compression and denoising

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 : [email protected], [email protected]

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].

mailto:[email protected]

mailto:[email protected]

ITSI Transactions on Electrical and Electronics Engineering (ITSI-TEEE)


127

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.



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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.



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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


an array.




Step 4: Compute the DWT of the data.



131

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


an array.



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.



132

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


an array.




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



133

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



Image size=147 X 81



134

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



Image size=147 X 81

Table 3: Coiflets Wavelet Compression

Sl.

No.

Decom

levels

Short Name

( w )

Compressed

Image ( % )

De-noising


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



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



135

Discrete Meyer Wavelet Compression



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



Image size=147 X 81

Table 6: Biorthogonal Wavelet Compression

Sl.

No.

Decom

levels

Short

Name

( w )

Compressed Image

( % )


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



Image size=147 X 81

Table 7: Reverse BiorthogonWavelet Compression

Sl. No.

Decom levels

Short Name

( w )

Compressed Image

( % )


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



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.



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


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


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


Image size=109 X 87, Wavelet Short name – „sym4‟



138

Table 11: Symlets Compression

Sl.

No.

Thresa hold

(thr)

Compressed

Image ( % )

De-noising


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


Image size=109 X 87, Wavelet Short name – „dmey‟

Table -12: Discrete Meyer Compression

Sl.

No.

Thresa hold

(thr)

Compressed

Image ( % )

De-noising


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


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


Image size=109 X 87, Wavelet Short name – „rbio6.8‟



139

Table 14: Reverse Biorthogonal Compression

Sl.

No.

Thresa hold

(thr)

Compressed

Image ( % )

De-noising


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



140

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



141

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



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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