overcome this problem is to compress the
information by removing the redundancies present in it.
This is the lossy compression scheme that is often used to
compress information such as digital images. The main
objective is to investigate the still image format
compression and de-noising using different wavelet
techniques.
The “Compression and Denoising – Comparative Analysis
from still Images using Wavelet Techniques” is
implemented in software using „MATLAB2012a‟ version
Wavelet Toolbox and 2-D DWT technique. The purpose is
to analyze still images using different wavelets families
such as Haar, Daubechies, Coiflets, Symlets, Discrete
Meyer, Biorthogonal and Reverse Biorthogonal. The
experiments and simulation is carried out on still image
.jpg formats.
This work tries to introduce wavelets and then some of its
applications and technique in image processing. The scope
of the work involves– Compression and de-noising, image
clarity and comparing the results of wavelet families, to
find the effect of the decomposition and threshold levels
and to find out energy retained (image recovery) and lost,
knowing the best wavelet and so on.
The wavelet differs from each other in image clarity and
energy retaining. Each method is compared and classified
in terms of its efficiency at different decomposition and
threshold levels. Therefore, the image recovery is good and
clarity, but the percentage of compression and retaining
the energy is different. In order to quantify the
performance of the de-noising, a noise is added to the still
image and given as input to the de-noising algorithm,
which produces an image close to the original image.
Keywords – Joint production experts group(.JPG), Two-
Dimensional Discrete Wavelet Transform
I. INTRODUCTION
The main objective of this research is to investigate
and provide a foundation for implementing wavelet-
based image processing algorithms using
MATLAB2012a. A complementary objective is to
analyze still images using different wavelet families
such as Haar, Daubechies, Coiflets, Symlets, Discrete
Meyer, Biorthogonal and Reverse Biorthogonal [ 9].
The research objective is to review the
compression, de-noising and decomposition &
reconstruction property of wavelet by using different
wavelet families to analyze image data. The purpose of
the investigation is to find the effect of the
decomposition & threshold level, knowing the best
wavelet on compression and de-noising, image clarity,
comparing the results of wavelet families, to find out
energy retaining(image recovery) and lost. Therefore
families of wavelets, the Haar, Daubechies, Coiflets,
Symlets, Discrete Meyer, Biorthogonal and Reverse
Biorthogonal are used. The image used in the analysis is
.jpg image format.
II. RESEARCH LIMITATIONS
In fact, the decomposition results depends on the
choice of analyzing wavelet i.e., its corresponding filters
that are used. The choice of mother wavelet depends
whether one needs to obtain better resolution in time or
frequency. The design and proper choice of the wavelet
function for diverse tasks comprises a considerable part
of wavelet research [13], [17].
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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III. RESEARCH APPROACH
In principle, the approach tries to know the best
wavelet for image compression and de-noising. The
simulations are conducted on still images(.jpg format)
using wavelets such as Haar, Daubechies, Coiflets,
Symlets, Discrete Meyer, Biorthogonal and Reverse
Biorthogonal. Image analysis and implementation for
still images are done using 2D DWT. The main
framework of this project involves a number of
experimental results such as Image compression,
Finding the effect of the decomposition & threshold
level, Finding energy retained(image recovery) and lost,
Image de-noising, Image clarity, Comparing the results
of wavelet families, Knowing the best wavelet.
IV. PROBLEM DEFINITION
Each wavelet having efficient image clarity, but
differs in compression and de-noising percentage rate,
hence this paper presents comparison of seven wavelets
analysis at decomposition and threshold levels. In this
research the following basic classes of problems will be
considered – Image Restoration, Image analysis, Image
Reconstruction, Image Compression and De-noising.
V. WAVELETS
The word “wavelet” is due to Morlet and
Grossmann in the early 1980s influenced by ideas from
both pure and applied mathematics. They used the
French word ondelette, meaning "small wave". Soon it
was transferred to English by translating "onde" into
"wave", giving "wavelet". Wavelets were developed
independently in the fields of mathematics, quantum
physics, electrical engineering, seismic geology and
medical technology etc. [7], [14]
Before 1930, the main branch of mathematics
leading to wavelets began with Joseph Fourier at 1807
with his theories of frequency analysis. The first
mention of wavelets appeared in the Ph.D. thesis of A.
Haar at 1909. A physicist Paul Levy investigated
Brownian motion at 1930, a type of random signal.
Another 1930s research effort by Little-wood, Paley and
Stein involved computing the energy of a function. The
development of wavelets was starting with Haar's work
in the early 20th
century. Pierre Goupillaud, Grossmann
and Morlet's formulation of what is now known as the
continuous wavelet transformation(CWT), Jan-Olov
Stromberg's early 1983s work on discrete wavelets.
Stephane Mallat gave wavelets in digital signal
processing at 1985 and multiresolution framework at
1989. In 1988, Daubechies referred orthogonal wavelets
with compact support, Nathalie Delprat introduced time-
frequency interpretation of the CWT at 1991, Newland
exhibited harmonic wavelet transformation at 1993 and
many others since.
Theory of wavelets has been developed essentially
in last twenty years. Approximation by wavelet
polynomials is progressing rapidly. A wavelet is a
wavelike oscillation with amplitude that starts out at
zero, increases, and then decreases back to zero. It can
typically be visualized as a "brief oscillation" like one
might see recorded by a seismograph or heart monitor.
Generally, wavelets are purposefully crafted to have
specific properties that make them useful for signal
processing. Wavelets can be combined, using a "shift,
multiply and sum" technique called convolution, with
portions of an unknown signal to extract information
from the unknown signal
The Fourier transform shows up in a remarkable
number of areas outside of classic signal processing.
Now a day the mathematics of wavelets is much larger
than that of the Fourier transform. Initial wavelet
applications involved signal processing and filtering.
However, wavelets have been applied in many other
areas including non-linear regression, image
compression, turbulence, human vision, radar
earthquake prediction and seismic wave etc.
Wavelets are mathematical functions that cut up
data into different frequency components and then study
each component with a resolution matched to its scale.
A wavelet transform is the representation of a function
by wavelets. More technically, a wavelet is a
mathematical function used to divide a given function or
continuous time signal into different scale components.
Usually one can assign a frequency range to each scale
component. Each scale component can then be studied
with a resolution that matches its scale. The wavelet will
resonate if the unknown signal contains information of
similar frequency - just as a tuning fork physically
resonates with sound waves of its specific tuning
frequency. This concept of resonance is at the core of
many practical applications of wavelet theory.
A recent literature on wavelet image processing
shows the focus on using the wavelet algorithms for
processing one-dimensional and two-dimensional
signals. Acoustic, speech, music and electrical transient
signals are popular in 1-D wavelet signal processing.
The 2-D wavelet signal processing involves mainly
noise reduction, signature identification, target
detection, signal and image compression and
interference suppression.
In this work we have tried to show the technique
how we use the wavelet in image processing. In this
technique we consider only the wavelet coefficients
which are mainly contribute to the given image or which
are having a special behavior.
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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The principles of wavelet and wavelet transforms are
explained in brief. Finally, the discrete wavelet
transform (DWT) will be introduced as well as issues of
its practical implementation for images.
Wavelet Definitions
A wavelet (t) is a function in the form of wave
that has effectively a limited extent. It has an average
value of zero. The wavelet function can be defined with
any function (t) that satisfies following conditions:
1) The integral of wavelet (t) equals zero and
therefore it must be oscillatory. In other words it
must be a wave.
(t)dt=0 (1)
-
2) It is square integral (t) or, equivalently, has finite
energy:
(t ) 2dt < (2)
-
the function (t) is a mother wavelet or wavelet if it
satisfies these two properties as well as the admissibility
condition defines later in this chapter. While the
admissibility condition is useful in formulating a simple
inverse wavelet transform, properties 1 and 2 suffice to
define the Continuous Wavelet Transform(CWT) [8],
and they capture essentially the reasons for calling the
function a wavelet. Property 2 implies that most of the
energy in (t) is confined to a finite duration. Property
1‟s suggestive of a function that is oscillatory or that has
a wavy appearance. Thus, in contrast to a sinusoidal
function, it is a “small wave” or a wavelet. The two
properties are easily satisfied and there is infinity of
functions that qualify as mother wavelets.
Wavelet properties
Compact support
The condition of (equation-2) implies that the basis
functions are non-zero only on a finite interval while the
sinusoidal basis functions of the Fourier transform are
infinite in extent.
Localization
The wavelet function must have localization both in
frequency and time. In other words the wavelet basis
functions must have zero average (equation-1) to allow
the WT to efficiently represent functions or signals,
which have localized features. There is one more
condition, which is the regularity condition stating that
the wavelet function should be smooth and concentrated
in both time and frequency domains.
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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Fig. 1: Several different families of wavelets
VI. THE DISCRETE WAVELET TRANSFORM
Dilations and translations of the “Mother function"
or “analyzing wavelet" Ф(x); define an orthogonal basis,
our wavelet basis:
Ф(s, l)(x) = 2-s/2
Ф(2-sx - l) (3)
The variables s and l are integers that scale and dilate
the mother function Ф to generate wavelets, such as a
Daubechies wavelet family [14]. The scale index s
indicates the wavelet's width, and the location index l
gives its position. Notice that the mother functions are
rescaled, or “dilated" by powers of two, and translated
by integers. What makes wavelet bases especially
interesting is the self-similarity caused by the scales and
dilations. Once we know about the mother functions, we
know everything about the basis [16], [18].
To span our data domain at different resolutions, the
analyzing wavelet is used in a scaling equation:
N-2
W(x)=(-1)kck+1Ф(2x+k) (4)
k= -1
Where W(x) is the scaling function for the mother
function Ф; and ck are the wavelet coefficients. The
wavelet coefficients must satisfy linear and quadratic
constraints of the form
N-1 N-1
ck=2, ckck+2l=2δl,0
k=0 k=0
Where δ is the delta function and l is the location index.
One of the most useful features of wavelets is the ease
with which a scientist can choose the defining
coefficients for a given wavelet system to be adapted for
a given problem. In Daubechies' original paper, she
developed specific families of wavelet systems that were
very good for representing polynomial behavior. The
Haar wavelet is even simpler, and it is often used for
educational purposes [14], [15].
Two - Dimensional Discrete Wavelet Transform
Decomposition
In the discrete wavelet transform, an image signal
can be analyzed by passing it through an analysis filter
bank followed by a decimation operation. This analysis
filter bank, which consists of a low pass and a high pass
filter at each decomposition stage, is commonly used in
image compression. When a signal passes through these
filters, it is split into two bands. The low pass filter,
which corresponds to an averaging operation, extracts
the coarse information of the signal. The high pass filter,
which corresponds to a differencing operation, extracts
the detail information of the signal. The output of the
filtering operations is then decimated by two.
A two-dimensional transform (figure-2a) can be
accomplished by performing two separate one-
dimensional transforms. First, the image is filtered along
the x-dimension and decimated by two. Then, it is
followed by filtering the sub-image along the y-
dimension and decimated by two. Finally, we have split
the image into four bands denoted by LL, HL, LH and
HH after one-level decomposition (figure-3b). Further
decompositions can be achieved by acting upon the LL
subband successively and the resultant image is split
into multiple bands as shown in figure-3c and figure-3d.
Fig. 2a : 2-D DWT
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Fig. 2b: 2-D IDWT
In mathematical terms, the averaging operation or
low pass filtering is the inner product between the signal
and the scaling function () as shown in equation-5
whereas the differencing operation or high pass filtering
is the inner product between the signal and the wavelet
function () as shown in equation-6
Average coefficients,
Cj (k) = < f(t), j, k (t) > = f(t), j, k (t) dt (5)
Detail coefficients,
dj (k) = < f(t), j, k (t) > = f(t), j, k (t) dt (6)
The scaling function or the low pass filter is defined as
j, k (t) = 2j/2 (2
j t – k) (7)
The wavelet function or the high pass filter is defined as
j, k (t) = 2j/2 (2
j t – k) (8)
where j denotes the discrete scaling index, k denotes the
discrete translation index.
The reconstruction of the image can be carried out
by the following procedure. First, we will upsample by a
factor of two on all the four subbands at the coarsest
scale, and filter the subbands in each dimension. Then
we sum the four filtered subbands to reach the low-low
subband at the next finer scale. We repeat this process
until the image is fully reconstructed shown in figure-2.
Fig. 3: 2-D DWT Decomposition: a) Original image, b) One
level decomposition, c) Two levels decomposition, d) Three
levels decomposition
VII. ALGORITHMS FOR IMAGE ANALYSIS USING
WAVELETS
Algorithm for Decomposition
Step 1: Start-Load the source image data from a file into
an array.
Step 2: Choose a Haar Wavelet.
Step 3: Decompose-choose a level N, compute the
wavelet decomposition of the signals at level N.
Step 4: Compute the DWT of the data.
Step 5: Read the 2-D decomposed image to a matrix.
Step 6: Retrieve the low pass filter from the list based on
the wavelet type.
Step 7: Compute the high pass filter i=1.
Step 8: i >= 1decomposed level, then if Yes goto step
10, otherwise if No goto step 9.
Step 9: Perform 2-D decomposition on the image i++
and goto to step 8.
Step 10: Decomposed image.
Step 11: End.
Algorithm for Reconstruction
Step 1: Start-Load the source image data from a file into
an array.
Step 2: Choose a Haar Wavelet.
Step 3: Decompose-choose a level N, compute the
wavelet decomposition of the signals at level N.
Step 4: Compute the DWT of the data.
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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Step 5: Read the 2-D decomposed image to a matrix.
Step 6: Retrieve the low pass filter from the list based on
the wavelet type.
Step 7: Compute the high pass filter i=decomp level.
Step 8: i <= 1, then if Yes goto step 10, otherwise if No
goto step 9.
Step 9: Perform 2-D reconstruction on the image and
goto to step 8.
Step 10: Reconstruction image.
Step 11: End.
Image Compression
The study is projected on lossy compression [17].
Images require much storage space, large transmission
bandwidth and long transmission time. The only way
currently to improve on these resource requirements is
to compress images, such that they can be transmitted
quicker and then decompressed by the receiver.
In image processing there are 256 intensity levels
(scales) of grey. 0 is black and 255 is white. Each level
is represented by an 8-bit binary number so black is
00000000 and white is 11111111. An image can
therefore be thought of as grid of pixels, where each
pixel can be represented by the 8-bit binary value for
grey-scale.
Fig. 4 : Data level in image
"Image compression algorithms aim to remove
redundancy in data in a way which makes image
reconstruction possible." This basically means that
image compression algorithms try to exploit
redundancies in the data; they calculate which data
needs to be kept in order to reconstruct the original
image and therefore which data can be 'thrown
away'. By removing the redundant data, the image can
be represented in a smaller number of bits, and hence
can be compressed [6], [13], [17].
Compression Procedure
The compression procedure contains three steps:
1. Decompose -Choose a wavelet, choose a level N.
Compute the wavelet decomposition of the signals
at level N.
2. Threshold detail coefficients, for each level from 1
to N, a threshold is selected and hard thresholding
is applied to the detail coefficients.
3. Reconstruct -Compute wavelet reconstruction using
the original approximation coefficients of level N
and the modified detail coefficients of levels from 1
to N.
The difference of the de-noising procedure is found
in step 2. There are two compression approaches
available. The first consists of taking the wavelet
expansion of the signal and keeping the largest absolute
value coefficients. In this case, you can set a global
threshold, a compression performance, or a relative
square norm recovery performance. Thus, only a single
parameter needs to be selected. The second approach
consists of applying visually determined level-
dependent thresholds.
Algorithm for Compression
Step 1: Start-Load the source image data from a file into
an array.
Step 2: Choose a Haar Wavelet.
Step 3: Decompose-choose a level N, compute the
wavelet decomposition of signals at level N.
Step 4: Threshold detail coefficients, for each level from
1 to N, a threshold is selected and hard thresholding is
applied to the detail coefficients
Step 5: Remove(set to zero) all coefficients whose value
is below a threshold(this is the compression step).
Step 6: Reconstruct, Compute wavelet reconstruction
using the original approximation coefficients of level N
and the modified detail coefficients of levels from 1 to N.
Step 7: Compare the resulting reconstruction of the
compressed image to the original image.
Step 8: End.
Image De-noising
There are various methods to help restore an image
from noisy distortions. Selecting the appropriate method
plays a major role in getting the desired image. The de-
noising methods tend to be problem specific. For
example, a method that is used to de-noise satellite
images may not be suitable for denoising medical
images. In this thesis, a study is made on the de-noising
algorithm and implemented in Matlab7. In order to
quantify the performance of the de-noising algorithm,
the image is taken and random noise [15] is added to it.
This would then be given as input to the de-noising
algorithm, which produces an image close to the original
image.
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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De-Noising Procedure
The two-dimensional de-noising procedure has the
same three steps and uses two-dimensional wavelet tools
instead of one-dimensional ones.
The general de-noising procedure involves three
steps. The basic version of the procedure follows the
steps described below.
Decompose - Choose a wavelet, choose a level N.
Compute the wavelet decomposition of the signals
at level N.
Threshold detail coefficients, for each level from 1
to N, select a threshold and apply soft thresholding
to the detail coefficients.
Reconstruct - Compute wavelet reconstruction
using the original approximation coefficients of
level N and the modified detail coefficients of levels
from 1 to N. Two points must be addressed: how to
choose the threshold, and how to perform the
thresholding.
Algorithm for Denoising
Step 1: Start-Load the source image data from a file into
an array.
Step 2: Choose a Haar Wavelet.
Step 3: Decompose-choose a level N, compute the
wavelet decomposition of the signals at level N.
Step 4: Add a random noise to the source image data.
Step 5: Threshold detail coefficients, for each level from
1 to N, a threshold is selected and soft thresholding is
applied to the detail coefficients.
Step 6: Reconstruct, Compute wavelet reconstruction
using the original approximation coefficients of level N
and the modified detail coefficients of levels from 1 to N.
Step 7: Compare the resulting reconstruction of the
denoised image to the original image.
Step 8: End.
VIII. EXPERIMENTAL RESULTS
Decomposition Results
Note: Experiments are conducted on Kumar image and
results noted on this image only .
Image Used (grayscale)=kumar.jpg, Image size=147X81
Fig. 5: Original Image
Fig. 6: 1st level Decomposition
Fig. 7: 2nd level Decomposition
Fig. 8: Reconstructed Image
Fig. 9: Decomposition approximations
The decomposition experiment is conducted using
Haar wavelet has two functions “wavelet” and “scaling
function”. They are such that there are half the
frequencies between them. They act like a low pass
filter and a high pass filter, a typical decomposition
scheme. The decomposition of the signal into different
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frequency bands is simply obtained by successive high
pass and low pass filtering of the time domain signal.
This filter pair is called the analysis filter pair. First, the
low pass filter is applied for each row of data, thereby
getting the low frequency components of the row. But
since the low pass filter is a half band filter, the output
data contains frequencies only in the first half of the
original frequency range. By Shannon's Sampling
Theorem, they can be sub-sampled by two, so that the
output data now contains only half the original number
of samples. Now, the high pass filter is applied for the
same row of data, and similarly the high pass
components are separated.
This is a non-uniform band splitting method that
decomposes the lower frequency part into narrower
bands and the high-pass output at each level is left
without any further decomposition. This procedure is
done for all rows. Next, the filtering is done for each
column of the intermediate data. The resulting two-
dimensional array of coefficients contains four bands of
data, each labeled as LL (low-low), HL (high-low), LH
(low-high) and HH (high-high). The LL band can be
decomposed once again in the same manner, thereby
producing even more sub bands. This can be done up to
any level, thereby resulting in a pyramidal
decomposition as shown in figures-2a, 2b & 3.
The LL band is decomposed thrice in figure-3. The
compression ratios with wavelet based compression can
be up to 300-to-1, depending on the number of
iterations. The LL band at the highest level is most
important, and the other 'detail' bands are of lesser
importance, with the degree of importance decreasing
from the top of the pyramid to the bands at the bottom.
This can be done to any image. Figures-3, shows how it
would work for an image at different levels. The Image
is reconstructed (figure-8) to retain as original image by
IDWT(figures-5).
Compression Results
The Wavelets work by looking at the values of
neighboring pixels, and splitting those values into an
approximation value and a detail value. If the pixel
values are similar then the value of the detail is small.
Thus an image with intensity values that only have small
changes between pixel values is easier to compress with
wavelets than those that have dramatic and irregular
changes. This is because with these images the
approximation signal will contain most of the energy
(image recovery); the detail signals will have values
close to zero and therefore not much energy.
Thresholding the detail signals will therefore have little
effect on the energy, but provide more zeros. So
compression can be obtained with little cost in energy
loss. Thus if an image contains a high frequency of a
certain intensity value, then this could help to provide a
good compression rate, but it depends on where they
are in the image. If they are all together then there will
be an area of the same intensity value, and this means
that the detail values will be zero. If they are randomly
spread throughout the image, next to pixels of dissimilar
intensities, then the fact that there was a high frequency
of certain intensity will not be enough to provide good
compression.
Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 1: Haar Wavelet Compression
Sl. No.
Decom levels
Short Name
( w )
Compressed Image
( % )
Denoising Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 One haar 99.87 71.71 100.00 43.63
2 Two haar 99.82 87.09 100.00 43.63
3 Three haar 99.83 90.01 100.00 43.63
4 Four haar 99.86 90.43 100.00 43.63
5 Five haar 99.86 90.46 100.00 43.63
6 Ten haar 100.00 90.48 100.00 43.63
Fig. 10:Haar Wavelet Compression
Daubechies Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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Table 2: Daubechies Wavelet Compression
Sl.
No.
Decom
levels
Short Name
( w )
Compressed
Image ( % )
De-noising
Compressed Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 One db10 99.90 73.25 100.00 44.08
2 Two db10 99.87 85.20 100.00 44.08
3 Three db10 99.91 86.07 100.00 44.08
4 Four db10 99.95 83.82 100.00 44.08
5 Five db10 99.98 80.08 100.00 44.08
6 Ten db10 100.00 68.11 100.00 44.08
Fig. 11: Daubechies Wavelet Compression
Coiflets Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 3: Coiflets Wavelet Compression
Sl.
No.
Decom
levels
Short Name
( w )
Compressed
Image ( % )
De-noising
Compressed Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 One coif4 99.91 73.70 100.00 46.86
2 Two coif4 99.90 85.52 100.00 46.86
3 Three coif4 99.93 86.66 100.00 46.86
4 Four coif4 99.97 84.68 100.00 46.86
5 Five coif4 99.99 82.31 100.00 46.86
6 Ten coif4 100.00 68.30 100.00 46.86
Fig. 12: Coiflets Wavelet Compression
Symlets Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 4: Symlets Wavelet Compression
Sl. No.
Decom levels
Short
Name
( w )
Compressed Image
( % )
De-noising Compressed
Image ( % )
Norm Rec
Nul Coeffs
Norm Rec
Nul Coeffs
1 One sym4 99.91 73.32 100.00 47.58
2 Two sym4 99.89 87.66 100.00 47.58
3 Three sym4 99.91 89.76 100.00 47.58
4 Four sym4 99.95 89.96 100.00 47.58
5 Five sym4 99.98 88.99 100.00 47.58
6 Ten sym4 100.00 83.30 100.00 47.58
Fig. 13: Symlets Wavelet Compression
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Discrete Meyer Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 5: Discrete Meyer Wavelet Compression
Sl.
No.
Decom
levels
Short
Name ( w )
Compressed
Image
( % )
De-noising
Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 One dmey 99.91 73.26 100.00 31.27
2 Two dmey 99.91 77.27 100.00 31.27
3 Three dmey 99.96 69.50 100.00 31.27
4 Four dmey 99.99 58.04 100.00 31.27
5 Five dmey 100.00 48.15 100.00 31.27
6 Ten dmey 100.00 41.28 100.00 31.27
Fig. 14: Discrete Meyer Wavelet Compression
Biorthogonal Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 6: Biorthogonal Wavelet Compression
Sl.
No.
Decom
levels
Short
Name
( w )
Compressed Image
( % )
De-noising Compressed
Image ( % )
Norm Rec
Nul Coeffs
Norm Rec
Nul Coeffs
1 One bior3.9 99.93 74.03 100.00 50.26
2 Two bior3.9 99.91 85.64 100.00 50.26
3 Three bior3.9 99.94 87.03 100.00 50.26
4 Four bior3.9 99.97 83.92 100.00 50.26
5 Five bior3.9 99.99 80.73 100.00 50.26
6 Ten bior3.9 100.00 65.52 100.00 50.26
Fig. 15: Biorthogonal Wavelet Compression
Reverse Biorthogonal Wavelet Compression
Threshold (thr) = 20, Image Used
(grayscale)=kumar.jpg,
Image size=147 X 81
Table 7: Reverse BiorthogonWavelet Compression
Sl. No.
Decom levels
Short Name
( w )
Compressed Image
( % )
De-noising Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 One rbio6.8 99.91 73.41 100.00 45.84
2 Two rbio6.8 99.89 85.78 100.00 45.84
3 Three rbio6.8 99.93 87.43 100.00 45.84
4 Four rbio6.8 99.97 86.75 100.00 45.84
5 Five rbio6.8 99.99 85.96 100.00 45.84
6 Ten rbio6.8 100.00 73.59 100.00 45.84
Fig. 16: Reverse Biorthogonal Wavelet Compression
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
136
Graph 1: Compression Comparison at Decomposition
Graph 2: Nul Coeffs Comparison(Compression) at
Decomposition
Graph 3: Nul Coeffs Comparison(Denoised
Compression) at Decomposition
At different decomposition levels:
The decomposition level changes the proportion of
detail coefficients in the decomposition. Decomposing a
signal to a greater level provides extra detail that can be
thresholded in order to obtain higher compression rates.
However this also leads to energy lose. The best trade-
off between energy loss and compression is provided by
decomposing to higher levels. Decomposing to fewer
levels mean provides better energy retention but not as
great compression when threshold level is lower. When
threshold level is higher provides better compression but
more energy loss.
The type of wavelet affects the actual values of the
coefficients and hence how many detail coefficients are
zero or close to zero and therefore how much energy
and zeros can be obtained. Wavelets that work well with
an image redistribute as much energy as possible into
the approximation subsignal, while giving a large
proportion of the coefficient value to describe details.
An image is a collection of intensity values and hence a
collection of energy varying. The image has a huge
effect on the compression and how well energy can
be compacted into the approximation subsignal.
As shown in the graphs 1-3, figures 10-16 & tables
1-7 the compression at different decomposition levels, it
is clearly seen at higher decomposition levels having
better energy retaining(image recovery) and
compression is not excellent. At higher levels of
decomposition the number of zeros decreasing and
hence the compression rate is low. The experiments are
conducted keeping threshold level constant and varying
the decomposition levels.
At de-noised compression the image retains same as
original but differs in number of zeros. The
Biorthogonal wavelet having good de-noised
compression rate then the other wavelets and Discrete
Meyer having very poor de-noised compression rate.
Therefore the Biorthogonal is the best wavelet in de-
noised compression at different decomposition levels.
Threshold Results
At different Thresholds levels:
To change the energy retained(image recovery) and
number of zeros values, a threshold value is changed.
When threshold values are changed i.e. increased the
energy lost but having good compression rate. The
threshold is the number below which detail coefficients
are set to zero. The higher the threshold value, the more
zeros can be set, but the more energy is lost as shown in
the graphs 4 to 7, figures 17 to 23 and tables 8 to 14.
The Biorthogonal wavelet having balance in energy
retaining(image recovery) and number of zeros as
threshold is changed and decomposition levels, hence
the Biorthogonal is the best wavelet in compression as
threshold increases and more efficient then the other
wavelets.
At de-noised compression the image retains same as
original but differs in number of zeros. All the wavelet
having good de-noised compression rate only the
Discrete Meyer having very poor de-noised compression
rate.
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
137
1. Haar
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „haar‟
Table 8: Haar Compression
Sl. No.
Thresa
hold
(thr)
Compressed
Image
( % )
De-noising
Compressed
Image ( % )
Norm Rec
Nul Coeffs
Norm Rec
Nul Coeffs
1 10 99.95 81.09 99.93 79.08
2 20 99.84 89.22 99.93 79.08
3 30 99.71 92.86 99.93 79.08
4 40 99.57 94.68 99.93 79.08
5 50 99.42 95.85 99.93 79.08
6 60 99.24 96.80 99.93 79.08
7 100 98.61 98.47 99.93 79.08
8 200 97.50 99.34 99.93 79.08
Fig. 17: Haar Compression
2. Daubechies
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „db10‟
Table 9: Daubechies Compression
Sl.
No.
Thresa
hold (thr)
Compressed
Image
( % )
De-noising
Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 100.00 69.46 99.95 79.07
2 20 99.98 78.26 99.95 79.07
3 30 99.97 83.15 99.95 79.07
4 40 99.95 85.89 99.95 79.07
5 50 99.92 88.36 99.95 79.07
6 60 99.88 90.08 99.95 79.07
7 100 99.75 93.73 99.95 79.07
8 200 99.44 96.34 99.95 79.07
Fig. 18: Daubechies compression
3. Coiflets
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „coif1‟
Table 10: Coiflets Compression
Sl.
No.
Thresa
hold (thr)
Compressed
Image
( % )
De-noising
Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 99.99 82.61 99.95 81.57
2 20 99.97 89.35 99.95 81.57
3 30 99.94 92.07 99.95 81.57
4 40 99.92 93.73 99.95 81.57
5 50 99.89 94.64 99.95 81.57
6 60 99.86 95.45 99.95 81.57
7 100 99.71 97.12 99.95 81.57
8 200 99.28 98.57 99.95 81.57
Fig. 19: Coiflets Compression
4. Symlets
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „sym4‟
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
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Table 11: Symlets Compression
Sl.
No.
Thresa hold
(thr)
Compressed
Image ( % )
De-noising
Compressed Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 99.99 80.89 99.96 81.36
2 20 99.98 87.82 99.96 81.36
3 30 99.97 90.76 99.96 81.36
4 40 99.95 92.45 99.96 81.36
5 50 99.93 93.64 99.96 81.36
6 60 99.90 94.60 99.96 81.36
7 100 99.81 96.23 99.96 81.36
8 200 99.52 97.94 99.96 81.36
Fig. 20: Symlets Compression
5. Discrete Meyer
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „dmey‟
Table -12: Discrete Meyer Compression
Sl.
No.
Thresa hold
(thr)
Compressed
Image ( % )
De-noising
Compressed Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 100.00 36.92 99.97 68.16
2 20 100.00 46.08 99.97 68.16
3 30 99.99 51.63 99.97 68.16
4 40 99.99 55.63 99.97 68.16
5 50 99.98 58.94 99.97 68.16
6 60 99.97 61.76 99.97 68.16
7 100 99.91 69.96 99.97 68.16
8 200 99.62 80.96 99.97 68.16
Fig. 21: Discrete Meyer Compression
6. Biorthogonal
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „bior6.8‟
Table -13: Biorthogonal Compression
Sl.
No.
Thresa
hold (thr)
Compressed
Image
( % )
De-noising
Compressed
Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 100.00 77.13 99.95 80.39
2 20 99.99 84.07 99.95 80.39
3 30 99.98 87.33 99.95 80.39
4 40 99.97 89.18 99.95 80.39
5 50 99.95 90.55 99.95 80.39
6 60 99.94 91.31 99.95 80.39
7 100 99.87 93.71 99.95 80.39
8 200 99.64 95.92 99.95 80.39
Fig. 22: Biorthogonal Compression
7. Reverse Biothogonal
Level (n)= 5, Image Used (grayscale)=kumar1.jpg,
Image size=109 X 87, Wavelet Short name – „rbio6.8‟
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
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Table 14: Reverse Biorthogonal Compression
Sl.
No.
Thresa hold
(thr)
Compressed
Image ( % )
De-noising
Compressed Image ( % )
Norm
Rec
Nul
Coeffs
Norm
Rec
Nul
Coeffs
1 10 100.00 75.86 99.96 78.99
2 20 99.99 83.12 99.96 78.99
3 30 99.98 86.80 99.96 78.99
4 40 99.96 89.03 99.96 78.99
5 50 99.95 90.34 99.96 78.99
6 60 99.94 91.29 99.96 78.99
7 100 99.87 93.61 99.96 78.99
8 200 99.60 96.16 99.96 78.99
Fig. 23: Reverse Biorthogonal Compression
Denoised Compressed Images
Original Image
Fig. 24: Denoised Compressed Images
Graph 4: Compression Comparison at Threshold
Graph 5: Nul Coeffs Comparison(Compression) at
Threshold
Graph 6: Nul Coeffs Comparison(Denoised
Compression) at Threshold
Graph 7: Denoised Compression Comparison at
Threshold
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
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The graphical tools automatically provide an initial
threshold based on balancing the amount of compression
and retained energy. This threshold is a reasonable first
approximation for most cases. However, in general to
refine the threshold by trial and error so as to optimize
the results to fit the particular analysis and design
criteria. The tools facilitate experimentation with
different thresholds, and make it easy to alter the
tradeoff between amount of compression and retained
signal energy.
Digital image is represented as a two-dimensional
array of coefficients, each coefficient representing the
brightness level in that point. We can differentiate
between coefficients as more important ones, and lesser
important ones. Most natural images have smooth color
variations, with the fine details being represented as
sharp edges in between the smooth variations.
Technically, the smooth variations in color can be
termed as low frequency variations, and the sharp
variations as high frequency variations. The low
frequency components (smooth variations) constitute the
base of an image, and the high frequency components
(the edges which give the details) add upon them to
refine the image, thereby giving a detailed image.
Hence, the smooth variations are more important than
the details.
Separating the smooth variations and details of the
image can be performed in many ways. One way is the
decomposition of the image using the DWT. Digital
image compression is based on the ideas of sub-band
decomposition or DWT‟s. Wavelets which refer to a set
of basis functions are defined recursively from a set of
scaling coefficients and scaling functions. The DWT is
defined using these scaling functions and can be used to
analyze digital images with superior performance than
classical short-time Fourier-based techniques. The basic
difference between wavelet-based and Fourier-based
techniques is that short-time Fourier-based techniques
use a fixed analysis window, while wavelet-based
techniques can be considered using a short window at
high spatial frequency data and a long window at low
spatial frequency data. This makes DWT more accurate
in analyzing image signals at different spatial frequency,
and thus can represent more precisely both smooth and
dynamic regions in image. The compression system
includes forward wavelet transform, a quantizer, and a
lossless entropy encoder. The corresponding
decompressed image is formed by the lossless entropy
decoder, a de-quantizer, and an inverse wavelet
transform. Wavelet-based image compression has good
compression results in both rate and distortion sense.
Therefore varying the threshold and decomposition
levels the image is processed to get good quality image
and to have best possible results [10].
De-Noising Results
Image Used (grayscale)=kumar.jpg, Image size=147 X
81
Haar Wavelet Denoising
Fig. 25: Haar Wavelet Denoising
Daubechies Wavelet Denoising
Fig. 26: Daubechies Wavelet Denoising
Coiflets Wavelet Denoising
Fig. 27: Coiflets Wavelet Denoising
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
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Symlets Wavelet Denoising
Fig. 28: Symlets Wavelet Denoising
Discrete Meyer Wavelet Denoising
Fig. 29: Discrete Meyer Wavelet Denoising
Biorthogonal Wavelet Denoising
Fig. 30: Biorthogonal Wavelet Denoising
Reverse Biorthogonal Wavelet Denoising
Fig. 31: Reverse Biorthogonal Wavelet Denoising
In order to quantify the performance of the de-noising
algorithm, the image is taken and random noise is added
to it. This would then be given as input to the de-noising
algorithm, which produces an image close to the original
image. The de-noising at lower level of decomposition
having reasonable clarity but at the higher levels the
image is not clear. It is found the best wavelet for de-
noising at decomposition levels is Biorthogonal wavelet.
The Haar wavelet is having very poor de-noising then
the other wavelet families.
IX. CONCLUDING ANNOTATIONS
Conclusion
The main objective is to analyze still images using
wavelets theory of different wavelets families such as
Haar, Daubechies, Coiflets, Symlets, Discrete Meyer,
Biorthogonal and Reverse Biorthogonal. The
experiments and simulation is carried out on .jpg format
images.
Paper concentrated on the decomposition and
reconstruction by DWT technique and the results that
were collected were values for percentage energy
retained and percentage number of zeros. These values
were calculated for a range of threshold and
decomposition values on all the images. The energy
retained describes the amount of image detail that
has been kept, it is a measure of the quality of the
image after compression. The number of zeros is a
measure of compression. A greater percentage of zeros
implies that higher compression rates can be obtained.
The decomposition level changes the proportion of
detail coefficients in the decomposition. Decomposing a
signal to a greater level provides extra detail that can be
thresholded in order to obtain higher compression rates.
However this also leads to energy lose. The best trade-
off between energy loss and compression is provided by
decomposing to higher levels. Decomposing to fewer
levels means provides better energy retention but not as
great compression when threshold level is lower. When
threshold level is higher provides better compression but
more energy loss.
To change the energy retained and number of zeros
values, a threshold value is changed. When threshold
values are changed i.e. increased, energy lost but having
good compression rate. The threshold is the number
below which detail coefficients are set to zero. The
higher the threshold value, the more zeros can be set,
but the more energy is lost.
The wavelets are efficient in image processing but
differ in compatibility of individuality of wavelet
families. All the wavelets having good de-noised
compression image with clarity, but differ in energy
ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)
ISSN (PRINT) : 2320 – 8945, Volume -1, Issue -6, 2013
142
retaining & percentage of zeros. The de-noising at lower
level of decomposition having reasonable clarity but at
the higher levels the image is not clear. It is found the
best wavelet for compression & de-noising at
decomposition & thresholding is Biorthogonal wavelet.
The Discrete Meyer wavelet is having very poor
compression and Haar wavelet is having very poor de-
noising then the other wavelet families.
Future Work
There are many possible extensions to this paper.
These include finding the best thresholding strategy,
finding the best wavelet for a given image, investigating
other complex wavelet families, the use of wavelet
packets in compression and de-noising [19].
The wavelets theory is new advanced topic to go
research in enormous field of different image formats
and also very interesting. Hence, therefore wavelets
theory can be implemented as applications to provide
better results in digital signal processing and digital
image processing.
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[2] Rafael C. Gonzalez, Richard E. Woods, Steven L.
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[3] Rudra Pratap - “Getting started with
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[4] Rafael C. Gonzalez, Richard E. Woods – “Digital
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