EMBEDDED ZEROTREE WAVELET COMPRESSION

Neyre Tekbıyık

And

Hakan Şevki Tozkoparan

Undergraduate Project Report submitted in partial fulfillment of

the requirements for the degree of Bachelor of Science (B.S.)

in

Electrical and Electronic Engineering Department Eastern Mediterranean University

June 2005

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Approval of the Electrical and Electronic Engineering Department ______________________________ Prof. Dr. Derviş Z. Deniz Chairman This is to certify that we have read this thesis and that in our opinion it is fully adequate, in cope and quality, as an Undergraduate Project.

______________________________ …….……………………..

Supervisor Members of the examining committee Name Signature 1. Prof. Dr. Haluk Tosun …….…………………………….. 2. Asst. Prof. Dr. Hüseyin Bilgekul …….……………………………. . 3. Asst. Prof Dr. Hasan Abou Rajab …….…………………………….. 4. Asst. Prof. Dr. Erhan Ince …….…………………………….

5. Asst. Prof. Dr. Aykut Hocanın …….……………………………. Date: …….……………………………..

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ABSTRACT

EMBEDDED ZEROTREE WAVELET COMPRESSION

by

Neyre Tekbıyık and Hakan Şevki Tozkoparan

Electrical and Electronic Engineering Department Eastern Mediterranean University

Supervisor: Asst. Prof. Dr. Erhan A. İNCE Keywords: EZW, Successive Approximation Quantization, Entropy Coding, Discrete Wavelet Transform Image compression is very important in many applications, especially for progressive

transmission, image browsing and multimedia applications. The whole aim is to obtain

the best image quality and yet occupy less space. Embedded zerotree wavelet

compression (EZW) is a kind of image compression that can realize this goal. EZW

algorithm is fairly general and performs remarkably well with most types of images.

Also, it is applicable to transmission over a noisy channel. This project will focus on

embedded zerotree wavelet (EZW) compression for images. The objective is to explain

the idea of EZW compression, design a matlab program which will apply an embedded

zerotree wavelet compression algorithm on an image. Eventually, we should achieve the

best image quality for a given bit rate in such a way that all encodings of the same image

at lower bit rates are embedded in the beginning of the bit stream for the target bit rate.

The report will introduce the concept of zerotree and show how zerotree coding can

efficiently encode[9] a significance map of wavelet coefficients by predicting the absence

of significant information across scales. We will reveal how successive approximation is

used in conjunction with zerotree coding and arithmetic coding to accomplish efficient

embedded coding. Finally, the MATLAB program that we are going to write, will enable

embedded zerotree wavelet algorithm. So, we will be able to apply this algorithm on an

image to see its effects by achieving some experimental results for various rates.

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Acknowledgments We would like to thank our supervisor, Asst. Prof. Dr. Erhan Ince, for his invaluable

guidance and support throughout the entire process of this work. His valuable critics were

largely responsible for the success of the final version. We would like to thank all our

families for their love and caring at all times. The number of people that have contributed

to our both personal and scientific life during the last four years is too high to list them

explicitly, so rather than risking forgetting anybody, we would just like to thank all of

you.

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Table of Contents

ABSTRACT ....................................................................................................................... 3

ACKNOWLEDGMENTS ................................................................................................ 4

TABLE OF CONTENTS.................................................................................................. 5

LIST OF FIGURES .......................................................................................................... 6

LIST OF TABLES ............................................................................................................ 7

LIST OF EQUATIONS… ................................................................................................ 8

1. INTRODUCTION......................................................................................................... 9

2. EMBEDDED ZEROTREE WAVELET CONCEPTS ............................................ 11

2.1 EMBEDDED CODING ................................................................................................. 11

2.2 WAVELET TRANSFORM............................................................................................ 12

2.2.1 Continuous Wavelet Transform ....................................................................... 15

2.2.2 Discrete Wavelet Transform ............................................................................ 16

2.2.3 DWT and Filter Banks..................................................................................... 19

2.3 ZEROTREES OF WAVELET COEFFICIENTS................................................................... 20

2.3.1 Significance Map Encoding ............................................................................. 20

2.3.2 Compression of Significance Maps Using Zerotrees of Wavelet Coefficients23

2.4 SUCCESSIVE APPROXIMATION ENTROPY-CODED QUANTIZATION.............................. 27

2.5 AN EXAMPLE ABOUT EZW ......................................................................................... 29

3. EXPERIMENTAL RESULTS....................................................................................32

4. CONCLUSION ............................................................................................................33

5. FUTURE WORK.........................................................................................................34

6. REFERENCES……………………………………………………………………….35

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List of Figures Figure 1. Embedded Zerotree Wavelet Compression Chart ........................................................ 10

Figure 2. Embedded Coding Scheme........................................................................................... 12

Figure 3. Demonstration of (a) a Wave and (b) a Wavelet .......................................................... 13

Figure 4. Wavelet Transform Diagram ........................................................................................ 15

Figure 5. First Stage of a discrete wavelet transform................................................................... 16

Figure 6. A two scale wavelet decomposition.............................................................................. 17

Figure 7. 2 scale wavelet decomposition (matlab simulation) ..................................................... 17

Figure 8. 3 scale wavelet decomposition ..................................................................................... 18

Figure 9. 3 scale wavelet decomposition (matlab simulation) ..................................................... 18

Figure 10. Original picture after reconstruction............................................................................ 19

Figure 11. Successive lowpass and highpass filtering ................................................................. 19

Figure 12. Reconstruction of the original signal from Wavelet Coefficients .............................. 20

Figure 13. Quantized Coefficients and Significance Map ........................................................... 21

Figure 14. Low bit rate image coder ............................................................................................ 22

Figure 15. Zig-zag manner Scanning order................................................................................. 25 Figure 16. Morton Scan order ...................................................................................................... 25

Figure 17. Flow chart for encoding a coefficient of the significant map..................................... 25

Figure 18. Example of 3 scale wavelet Transform of an 8 x 8 image.......................................... 28

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List of Tables Table 1. First Dominant pass at threshold T = 32 ........................................................................ 30

Table 2. Results of the Example................................................................................................... 31

Table 3. Experimental Results For bird.gif….. ............................................................................ 33

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List of Equations

Figure 1. Wavelet Transform Formula…………………………................................................. 14

Figure 2. Wavelet Reconstruction Formula………………………….. ....................................... 14

Figure 3. Continuous Wavelet Transform Formula……………………. .................................... 15

Figure 4. Simple Representation of the formula of CWT………….. .......................................... 15

Figure 5. Entropy of the symbols H…………………………….. ............................................... 22

Figure 6. Cost of encoding the bit stream…………………………. ........................................... 23

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1. Introduction

The embedded zerotree wavelet algorithm (EZW) [1] is an effective image

compression algorithm. This new technique produces a fully embedded bit stream for

image coding. Also, the compression performance of this algorithm is competitive with

virtually all known techniques. Furthermore, this technique requires definitely no

training, no pre-stored codebooks or tables and requires no preceding knowledge of the

image source. The EZW algorithm is based on four principal concepts: a discrete wavelet

transform or hierarchical sub-band decomposition, prediction of the absence of

significant information across scales, entropy coded successive-approximation

quantization and universal lossless data compression which is accomplished via adaptive

arithmetic coding. The EZW algorithm includes: a discrete wavelet transform (DWT) [4],

zerotree coding, successive approximation and a priorization protocol which helps us to

determine the order of importance due to various characteristics. In addition, adaptive

arithmetic coding is used to achieve a fast and efficient method for entropy coding. We

want to call attention to the following sentence. ‘This algorithm runs consecutively and

stops whenever a target bit rate or a target distortion is met. Certainly, one of the most

significant parts of this algorithm is the encoding process. As mentioned above, in our

case, discrete wavelet transform is used. The original image is passed through discrete

wavelet transformation to produce transform coefficients. The aim of this transformation

is to produce de-correlated coefficients and remove dependencies between samples as

much as possible. Besides, the more significant bits of precision of most coefficients are

allowed by this transformation to be efficiently encoded as part of exponentially growing

zerotrees. This transformation is considered to be lossless. The produced transform

coefficients are then quantized to produce a stream of symbols which will be used in

compression. Almost all of the information loss occurs in the quantization[10] stage. For

embedded coding, the successive approximation quantization is used which allows the

coding of multiple significance maps using zerotrees, and allows the encoding and

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decoding to stop at any point. Therefore, in quantization stage the threshold value is

arranged to determine significance. The coding process allows the entropy coder to

incorporate learning into the bit stream itself. As a final part of encoding, the data

compression stage takes the stream of symbols and attempts to represent the data stream

with no loss as efficiently as possible. In decoding operation converse of these concepts

will be applied. In the decoding process after receiving the compressed image data,

entropy decoding, de-quantization and inverse DWT will be applied consecutively. In the

decoding operation, each decoded symbol refines and reduces the width of the

uncertainty interval in which the true value of the coefficient may occur. A brief literature

search has only found two published results where authors generate an actual bit stream

that claims higher Peak Signal to Noise Ratio (PSNR) performance at rates between 0.25

and 1 bit/pixel, the latter of which is a variation of the EZW. The performance of the

EZW coder was also compared widely available version of JPEG. It is seen that JPEG

does not allow the user to select a target bit rate.

EZW algorithm is an evident advantage. The user can choose a bit rate and encode[3] the

image to exactly the wanted bit rate. Below you will find the step-by-step explanations on

how one can apply the EZW coding algorithm:

Figure 1. Embedded Zerotree Wavelet Compression Chart

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2. Embedded Zero Tree Wavelet Concepts 2.1 Embedded Coding

The embedded zerotree wavelet algorithm (EZW) is a simple [1] and remarkably

effective, image compression [5] algorithm. This coding [12] algorithm producing an

embedded code has the property that the bits in the bit stream are generated in order of

importance. This property allows all the low rate codes to be included at the beginning of the

bit stream. The embedded code represents a sequence of binary decisions that distinguish an

image from the “null” image, or all gray, image. Using an embedded code, an encoder can

terminate the encoding at any point and allows a target rate or distortion metric to be met

exactly. When the target is met the encoding simply stops. The embedded coder can cease at

any time and provide the best representation of an image achievable within its framework.

The binary coded stream can realize progressive transmission using multi-threshold

embedded zerotree wavelet coding. Therefore, the coding rate and distortion can be

controlled accurately. According to the order of importance of the bit plane, the most

important bit plane is coded first. The coding can end at any time when the bit budget is

reached or compression ratio is attained. When transmission occurs, the most important bit

plane is transmitted first, and the least important is transmitted last. Since the information

which is needed to represent an image coded at some rate always contains the needed

information for the same image coded at lower rates, it is possible to choose a fixed target bit

rate. Hence, the decoder can cease decoding the bit stream at any point, and simulate an

image coded at a lower rate corresponding to the truncated bit stream. This means that, the

decoder can interrupt the decoding process at any point in the bit stream, and still reconstruct

the image. Thus, a compression scheme which generates an embedded code can start sending

over the network the coarser version of the image first. Then, it continues with the

progressive transmission of the refinement details.

The schematic diagram of an embedded image coding scheme that employs the image

transform, can be seen in Figure 2. The image coder constitutes of three main blocks which

are: transform, quantizer, and entropy coder [7]. The purpose of an image transform (T)

operated on an image (I) is to decorrelate the image data. The output C of the transform can

be attained or calculated as C = TI.

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Figure 2. Embedded Coding Scheme

In the case of the DWT (image transform), C is the subband decomposed version of the

original image. It consists of the wavelet coefficients at different scales, and the lowest

frequency contents of the image at the coarsest scale. The image I can be perfectly recovered

by operating the inverse transform on C. The next block in Figure is a quantizer. It quantizes

the transformed image into a sequence of integers. Output of the quantizer is coded using an

entropy coding scheme.

In addition, the EZW encoder is based on progressive encoding which is also known

as embedded encoding to compress an image into a bit stream with increasing accuracy. This

means that when more bits are added to the stream, as a result, the decoded image will

contain more detail. Every digit which is added increases the accuracy of the number,

however we can stop at any accuracy we like. It can be said, embedded coding is similar in a

way to the binary finite precision representations of real numbers. All real numbers can be

represented by a string of binary digits. For each digit added to the right, more precision is

added so that the decoded image will have more detail.

It is known that, typical image coders require extensive training for both quantization

and generation of nonadaptive entropy codes. However, the embedded coder attempts to be

universal by incorporating all learning process in to the bit stream.

2.2 Wavelet Transform The Wavelet Transform enables a time-frequency representation of the signal. It was

developed to overcome the short coming of the Short Time Fourier Transform (STFT), which

can also be used to analyze non-stationary signals. While STFT gives a constant resolution at

all frequencies, the Wavelet Transform uses multi-resolution technique by which different

frequencies are analyzed with different resolutions.

A wave is an oscillating function of time or space and is periodic. In contrast, wavelets are

localized waves. They have their energy concentrated in time or space and are suited to

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analysis of transient signals. While Fourier Transform and STFT use waves to analyze

signals, the Wavelet Transform uses wavelets of finite energy. A demonstration of a wave

and wavelet can be seen below in Figure 3.

(a) (b)

Figure 3. Demonstration of (a) a Wave and (b) a Wavelet [16]

The wavelet analysis is done similar to the STFT analysis. The signal to be analyzed is

multiplied with a wavelet function just as it is multiplied with a window function in STFT,

and then the transform is computed for each segment generated. However, unlike STFT, in

Wavelet Transform, the width of the wavelet function changes with each spectral component.

The Wavelet Transform, at high frequencies, gives good time resolution and poor frequency

resolution, while at low frequencies, the Wavelet Transform gives good frequency resolution

and poor time resolution.

Wavelet analysis [2] is a technique to transform an array of N numbers from their actual

numerical values to an array of N wavelet coefficients. Each wavelet coefficient represents

the closeness of the fit (or correlation) between the wavelet function at a particular size and a

particular location within the data array. By varying the size of the wavelet function (usually

in powers-of-two) and shifting the wavelet so it covers the entire array, you can build up a

picture of the overall match between the wavelet function and your data array. Since the

wavelet functions are compact (hence the term wavelet), the wavelet coefficients only

measure the variations around a small region of the data array. This property makes wavelet

analysis very useful for signal or image processing; the "localized" nature of the wavelet

transform allows you to easily pick out features in your data such as spikes (for example,

noise or discontinuities), discrete objects (in, for example, astronomical images or satellite

photos), edges of objects, etc.

In general wavelets are functions that can be used to efficiently represent other functions.

Wavelet representation is a recent technique and connected to image representation. Simply

we can say wavelets allow complex filters to be constructed for digital data which can

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remove or enhance selected parts of the signal. If we compare Wavelet Transform with the

Fourier transform (FT), FT is less useful in analyzing non-stationary data, where there is no

repetition within the region sampled. Wavelet transforms (of which there are, at least

formally, an infinite number) allow the components of a non-stationary signal to be analyzed.

Wavelets also allow filters to be constructed for stationary and non-stationary signals.

The wavelet Transform formula could be seen below in Equation 1:

( ) ( ) 0,1

, >

−= ∫

∞

∞−

sdts

th

stxsX

ττ

Equation 1. Wavelet Transform Formula

Where; s = scale , τ = time shift

For reconstruction formula can be seen below in Equation 2.

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1),(

1)(

s

dds

s

th

ssXtx

Cτττ

−= ∫ ∫

∞ ∞

∞−Ψ

Equation 2. Wavelet Reconstruction Formula

Where h(τ) is called the mother wavelet.

Wavelet based lossy compression techniques have three steps in general [1];

(i)Transform: data are first transformed into wavelet domain. This step is invertible.

(ii)Quantization: the wavelet coefficients are quantized to a finite alphabet. This step is not

invertible, thus introduces the so called quantization noise.

(iii)Entropy Coding: the resulting symbols after quantization are further entropy coded to

reduce the bit rate. This step is also invertible.

For a Wavelet Transform a simple diagram is as shown below in Figure 4.

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Figure 4. Wavelet Transform Diagram[15]

Now we are going to explain two wavelet[8] transforms called; a) Continuous Wavelet Transform b) Discrete Wavelet Transform 2.2.1 Continuous Wavelet Transform

The Continuous Wavelet Transform can be found from the below equation, where x(t)

is the signal to be analyzed where ψ(t) is the mother wavelet. All the wavelet functions used

in the transformation are derived from the mother wavelet through shifting and scaling.

Continuous wavelet transform formula is as shown in Equation 3.

dts

ttx

ssX WT ∫ Ψ −= )(.)(

1),(

* ττ

Equation 3. Continuous Wavelet Transform Formula

Simply we can show as a simple version as below in Equation 4,

∫∞

∞−

Ψ= dttpositionscaletfpositionscaleC ),,()(),(

Equation 4. Simple Representation of the formula of CWT

The mother wavelet used to generate all the basis functions is designed based on some

desired characteristics associated with that function. The translation parameter τ relates to the

location of the wavelet function as it is shifted through the signal. Thus, it corresponds to the

time information in the Wavelet Transform. The scale parameter s is defined as | 1/frequency

| that corresponds to frequency information. Scaling either expands or compresses a signal.

Large scales, low frequencies, expands the signal and provide detailed information hidden in

the signal, while small scales, high frequencies, compress the signal and provide global

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information about the signal. The above analysis becomes very useful as in most practical

applications, high frequencies, low scales do not last for a long duration while low

frequencies, high scales usually last for entire duration of the signal.

2.2.2 Discrete Wavelet Transform

In the project we used discrete wavelet transform while compiling our matlab

program. [4]. In CWT, the signals are analyzed using a set of basis functions which relate to

each other by simple scaling and translation. In the case of DWT, a time-scale representation

of the digital signal is obtained using digital filtering techniques.

The signal to be analyzed is passed through filters with different cutoff frequencies at

different scales. Discrete wavelet transform is defined by a square matrix of filter

coefficients, transforming an array into a new array, usually of the same length. If correctly

constructed, the matrix is orthogonal, and in this case not only the transform but also its

inverse can be easily implemented. The wavelet transform resembles the Fourier transform in

many respects. Also it’s identical to a hierarchical sub-band system, where the sub-bands are

logarithmically spaced in frequency and represent octave-band decomposition. To begin the

decomposition, the image is divided into four sub-bands and critically sub-sampled as shown

below in Figure 5.

Figure 5. First stage of a discrete wavelet transform

Each coefficient represents a spatial area corresponding to approximately a (2 × 2) area of the

original image. The low frequencies represent a bandwidth approximately corresponding to

w, which can take values between 0 and π/2, whereas the high frequencies represent the band

from π/2 to π. The four sub-bands arise from separable application of vertical and horizontal

filters. The sub-bands labeled LH1, HL1, and HH1 represent the finest scale wavelet

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coefficients. To obtain the next coarser scale of wavelet coefficients, the sub-band LL1 is

further decomposed and critically sampled as shown below in Figure 6. with the output we

get from the matlab program by using the dwt2 function.

Figure 6. A two scale wavelet decomposition

Figure 7. Two scale wavelet decomposition (matlab simulation)

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Figure 8. Three scale wavelet decomposition

Figure 9. Three scale wavelet decomposition (matlab simulation)

In the project we used 3 scale wavelet decomposition and 8 x 8 analysis of the matrixes. After

finishing the encoding in the very last step after the decoding operations we used wavelet

reconstruction or synthesis using the idwt2 function to get the original picture as below in

Figure 10.

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Figure 10. Original Picture after reconstruction

2.2.3 DWT and Filter Banks

Filters are one of the most widely used signal processing functions. Wavelets can be

realized by iteration of filters with rescaling. The resolution of the signal, which is a measure

of the amount of detail that are found in the signal and the scale is determined by upsampling

and downsampling operations.

The DWT is computed by successive lowpass and highpass filtering of the discrete time-

domain signal as shown in the figure below in Figure 11.. This is called the Mallat algorithm

or Mallat-tree decomposition. In the figure, the signal is denoted by the sequence x[n], where

n is an integer. The low pass filter is denoted by G0 and the high pass filter is denoted by H

0.

At each level, the high pass filter produces detail information, d[n], while the low pass filter

associated with scaling function produces coarse approximations, a[n].

Figure 11. Successive lowpass and highpass filtering[16]

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At each decomposition level, the half band filters produce signals spanning only half

the frequency band. This doubles the frequency resolution as the uncertainity in frequency is

reduced by half. In accordance with Nyquist’s rule if the original signal has a highest

frequency of ω, which requires a sampling frequency of 2ω radians, then it now has a highest

frequency of ω/2 radians. It can now be sampled at a frequency of ω radians thus discarding

half the samples with no loss of information. This decimation by 2 halves the time resolution

as the entire signal is now represented by only half the number of samples. Thus, while the

half band low pass filtering removes half of the frequencies and thus halves the resolution,

the decimation by 2 doubles the scale. The filtering and decimation process is continued until

the desired level is reached. The maximum number of levels depends on the length of the

signal. The DWT of the original signal is then obtained by linking all the coefficients, a[n]

and d[n], starting from the last level of decomposition.

Figure 12. Reconstruction of the original signal from Wavelet Coefficients[16]

The above Figure 12. shows the reconstruction of the original signal from the wavelet

coefficients. Basically, the reconstruction is the reverse process of decomposition. The

approximation and detail coefficients at every level are upsampled by two, passed through the

low pass and high pass synthesis filters and then added. This process is continued through the

same number of levels as in the decomposition process to obtain and Hthe original signal.

The Mallat algorithm works equally well if the analysis filters, G0 and H

0, are exchanged with

the synthesis filters, G1 1

.

2.3 Zerotrees of Wavelet Coefficients

2.3.1 Significance Map Encoding

The EZW algorithm recognized that a significant fraction of the total bits needed to

code an image were required to code position information, which EZW called significance

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maps. A significance map is a binary function whose value determines whether each

coefficient is significant or not significant. If a coefficient is not significant, it is assumed to

quantize to zero. Hence, a decoder knows the significance map needs no further information

about that coefficient. Otherwise, the coefficient is quantized to a non-zero value. The EZW

algorithm determined a very efficient way to code significance maps by coding the location

of the zeros. It was proven experimentally that zeros could be predicted very precisely across

different scales in the wavelet transform. The EZW algorithm is based on the hypothesis that

“if a wavelet coefficient at a coarse scale is insignificant with respect to a given threshold T,

then all wavelet coefficients of the same orientation in the same spatial location at finer

scales are likely to be insignificant with respect to given threshold.”

For each threshold T, the positions of the significant, and insignificant coefficients,

are indicated in significance maps. The values of "0" indicate the positions of the

insignificant coefficients, whereas the values of "1" are indicating the positions of the

significant coefficients. A sample for both quantized coefficients and a significance map is

shown in figure. A modality to encode the significance maps is the Zerotree Representation.

It allows us to predict insignificant coefficients across the scales efficiently. By using this

technique, the cost of encoding the significance maps is reduced by grouping the insignificant

coefficients in trees growing exponentially across the scales, and by coding these coefficients

with zerotree symbols.

Quantized Coefficients Significance Map

Figure 13. Quantized Coefficients and Significance Map

The coding of the significance map which means the positions of those coefficients

that is going to be transmitted as nonzero values is one of the important views of low bit rate

64 56 48 32 24 16 0 0 56 48 40 24 16 23 0 0 40 40 30 24 16 8 0 8 32 32 32 24 24 16 0 0 24 24 16 8 0 0 8 0 16 16 8 0 0 8 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

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image coding. After quantization followed by entropy coding, the zero symbol which occurs

with the highest probability, should be extremely high in order to attain very low bit rates.

Thus, a large fraction of the bit budget is spent to encode the significance map. This results in

an important progress in encoding the significance map, and a higher efficiency in

compression.

To understand the importance of significance map encoding, a typical transform

coding system may be considered. This consideration is required to provide a person with a

sense of the relative coding costs of the position information which is contained in the

significance map relative to sign and amplitude information. A typical low bit-rate image

coder has three basic components: a transformation, a quantization and data compression. A

typical low bit rate image coder can be seen in Figure 14.

Figure 14. Low bit rate image coder

Transformation is required to produce transform coefficients, and to produce

coefficients that are decorrelated. Most of the time, it is considered to be lossless.

Quantization produces stream of symbols by quantizing the transform coefficients. Each of

these stream of symbols corresponds to an index of a particular quantization bin. Finally, the

aim of the data compression stage is to losslessly represent the data stream by taking the

stream of symbols. The aim of quantization is to make the entropy of the resulting

distribution of bin indexes small enough so that the symbols can be entropy coded at some

target low bit rate. For each nonzero index encoded, the entropy code will need at least one

bit for the sign. An entropy code can be designed based on modeling probabilities of bin

indices. Thus, the entropy of the symbols H is expressed as:

[ ]H NZpppppH +−+−−−−= 1)1()1()1( loglog

22

Equation 5. Entropy of the symbols H

In this equation, p is the probability that a transform coefficient is quantized to zero,

whereas HNZ represents the conditional entropy of the absolute values of the quantized

coefficients conditioned on them being zero. Besides, the first two terms of this sum

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represents the first order binary entropy of the significance map. Therefore, the true cost of

encoding the actual bit stream becomes:

ValuesNonzeroofCostMapceSignificanofCostCostTotal +=

Equation 6. Cost of encoding the bit stream

Hence, the conclusion is that the cost of determining the positions of the several

significant coefficients represents an important portion of the bit budget at low rates. As the

rate decreases, the cost of determining the positions has a probability to become an increasing

fraction of the total cost.

2.3.2 Compression of Significance Maps Using Zerotrees of Wavelet Coefficients

For improvement of the compression of significance maps of wavelet coefficients, a

data structure called zerotree[13] is defined. A zerotree is a quad-tree of which all nodes are

equal to or smaller than the root. The tree is coded with a single symbol. Then, it is

reconstructed by the decoder as a quadtree filled with zeroes. The zerotree is based on a

hypothesis which says “if a wavelet coefficient at a coarse scale is insignificant with respect

to a threshold T, then all wavelet coefficients of the same orientation in the same spatial

location at the finer scale are likely to be insignificant with respect to the same threshold”.

An Embedded Zerotree Wavelet (EZW) uses a series of decreasing thresholds and

compares the wavelet coefficients with those thresholds. If the magnitude of a coefficient is

smaller than a given threshold T, the node is called insignificant or 0 with respect to given

threshold. Otherwise, the coefficient is significant or 1. The EZW uses four symbols to

represent a tree node: POS (positive significant), NEG (negative significant), IZ (isolated

zero node) and ZTR (zerotree) instead of 0 and 1. Embedded zerotree wavelet coding can be

seen below which is done in 5 steps.

Initialization:

)))(max(2(log0 2 coeffsfloorT =

k=0 Dominant List = All Coefficients Subordinate List = [] SignificantMap for each coefficient x in the Dominant List if | x | kT≥

if x > 0 set symbol POS

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else set symbol NEG else if x is non-root part of a zerotree set symbol ZTD(ZeroTree Descendant) if x is zerotree root set symbol ZTR otherwise set symbol IZ Dominantpass If symbol(x) is POS or NEG(it is significant) Put symbol(x) on the Subordinate List Remove x from the Dominant List SubordinatePass For each entry symbol(x) in Subordinate List if value(x) is the element of Bottom Half of [kT , kT2 ]

output “0” else output “1” Update

=+1kT2kT

k=k + 1 Go to Significance Map

Furthermore, in a hierarchical subband system (except the highest frequency

subbands), every coefficient at a specific scale can be related to a set of coefficients at the

next finer scale of similar orientation. The coefficient at the coarse scale is called the parent,

and all coefficients corresponding to the same spatial location at the next finer scale of

similar orientation are called children. Similarly, we can define the concepts as descendants

and ancestors. For a given parent, the set of all coefficients at all finer scales of similar

orientation corresponding to the same location are called descendants. Similarly, for a given

child, the set of coefficients at all coarser scales of similar orientation corresponding to the

same location are called ancestors. It is known that, every coefficient is a parent for the

previous related coefficients and the last coefficient in the lowest resolution is the parent of

the whole image. If a parent is insignificant in a low-resolution image (when compared with

threshold), than we can assume that all its children are also insignificant (when compared

with threshold).When an insignificant parent node is detected, most probably the descendents

are also insignificant. Thus, most of the time, a zerotree significance map symbol is

generated. However, because p which is the probability of this event, is close to 1, its

information content which is −p log p, is very small. So most of the time, a very small

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amount of information is transmitted, and this keeps the average bitrate required for the

significance map small.

In the wavelet decomposition, coefficients that are spatially related across scale can

be compactly described using tree structures. With the exception of the low resolution

approximation (LLK) and the highest frequency bands (HL1, LH1, and HH1 ), each

coefficient (parent) at level j of the decomposition correlates to 4 coefficients (children) at

level j −1 of the decomposition which are at the same frequency orientation. For the LLK

band, each parent coefficient correlates with 3 children coefficients, one in the HLK, one in

the LH K , and one in the HH K bands. For example, level HH3 includes the children of HH2.

Also, level HH2 in turn is the parent to the sub band level HH1. In scanning order of the sub-

bands for encoding a significance map, parents must be scanned before the children and also

all positions in a given sub-band are scanned before the scan moves to the next sub-band. A

sample of the scanning order can be seen below in figure, the scanning order is in [6] Zigzag

manner. In our project, we used Morton scan order. Morton scan order is shown in below.

Figure 15. Zig-zag Manner Scanning Order [1]

Figure 22. Morton Scan Order [3]

Given a threshold T to determine whether or not a coefficient is significant, a

coefficient is part of a zero-tree if it is zero or insignificant with respect to the threshold T.

26

Also, it is an element of zerotree, if all of its descendants are zero or insignificant with

respect to the threshold T. If a coefficient is not a part of another zero-tree starting at a

coarser scale, then it is also a zero-tree root. Hence, for a given threshold, any wavelet

coefficient could be represented in one of the four data types. As mentioned before, these are

POS, NEG, ZTR, and IZ, corresponding to positive coefficients, negative coefficients,

zerotree roots,

and isolated zeros, respectively. Zerotree coding is lossless and zerotree symbols are assigned

to wavelet coefficients as the coefficient file is scanned. If a coefficient is found to be zero or

insignificant, then corresponding coefficients in lower resolutions are checked. If all of these

children are zero, this coefficient is coded as a (zerotree symbol) ZTR. Besides, if one or

more of the children of an insignificant node is significant, that means a symbol for isolated

zero (IZ) is transmitted. The probability of this event is lower most of the times but it is

necessary to be able to avoid losing significant information down the tree.

In this manner the significance map can be efficient represented as a string of symbols

from a 4-symbol alphabet (POS, NEG, ZTR, and IZ) which is then entropy coded. A sample

coefficient decision procedure and the encoding process of a coefficient of the significance

map can be seen below in Figure 17.

Figure 17. Flow chart for encoding a coefficient of the significant map

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If a significant coefficient is detected or found out, the coefficient is moved to the end

of the list of significant coefficients, and the node value is zeroed out in the wavelet tree.

Also, the zerotree symbols can be transmitted many times for a given coefficient, until it rises

above the sinking threshold that it will be tagged as a significant coefficient. After this point,

zerotree information is not going to be transmitted for this coefficient anymore. These trees

of zero coefficients can be coded by only coding the root coefficient of the tree. This provides

the compression that none of the coefficients below the root need to be coded. In general,

zerotree coders combine the idea of cross-band correlation with the notion of coding zeros

jointly, for generation of very powerful compression algorithms. Aside from the nice energy

compaction properties of the wavelet transform, this coder attains its compression ratios by

joint coding of zeros.

2.4 Successive Approximation Entropy-Coded Quantization

The successive approximation quantization gives an important characteristic to those

algorithms: they are all embedded, that is, the bit stream they generate for one rate contains

the bit – streams for all achievable lower rater. Shapiro characterizes an embedded code by

defining two properties:

a) When coding the same data with different rates, the two resulting

representations must match exactly for the extent of the smaller one. In other

words, the smaller one must be reproduced exactly in the beginning of the

larger one. This way, the coded representation for a given data rate contains all

the representations for smaller data rates. Shortly, the representations get more

precise as we add more symbols to them.

b) It should obtain a good representation for a given data rate.

To perform the embedded coding in the program we applied successive-approximation

quantization (SAQ). As will be seen, SAQ is related to bit-plane encoding of the magnitudes.

The SAQ sequentially applies a sequence of thresholds to determine significance, where the

thresholds are chosen so that T(i) = T (i-1)/2. The initial threshold T(0) is chosen so that |X(j)|

< 2T(0) for all transform coefficients Xj. In the program, during the encoding and decoding,

two separate lists of wavelet coefficients are present. Dominant list contains the coordinates

of the coefficients that have not yet been found to be significant in the same order as the

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initial scan. This scan is such that the sub-bands are ordered, and with each sub-band, the set

of coefficients are ordered. The subordinate list contains the magnitudes of those coefficients

that have been found to be significant and for each threshold, each list is scanned once.

During a dominant pass, coefficients with coordinates are compared to the threshold Ti

to determine their significance, and if significant, their sign. This significance map then later

zerotree coded. Each time a coefficient is encoded as significant as positive or negative then

its magnitude is appended to the subordinate list, and the coefficient in the wavelet transform

array is set to zero so that the significant coefficient does not prevent the occurrence of a

zerotree on future dominant passes at smaller thresholds.

Then a dominant pass is continued with subordinate pass is performed on the

subordinate list, which contains all pixel values previously found to be significant, in which

all coefficients on the subordinate list are scanned and the specifications of the magnitudes

available to the decoder are refined to an additional bit of precision. For each magnitude on

the subordinate list, this refinement can be encoded using a binary alphabet with a “1”

symbol indicating that the true value falls in the upper half of the old uncertainty interval and

a “0” symbol indicating the lower half. The string of symbols from this binary alphabet that is

generated during a subordinate pass is then entropy coded. After the completion of a

subordinate pass the magnitudes on the subordinate list are sorted in decreasing magnitude.

The process continues to alternate between dominant passes and subordinate passes where the

threshold is halved before each dominant pass.

Thus, the algorithm for the successive quantization for embedded zero tree encoding

procedure can be seen below :

1. Choose a high quantisation step T (half the max value)

2. Until finest quantisation step, repeat

2.1 Quantise wavelet coefficients using T

– ie. if (w(x,y)<T) w'(x,y)=0 ; else w'(x,y)=1;

– Creates a binary image of the most significant wavelet coefficients

2.2 Encode binary image (w') using zero-trees and output

– (This is the highest order bits of the remaining data)

2.3 Subtract the encoded data from the remaining data

– w(x,y)=w(x,y) – w'(x,y)*T

2.4 half the quantisation step size

– T=T/2

3. End

29

In the decoding operation, by reading the subordinate symbol corresponding to a

significant pixel and knowing the threshold, the decoder is able to determine the range in

which the pixel lies and reconstructs the pixel value to the midpoint of that range, each

decoded symbol which in both during a dominant and a subordinate pass, refines and reduces

the width of the uncertainty interval in which the true value of the coefficient may occur. The

reconstruction value used can be anywhere in that uncertainty interval. The encoding stops

when some target stopping condition is met, such as when the bit budget is exhausted. The

encoding can cease at any time and the resulting bit stream contains all lower rate encodings.

Terminating the decoding of an embedded bit stream at a specific point in the bit stream

produces exactly the same image that would have resulted had that point been the initial

target rate. This ability to cease encoding or decoding anywhere is extremely useful in

systems that are either rate or distortion constrained.

2.5 An Example About EZW

A 3-scale wavelet transform of an 8 x 8 image is given below [1]. We are going to discuss

about only LL2 section for simplification.

63 -34 49 10 7 13 -12 7

-31 23 14 -13 3 4 6 -1

15 14 3 -12 5 -7 3 9 -9 -7 -14 8 4 -2 3 2

-5 9 -1 47 4 6 -2 2 3 0 -3 2 3 -2 0 4 2 -3 6 -4 3 6 3 6 5 11 5 6 0 3 -4 4

Figure 18. Example of 3-scale Wavelet Transform of an 8 x 8 Image

We will choose our initial threshold as To=32, The table below, Table 1. ,shows the

processing on the first dominant pass.

30

Subband Coefficient

Value

Symbol Reconstruction

Value

LL3 63 POS 48

HL3 -34 NEG -48

LH3 -31 IZ 0

HH3 23 ZTR 0

Table 1. First Dominant Pass at Threshold T = 32

The top left corner coefficient 63 is the biggest value among all coefficients and represents

the DC coefficient. According to the largest coefficient magnitude, the first dominant pass

threshold T can be any number between 31.5 and 63. We Choose T0 = 32. The lowest

frequency coefficient with magnitude 63 is greater than the threshold and meaning a positive

symbol. At the decoder end, this coefficient will be decoded to the center value of the

uncertainty interval [32, 64), which is 48. Next, according to the scan order, the coefficient -

34 is to be processed absolute value is taking in order to process. Its magnitude is also greater

than 32 and then putting a negative symbol behind. It will be decoded to -48 in the

reconstruction of the image. The coefficient of LH3 is -31, which is insignificant with respect

to the threshold 32. But it has a descendant of magnitude 47 in LH1. Therefore, -31 is an

isolated zero. So far, the 1st dominant pass has completed the process on the finest scale and

is going to move to the 2nd scale. At HL2 subband, except 49 is a positive significance, all

other three values are zerotree root. At LH2 subband, all coefficients are insignificant.

However, 14 has a significant child. Therefore, 14 is an isolated zero, like -31, and all other

three are zerotree roots. Coefficients in HH2 subband are all zerotree roots since themselves

and their descendants are all insignificant. Then we move to the finest scale. Both HL1 and

HH1 have all zeros. At LH1 subband, except 47 is a positive significant, all others are zeros.

This completes the 1st dominant pass of the sample 8×8 image. After the 1st dominant pass,

the 1st subordinate pass is applied to the significance map. The subordinate pass only looks at

the nonzero values and refine them. The uncertainty intervals are refined as [32, 48) and [48,

64). Any significant coefficients greater than 48 will be encoded as symbol “1”, and as

symbol “0” if lying between 32 13 and 48. Each symbol “1” will later be decoded as 56,

which is the center of the interval [48, 64), while symbol “0” will later be decoded as 40, the

center of the interva l [32, 48). During the 2nd dominant pass, the threshold value is set to be

16 since all the coefficients to be processed are those insignificant ones after the 1st dominant

31

pass, which are between [0, 32). The procedure is the same as the 1st dominant pass and

results are shown in below. The 2nd dominant pass processes all the significant coefficients

after the 2nd dominant pass, including those from the 1st dominant pass. The uncertainty

intervals are refined as [16, 24), [24, 32), [32, 40), [40, 48), [48, 56) and [56, 64). In other

words, uncertainty intervals starts from the threshold value, and has interval length 16/2 = 8.

Results can be seen on following page in Table 2.

Table 2. Results of the example

Subband

Coefficient

Value

1st Dominant

Pass

(T=32)

1st Subordinate

Pass Interval

center (56 40)

2nd

Dominant

Pass

(T=16)

2nd Subordinate Pass

(60 52 44 36 28 20)

LL3 63 POS 48 1 56 0 60

HL3 -34 NEG -48 0 40 0 36

LH3 -31 IZ 0 0 NEG -24 28

HH3 23 ZTR 0 0 POS 24 20

HL2 49 POS 48 1 56 0 52

HL2 10 ZTR 0 0 ZTR 0 0

HL2 14 ZTR 0 0 ZTR 0 0

HL2 -13 ZTR 0 0 ZTR 0 0

LH2 15 ZTR 0 0 ZTR 0 0

LH2 14 IZ 0 0 ZTR 0 0

LH2 -9 ZTR 0 0 ZTR 0 0

LH2 -7 ZTR 0 0 ZTR 0 0

HL1 7 Z 0 0 ZTR 0 0

HL1 13 Z 0 0 ZTR 0 0

HL1 3 Z 0 0 ZTR 0 0

HL1 4 Z 0 0 ZTR 0 0

HL1 -1 Z 0 0 ZTR 0 0

HL1 47 POS 48 0 40 0 44

HL1 -3 Z 0 0 ZTR 0 0

HL1 -2 Z 0 0 ZTR 0 0

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2. Experimental Results

We applied the embedded zero-tree wavelet algorithm (EZW) on the bird image by adjusting

the Threshold and Last step level values. Below you will find results for the reconstructed

images at different compression rates. It is clear that as the number of significant coefficients

kept increases the compression decreases and hence the PSNR improves.

Original image

reconstructed image

Threshold to stop at : 8

Last Step Level : 4 Sum : 4710 PSNR : + 28.82 dB Compression : 13.91 : 1

Original image

reconstructed image

Threshold to stop at : 4

Last Step Level : 5 Sum : 8065 PSNR : +32.2597 dB Compression : 8.13 : 1

33

Original image

reconstructed image

Threshold to stop at : 1

Last Step Level : 7 Sum : 27017 PSNR : +39.5650 dB Compression : 2.43:1

Original image

reconstructed image

Threshold to stop at : 1

Last Step Level : 8 Sum : 42274 PSNR : +42.9387dB Compression : 1.55 : 1

Table .3 Experimental results for bird.gif

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4. Conclusion

This project is about Embedded Zerotree Wavelet (EZW) algorithm which is

considerably effective and important compression algorithm. This algorithm is devised by

Shapiro[1] in 1993, and it is an image compression algorithm that the bits in the bit stream

are generated in order of importance. The algorithm is based on the selection of significant

coefficients with respect to given thresholds. In this project, we tried to apply the embedded

zero tree wavelet compression on an image. In our program, we used an encoder to encode

the image by using three level decomposition, with discrete wavelet transform, and then

quantized the image. We checked the importance of coefficients by comparing with the

threshold which is done in the beginning of each dominant pass and taking the values which

are greater than the threshold. Then we used the decoding algorithms, and we used inverse

discrete wavelet transform to get the reconstructed image.

In our project, we tried our best to understand and improve EZW algorithm to be able

to apply it for various images. We wrote this report according to our matlab program. The

Matlab program that we have is the improved version of the matlab program which was

compiled by Xiangang Li. The main purpose of the program was to encode and decode the

only one 8 x 8 image matrix. After we had analyzed the program, we improved the program for

various of scaled images and added discrete wavelet transform algorithm (DWT) and inverse discrete

wavelet transform (IDWT) algorithm, to make the program as told in Shapiro’s [1]. We added

Discrete Wavelet transform and Inverse Discrete Wavelet Transform to the matlab program for

decomposition. Also, we adjusted the old program so that now, we are able to use this

algorithm for 256 x 256 images or different sized images. The advantages of this program are

that, it yields a fully embedded bit stream, providing competitive compression performance,

more compression in the first steps, and we had a precise rate controlling by changing the

threshold and last step values. Also the picture is getting more clear as we continue changing

the steps of the program.

While compiling the Matlab code, first we came across with some difficulties in the

analysis of the program. Then, we had some extra difficulties while we were trying to

develop our matlab program. However, these difficulties made us do more research and force

ourselves to do our best. Also, this project made us to learn and search the required matlab

codes needed for the program and helped us to improve our research abilities in the specified

manner. This project has taught us many valuable things, and we have gained significant

experience about EZW. It also showed us the importance of working in groups for future

projects since we will all be forced to work in groups to achieve the projects, and it showed

us how people can work together more efficiently and effectively for a project. Furthermore,

35

we learned the importance of timing and systematical work, which helps making the project

to yield in a successful way.

Finally, we want to say that this project is the result of work done by two Electrical and

Electronics Engineering students, because we worked and studied very hard to accomplish

our task by doing this project. To sum up, we have learned the importance of EZW and its

application of encoding, decoding and the compression of the image that we have done in this

project. EZW is one of the best compression algorithms and can be used in various fields

which can form one of the basis of image compression algorithms in the future.

5. Future Work

EZW is a remarkably effective compression algorithm. Therefore this algorithm can

be beneficial for many areas. We have chosen to apply this algorithm to different sized

images to encode, compress, decode, and reconstruct them without using extra codings such

as Arithmetic coding or Huffman coding. However, in future this project can be improved by

adding Arithmetic coding or Huffman coding to attain better compression results to achieve

high compression ratios. In addition to this, instead of DWT, different transforms can be used

for this project to obtain better results or to see the differences between these applications.

Also, an additional algorithm can be added to our matlab code to send an image through

noisy channel and receive images without noise. The program can be changed or written

again so that if a noisy image comes to the decoder, decoder will be able to do denoising and

reconstruct the image without having noise on it.

Furthermore, this project can be improved that it can be applied for video

compression using EZW. Because a video consists of various images taken sequentially, this

algorithm can be used to compress each frame or image of a video. Hence, the video

compression can be accomplished. Additionally, by using this project people can change this

algorithm to apply it to 3D images. In future, compression speed can also be increased. In this

situation, people can send their images faster via channels. Sending and receiving images

faster is very important according to different areas. For example, EZW can be used in

medicine or medical applications to send the images of the patients to other doctors quickly.

Thus, they can find quick solutions for the benefit of their patients. Finally, EZW algorithm

can be used in cell phones to send large sized images or even videos via cell phone by

compressing them.

36

References

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[2] Nasir, M. R., Rolan, G. W., François, G. M., Ronald R. C., December 2003, “Adaptive Wavelet Packet Basis Selection for Zerotree Image Coding”, IEEE Transactions on Signal Processing, vol. 12, no. 12

[3] C, Valens. (1999-2004). EZW encoding. Retrieved October 2, 2004 from the World Wide Web: http://perso.wanadoo.fr/polyvalens/clemens/ezw/ezw.html

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