+ (50% contribution by each member)

Size: px

Start display at page:

Download "+ (50% contribution by each member)"

Betty Stevens
6 years ago
Views:

1 Image Coding using EZW and QM coder ECE 533 Project Report Ahuja, Alok + Singh, Aarti + + (50% contribution by each member)

2 Abstract 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. Overview 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.

3 Statement of Work 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.

4 EZW based Image Coding 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.) Discrete Wavelet Transform 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.

5 Significance Map Encoding 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 LL 3 ) has three children (one each in HL 3, LH 3 and HH 3 ). Scanning of the coefficients is performed so that no child is scanned before its parent.

6 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. Successive-Approximation Entropy-Coded Quantization 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 T 0, chosen so that all transform coefficients are < 2T 0. Successive thresholds are obtained as T i = T i-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, T i. 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

7 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 T i /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 T i-1 =T i /2 before each dominant pass.

8 QM coder The QM coder is a binary arithmetic coder, implying that it codes a stream of only two symbols 0 and 1. A source with multiple 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 finiteprecision 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. Input binary alphabet 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. Symbol stream 0 1 L 0 1 H 0 1 Z 0 1 I 0 1 P N

9 Symbol ordering and interval subdivision 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. C+A LPS A.Q e C+A.(1-Q e ) MPS A.(1-Q e ) C 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 ( Qe) A 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. Coding a symbol changes the interval and code stream as follows: After MPS: C is unchanged A = A ( 1 Qe) After LPS: C = C + A ( 1 Qe) A = A Qe Renormalization 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

10 for A by confining A within the limits (since 1.5 is double of 0.75) and also aids to avoid the multiplication A.Qe as explained below. Elimination of multiplications by approximation We desire to have A 1, so that A.Qe Qe. Keeping A bounded in the range 0.75 = A <1.5 achieves this. After MPS: After LPS: C is unchanged A = A 1 ( Qe) = A A Qe A Qe ( 1 ) C = C + A Qe = C + A A Qe C + A Qe A = A Qe Qe Important note: Since after every LPS occurrence A Qe < 0.5, according to the discussion on renormalization, a renormalization occurs everytime an LPS occurs. Integer representation In the representation chosen for QM-coder, the upper bound for interval A (1.5) is defined as X and lower bound (0.75) as X (The register for A is 16 bit and is initialized as X 0000 (equivalent to X ). This makes multiplication by shifts easier). Thus the correspondence between decimal representation at implementation is given by (1/0.75)(X 8000 ). Conditional exchange When Qe is large, the size of the MPS subinterval may become less than the size of the LPS subinterval i.e. A-Qe < Qe. To avoid the conflict between the relation of probability and subinterval size for a symbol, the assignment of LPS and MPS to the two

11 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. Adaptive Probability Estimation 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).

12 MPS renormalization LPS Qe(S i ) MPS(S i ) Qe(S j ) MPS(S j ) Qe(S k ) MPS(S k ) Index(S) Qe(S i ) < Qe(S j ) < Qe(S k ) Fig. Markov-chain modeling of probability estimation Differences between the Q and QM coder: 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.

13 Results 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 = Lena.bmp 512x512x8 = Barbara.png 512x512x8 = 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.

14 Conclusion 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.

15 References EZW: 1. Embedded Image Coding using Zerotrees of Wavelet Coefficients, J. M. Shapiro, IEEE trans. on Sig. Proc., vol. 41, No. 12, Dec 1993, pp 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 QM-coder: 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 Probability estimation for the Q-coder, W. B. Pennebaker and J. L. Mitchell, IBM J. Res. Develop., vol. 32, No. 6, Nov. 1988, pp An introduction to arithmetic coding, Glen G. Langdon, IBM J. Res. Develop., vol. 28, No. 2, March 1984, pp

16 Summary of Code EZW encoder: Files required: EZW.m dominant_pass.m subordinate_pass.m process_ll_element.m process_element.m zerotree.m decodeezw.m QM coder: 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

Module 5 EMBEDDED WAVELET CODING. Version 2 ECE IIT, Kharagpur

Module 5 EMBEDDED WAVELET CODING Lesson 13 Zerotree Approach. Instructional Objectives At the end of this lesson, the students should be able to: 1. Explain the principle of embedded coding. 2. Show the