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EECCEE 553333 PPrroojjeecctt RReeppoorrtt

AAhhuujjaa,, AAllookk ++

SSiinngghh,, AAaarrttii ++

+(50% contribution by each member)

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This project involves Matlab implementation of the Embedded Zerotree Wavelet (EZW)

coding algorithm followed by a QM coder. The Embedded Zerotree Wavelet (EZW)

algorithm is a wavelet-based image compression algorithm that yields a fully embedded code

as well as remarkable compression efficiency. An embedded code has the property that bits

in the bit stream are generated in order of importance, hence the encoding can be truncated at

any point to meet a specified bit rate. The output of the algorithm is a stream of symbols that

can be further compressed using an adaptive binary arithmetic coder like the QM coder.

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This report starts with a brief statement of the work performed the objectives, material

studied and implemented. The main part of the report is divided into two broad sections

covering the concepts behind EZW algorithm and the QM coder. Specifically we have tried

to present issues that we found intricate, in a simpler manner and also tried to emphasize

issues that are not clearly stated in the references. The next section contains the results

followed by conclusion and discussion.

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This project is somewhere between an implementation and study project. The project was

started with an aim to understand and implement the EZW coding algorithm proposed by J.

M. Shapiro (1). We tried to understand the scope of improvement in the EZW algorithm (2)

and better ways to implement it eg. using SPIHT (3).

Also we tried to focus on investigating adaptive arithmetic coders, specifically Q and QM

coders that implemented finite precision representation for the interval size as well as

provided computational simplicity. Starting with papers on adaptive binary arithmetic coder

(7) to Q coders and its adaptive probability estimation (5), (6), we finally arrived at the QM

coder (4), which is used in the JPEG standard for image compression. Due to the many

improved features of the QM coder (as discussed later), we decided to implement it in

Matlab. Results of a test sequence provided in (4) were reproduced.

Finally, the QM-coder was interfaced and used alongwith the EZW coder.

EEZZWW bbaasseedd IImmaaggee CCooddiinngg

The Embedded Zerotree Wavelet (EZW) algorithm is based on four key concepts

(1) A discrete wavelet transform or hierarchical subband decomposition.

(2) Prediction of absence of significant information in finer level wavelet coefficients

based on coarser level coefficients.

(3) Entropy-coded successive-approximation quantization.

(4) Lossless data compression based on an adaptive arithmetic coder. (In this project, the

coder used is the QM coder and we devote a separate section to it.)

DDiissccrreettee WWaavveelleett TTrraannssffoorrmm is carried out by decomposing the image into four subbands

(LL, LH, HL and HH) using separable wavelet filters and critically subsampling the output.

The next coarser level of coefficients are obtained by decomposing the low frequency

subband LL.

Fig. A two-level subband decomposition (1)

The goal of the transform is to produce coefficients that are decorrelated. This results in

concentration of energy in a few coefficients while most are insignificant enough to be

discarded, thus offering opportunity for compression.

SSiiggnniiff iiccaannccee MMaapp EEnnccooddiinngg The coefficients are compared to a threshold to determine if

they are significant enough. The zerotree is based on the hypothesis that if a wavelet

coefficient is insignificant with respect to a 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 T.

More specifically, in hierarchical subband decomposition, with the exception of the highest

frequency subbands, every coefficient at the given level (parent) can be related to a set of

coefficients (children)at next finer level of similar orientation and spatial location. For a

given parent, the set of coefficients at all finer scales of similar orientation corresponding to

the same location are called descendants.

Fig. Parent-child dependencies (1)

As can be seen each parent has three children except for the lowest frequency subband where

the relationship is defined so that each parent node (in LL3) has three children (one each in

HL3 , LH3 and HH3). Scanning of the coefficients is performed so that no child is scanned

before its parent.

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

coefficient is an element of a zerotree for threshold T if itself and all of its descendents are

insignificant wrt T. An element of a zerotree is a zerotree root if it is not the descendent of a

previously found zerotree root. Thus, a zerotree root indicates that the insignificance of the

coefficients at all finer levels is completely predictable.

Thus, the output of the EZW coder consists of the following symbols:

1. Zerotree root, Z.

2. Isolated zero, I. (means that coefficient is insignificant but has some significant

descendents).

3. Positive significant, P.

4. Negative significant, N.

Two more symbols L and H are generated which are discussed in the next section.

SSuucccceessssiivvee--AApppprrooxx iimmaatt iioonn EEnnttrrooppyy--CCooddeedd QQuuaanntt iizzaatt iioonn is applied to achieve embedded

coding. The encoding can be stopped when a specified bit-rate is met. Successive

approximation quantization determines significance by starting with an initial threshold T0,

chosen so that all transform coefficients are < 2T0. Successive thresholds are obtained as Ti =

Ti-1/2. The encoding (and decoding) involves two main passes

1. Dominant pass During a dominant pass, coefficients that have not yet been found to

be significant are compared against the threshold, Ti. The scanning order follows

treating parents before children, eg. raster scan, however those coefficients are

omitted which are descendents of a zerotree root found before. If the coefficient is

found significant, it is appended to the subordinate list, and is set to zero in the

wavelet transform array to prevent it from effecting future dominant passes at smaller

thresholds. Coefficients found insignificant are coded as either zerotree root or

isolated zero based on the previous discussion.

2. Subordinate pass follows the dominant pass and outputs an L or H which further

refines the specification of the magnitude of the coefficient put out to the decoder. i.e.

it designates whether the coefficient is greater than Ti/2 (H) or less (L). Thus it

reduces the uncertainty interval of the magnitude of the coefficient (or equivalently

the quantizer step-size) by half. In the decoder, the reconstruction value used can be

anywhere in that uncertainty interval. The magnitudes on the subordinate list are then

sorted in decreasing order of magnitude.

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

threshold is halved Ti-1=Ti/2 before each dominant pass.

QQMM ccooddeerr

The QM coder is a binary arithmetic coder, implying that it codes a stream of only two

symbols 0 and 1. A source with mult iple symbols can also be coded by decomposing each

symbol using a binary decision tree.

Advantages of a binary arithmetic coder lie in that it generates compressed data as a finite-

precision fraction which identifies an interval on the number line. Each symbol is encoded or

decoded on the basis of a probability estimate which determines where the binary arithmetic

coder splits the interval into two subintervals. The current symbol determines which

subinterval becomes the new interval. When the size of the new interval drops below a

minimum value, renormalization shifts the precision until it is greater than or equal to the

minimum size. With each shift, a bit is produced for the compressed data stream.

IInnppuutt bbiinnaarryy aallpphhaabbeett was generated using a tree of binary decisions. As noted previously

the output of the EZW code was a stream of symbols containing the symbols: P, N, Z,

I, H and L. The following binary decision tree was used based on the probability of

occurrence of these symbols.

L

0 1

H

0 1

0 1

Z 0 1

I 0 1

P N

Symbol stream

SSyymmbbooll oorrddeerriinngg aanndd iinntteerrvvaall ssuubbddiivv iissiioonn When the QM coder codes a binary decision (0

or 1), it does not directly assign intervals to these symbols. Rather, it assigns intervals to the

more probable symbol (MPS) and the less probable symbol (LPS) such that the LPS

subinterval is always above the MPS subinterval.

If the interval is A and the LPS probability estimate is Qe, the MPS probability estimate

should ideally be (1-Qe). The lengths of the respective subintervals are then A Qe and

( )A Qe -1 . In the QM coder, code stream C points to the bottom of the current interval so

that we need to add to the code stream only when an LPS occurs.

CCooddiinngg aa ssyymmbbooll changes the interval and code stream as follows:

After MPS: ( )QeAAC -= 1 unchanged is After LPS: ( ) QeAAQeACC =-+= 1

RReennoorrmmaall iizzaatt iioonn is done by doubling (shifting left) A, and in accordance C, each time the

interval falls below some minimum value (0.75). This enables finite precision representation

C+A

C+A.(1-Qe)

C

A.Qe

A.(1-Qe)

LPS

MPS

for A by confining A within the limits 0.75-1.5 (since 1.5 is double of 0.75) and also aids to

avoid the multiplication A.Qe as explained below.

EElliimmiinnaatt iioonn ooff mmuulltt iipplliiccaatt iioonnss bbyy aapppprrooxx iimmaatt iioonn We desire to have A 1, so that A.Qe

Qe. Keeping A bounded in the range 0.75 = A

intervals is interchanged. Since A-Qe < Qe < 0.5, both subintervals are less than 0.5 and

renormalization must occur. Thus conditional exchange is always done after a

renormalization.

AAddaapptt iivvee PPrroobbaabbiilliittyy EEsstt iimmaatt iioonn The estimation process is based on a form of approximate

counting in which the interval register normalization is used to estimate the MPS and LPS

symbol counts. Whenever a renormalization occurs i.e. after every LPS and after every MPS

that requires renormalization, a new probability estimate is obtained from a look-up table that

provides a bigger Qe value when LPS renormalization occurs and a smaller Qe value when a

MPS renormalization occurs. Thus, the distribution of probability estimates gets centered

about the desired value.

Markov-chain modeling of probability estimation The estimation process can be regarded

as a markov-chain where each state represents one probability estimate (and also contains the

sense of MPS) i.e. each state S can be thought of as a structure that contains a Qe(S) and a

MPS(S). An index Index(S) points to the current state. Index(S) is initialized to the first state

in the look-up table which corresponds to a Qe(S) ~ 0.5 (i.e. assuming initially that both

symbols are almost equally probable). The state machine has mirror symmetry about the

change in the sense of MPS. The look-up table used for the QM-coder estimation state

machine thus consists of the state index, associated Qe, NMPS (Next state index after MPS

renormalization), NLPS (Next state index after LPS) and Switch (if 1 indicates MPS sense

needs to be switched).

Fig. Markov-chain modeling of probability estimation

DDiiff ffeerreenncceess bbeettwweeeenn tthhee QQ aanndd QQMM ccooddeerr::

1. Interval subdivision is improved in the QM coder by introducing conditional

exchange where the sense of MPS is reversed if the MPS subinterval becomes less

than the LPS subinterval. This leads to better compression.

2. QM coder allows carry to be resolved completely before transmitting next byte thus

preventing carry propagation problems.

3. The QM coder uses an initial (fast attack states) to arrive quickly at approximately the

right probability estimate and has more states in the probability estimation machine

thus is more responsive to unstable statistics.

Qe(Sj) MPS(Sj)

Qe(Sk) MPS(Sk)

Qe(Si) MPS(Si)

Index(S)

MPS renormalization

Qe(Si) < Qe(Sj) < Qe(Sk)

LPS

RReessuullttss

The EZW algorithm was implemented and it can be seen how Wavelet transform

concentrates all the energy in few significant coefficients. This feature is exploited by the

zerotree coding to discard the insignificant coefficients and achieve good compression.

The QM coder was also tested using the sequence given in (4) and detailed results compared

with the one given. The QM coder works as a perfect lossless coder.

The following compression ratios were obtained using a combination of the EZW and QM

coder:

Image Bits needed for image transmission Size x 8

compressed stream length (output of QM coder)

Compression ratio

Fruits.png 512x512x8 = 2097152 129472 16.2

Lena.bmp 512x512x8 = 2097152 138600 15.1

Barbara.png 512x512x8 = 2097152 196789 10.6

Clearly, lesser are the details the more is the compression efficiency since fewer coefficients

are required to capture those details.

Reconstruction results on images however could not be obtained due to some sign

manipulation problem in the EZW decoder. However the magnitudes of the reconstructed

coefficients agreed pretty well with the original coefficients.

CCoonncclluussiioonn

This project was a good learning experience about the power of wavelet coding-based

compression techniques for image compression. We were able to understand and reproduce

the results in references (1) and (4) by implementing EZW and QM coder in Matlab.

However, given the time, many concepts could not be explored and would form topic of

future work. Investigating the effect of different wavelet filters on compression and

reconstruction as well as the concept of embedded bit-coding to achieve the desired bit-rate

are some of these. Further there are many other coders like MQ etc. that were not looked at

and would be interesting to study and compare.

RReeffeerreenncceess

EEZZWW::

1. Embedded Image Coding using Zerotrees of Wavelet Coefficients, J. M. Shapiro, IEEE

trans. on Sig. Proc., vol. 41, No. 12, Dec 1993, pp. 3445-3462.

2. http://perso.wanadoo.fr/polyvalens/clemens/ezw/ezw.html

3. A New, Fast, and Efficient Image Codec Based on Set Partitioning in Hierarchical

Trees, A. Said and W. A. Pearlman, IEEE Trans. Circuits Syst. Video Technol., vol. 6,

June 1996, pp. 243-249.

QQMM--ccooddeerr::

4. JPEG Still Image Data Compression Standard, W. B. Pennebaker & J. L. Mitchell.

5. An overview of the basic principles of the Q-coder adaptive binary arithmetic coder,

W. B. Pennebaker, et al., IBM J. Res. Develop., vol. 32, No. 6, Nov. 1988, pp. 717-725.

6. Probability estimation for the Q-coder, W. B. Pennebaker and J. L. Mitchell, IBM J.

Res. Develop., vol. 32, No. 6, Nov. 1988, pp. 737-751.

7. An introduction to arithmetic coding, Glen G. Langdon, IBM J. Res. Develop., vol. 28,

No. 2, March 1984, pp. 135-149.

SSuummmmaarryy ooff CCooddee

EEZZWW eennccooddeerr ::

Files required: EZW.m

dominant_pass.m

subordinate_pass.m

process_LL_element.m

process_element.m

zerotree.m

decodeEZW.m

QQMM ccooddeerr ::

Files required: Arithmetic_coder.m

Code_MPS.m

Code_LPS.m

makestring.m

Renorm.m

Update_Index_S_after_LPS.m

Update_Index_S_after_MPS.m

byteout.m

Arithmetic_decoder.m

Cond_MPS_exchange_S.m

Cond_LPS_exchange_S.m

Renorm_d.m

byte_in.m