Coding the Wavelet Spatial Orientation Tree with Low Computational Complexity

Coding the Wavelet Spatial Orientation Tree with Low Computational Complexity Yushin Cho 1, Amir Said 2, and William A. Pearlman 1 1 Center for Image Processing Research Department of Electrical, Computer, and Systems Engineering Rensselaer Polytechnic Institute 110 8th St. Troy, New York 12180-3590 Phone: 518-276-6082, Fax: 518-276-8715, E-mail: pearlw@ecse.rpi.edu 2 Imaging Technology Department Hewlett-Packard Laboratories Palo Alto, CA 1 Abstract A very fast, low complexity algorithm for resolution scalable and random access decoding is presented. The algorithm is based on AGP (Alphabet and Group Partitioning) [1], where non bit-plane coding is used for speed improvement. The two-stage differential coding method of the dynamic range of coefficients magnitudes is devised. The dynamic range of coefficients magnitudes in a spatial orientation tree is hierarchically represented. The hierarchical dynamic range coding naturally enables resolution scalable representation of wavelet transformed coefficients. A zerotree representation is also implicitly exploited by our dynamic range coding scheme. Experiments show that the our suggested coding model lessens the computational burden of bit-plane based image coding, three times in encoding and six times in decoding. 1

2 Introduction We wish to satisfy the two criteria of wavelet based image coding : the computational complexity and resolution scalability. A quality or rate scalable coding scheme, represented by EZW, SPIHT, and JPEG2000 (EBCOT), receives every attention in image coding society in recent years. However, there are some application areas where the emphasis is put on other things. The speed is one of them. In this article, we put the highest priority on the time complexity of an image coding algorithm, which often is the most important virtue in real applications. The original EZW and SPIHT do not support resolution scalability since they do not code the resolution boundaries. Random access decoding functions are not supported by them too. The suggested algorithm, PROGRES (Progressive Resolution Decompression) is a extremely fast version of SPIHT that has full capability of resolution scalable random access decoding. It chooses a quality factor for lossy compression of any size image and can decompress a given arbitrary region to any resolution very quickly. It is an excellent choice for remote sensing and GIS applications, where rapid browsing of large images is necessary. 2.1 Previous Works The bit-plane coding scheme, exploited by EZW [2], SPIHT, JPEG2000, and others enables the embedded coding and so the precise rate control of the image compression is possible as a result. However, there is an overhead in computations since only one bit of every coefficient being coded is actually necessary for significance decision. Thus, in modern computers which have byte-addressable CPUs, all remaining bits except the one bit are the culprits of wastes of computing resources. The JPEG2000 (EBCOT) [?] [4] and the FS-SPIHT (Fully Scalable SPIHT) by Danyali and Mertins [8] supports both resolution and SNR scalability. They provide a bitstream that can be easily adapted (reordered) to a given bandwidth and resolution requirements by a simple transcoder (parser). It adds spatial scalability feature without sacrificing the SNR (or rate, quality) embeddedness property. However, both of them can not be qualified in time-critical encoding and decoding application because of the overhead from bit-plane coding. There was a hybrid approach between pure bit-planed based embedded coding and non bit-plane based coding [5] [7]. In these methods, once a coefficient is classified as significant during a certain bit-plane pass, all less significant bits are encoded together so that a following refinement pass is not needed. AGP (Alphabet and Group Partitioning) [1] introduced the general methodology

of entropy coding by sample-set partitioning, which is faster than bit-plane based image coding scheme. The issue of resolution scalability was not discussed in the article. In the article, it is mentioned that the main difference between AGP and the coding scheme in 2D-SPIHT [3] is that the latter was designed for embedded coding, while the former is directly applied to the hierarchical trees of already quantized wavelet coefficients. Though it is not mentioned in the article, the execution speed of coding is indeed faster than original 2D SPIHT with small loss of quality. The recent work, LTW (Lower-Tree Wavelet) by Oliver and Malumbres [11] presented another solution for resolution scalable wavelet image coding with low complexity, which also uses non-embedded coding. The lower-trees are equivalent to the zerotrees of pre-quantized wavelet coefficients, where the quantization step size is 2 rplane (rplane is the number of most insignificant bit-planes). Arithmetic coded symbols for zerotree, isolated zero, and number of bits to represent the magnitude of coefficients are used. It showed better compression performance than SPIHT and JPEG2000 (JASPER), with 2.5 times faster speed than SPIHT. 2.2 Suggested Idea We reduce the computational burden by not using the bit-plane based approach as shown in AGP. With the sacrifice of quality scalability, we can design three times faster encoding algorithm and six times faster decoding algorithm. The major modification of this work from the original SPIHT is that it is based on the sample-set partitioning entropy coding scheme in the AGP (Alphabet and Group Partitioning) [1]. As in AGP, the presented method is not using a bit-plane coding. The difference of the presented method from a sample-set partitioning in AGP is found in the coding method of coefficients in spatial orientation trees of wavelet coefficients. Instead of binary masks (indicating which descendent has the max value) and maximum values in AGP, the two-stage differential coding method of the dynamic range of coefficients magnitudes is devised. Viewing a spatial orientation tree as a set, a set is partitioned into subsets and each subset will have different dynamic range. Since the goal of compression is to save bits, we want to reduce the number of bits to represent the dynamic ranges of each set as much as possible. Here, we can use the conditioned, common information of dynamic range reduction after each set partitioning is done, which leads to saving of bit resources. The two-stage differential coding is applied to every partitioning event of spatial orientation tree. This works well if the magnitude of wavelet coefficient is decreasing fast along increasing frequency subbands. And it is observed as true that the coefficient energy is fast decreasing in natural images. The amount of expected savings in coding is

proportional to the decreased number of bits to represent the decreased dynamic range, which occurs at almost every partitioning. 3 Coding the Dynamic Ranges of Coefficients Magnitudes in the Set 3.1 Representing the Dynamic Range of Coefficients in the set The energy in the wavelet transformed image can be defined as amplitude in each spatial location and frequency band. It is represented as magnitude of coefficients. An wavelet coefficient c i,j at location (i, j) has its value, represented by a sign and a magnitude of the coefficient. The location indices (i, j) for coefficients are defined over the wavelet domain. The sign is represented by one bit but the number of required bits should be known to represent the magnitude compactly. The set or spatial orientation tree at location (i, j) is denoted as s i,j. The dynamic range of a coefficient magnitude is represented by the number of bits, k, where the magnitude varies in the range of [0, 1,, 2 k 1]. For example, if the dynamic range of a coefficient is 3, it can have the values varying from -7 to +7 with additional sign information. Each set (a spatial orientation tree) will have different dynamic range of magnitudes in it, based on the activity of its coefficients. We define the dynamic range of the set s i,j as r i,j = log 2 (max cm,n si,j c m,n + 1), which amounts for how many bits are required to represent every coefficient magnitude in the set. For example, if the maximum magnitude is 7 in the set s 0,1, then the dynamic range r 0,1 is 3. 3.2 Sample-Set Partitioning Algorithm from AGP The recursive algorithm of sample-set partitioning entropy coding from AGP, which is our basis, is briefly described as: function SetPartitioning (a set) 1. Encode the maximum value of the set. 2. Partition the set into a fixed number of n subsets 3. if the set cannot be partitioned any more, return. 4. Encode the n-bits binary mask, which indicates whether each subset has the maximum value.

5. for each subset i = 1 to n (a) SetPartitioning (subset i) The condition in step 3 is met if the set is a single sample, i.e. a size 1x1 set in 2D image. 3.3 Differential Coding of Energy Range in the quadrisected set Most of the energies in children subsets are usually smaller than their parent set. When a parent set is partitioned into its children subsets rooted in the next higher resolution band, each subset will have different range of coefficient magnitudes. And, by the property that higher frequency subbands tend to have lower energy or smaller magnitude coefficients, child subset in higher resolution will be likely to have shorter dynamic range of magnitudes than their parent set. In the highest resolution or frequency bands, many coefficients can be represented by a very small dynamic range, such as [0,1], which can be represented with only one bit. dynamic range r parent,max resolution k r i,j resolution k+1 four subsets d base r max r 2i,2j+1 Partition r 2i,2j r 2i+1,2j r 2i+1,2j+1 d local, 2i+1,2j+1 resolution a set, s i,j s 2i,2j s 2i,2j+1 s 2i+1,2j s 2i+1,2j+1 resolution k resolution k+1 Figure 1: Two-Stage Differential Coding of Dynamic Ranges : the dynamic range for the subset s 2i+1,2j+1 is reconstructed by : r 2i+1,2j+1 = r i,j d base d local,2i+1,2j+1, where the information of r i,j d base is common to every subset s m,n. Thus, it is a good idea to code the difference of dynamic ranges of energy between a parent set and its child subset to represent the dynamic range of energy in each subset. The differential coding for children is performed in two stages as shown in Figure 1. Assuming a parent set s i,j is partitioned into four subsets, {s 2i,2j, s 2i,2j+1, s 2i+1,2j, s 2i+1,2j+1 }, d base = r i,j max(r 2i,2j, r 2i,2j+1, r 2i+1,2j, r 2i+1,2j+1 )

is coded first. And then, for each child subset s m,n, d local,m,n = d base r m,n is coded next. So, given the information of r i,j and d base, the dynamic range of s m,n can be reconstructed if d local,m,n is known, i.e. r m,n = r i,j d base d local,m,n, where the information of r i,j d base is common to every subset s m,n. The coded information for the tree s i,j with two resolution scales will be : (r i,j, c i,j ), d base, (d local,2i,2j, c 2i,2j ), (d local,2i,2j+1, c 2i,2j+1 ), (d local,2i+1,2j, c 2i+1,2j ), (d local,2i+1,2j+1, c 2i+1,2j+1 ). For the subsets rooted in highest resolution, which are coefficients in HL1 or LH1 or HH1, d local,m,n is not exploited. Instead, they are represented by r max bits of unary code, i.e. the dynamic range is r max. There is a reason why we choose d base rather than r max. From our experience, it is more probable that d base r max, i.e. P (d base r max ) > 0.5 in any wavelet transformed image. And P (d base r max ) is getting closer to 1 for lower bit rates. This explains that coding d base will cost less bits than coding r max. The goal of differential coding of dynamic ranges is to save as many bits as possible, using the property that every subset from the same parent set shares the conditioned information, i.e. the shorter dynamic ranges than their parent s. The amount of bit saving by using d base for each group of four subsets is simply 3 d base bits. 4 Coding algorithm The encoding algorithm of PROGRES is given here. We assume that LL subband has one wavelet coefficient. Thus, the algorithm works on size 2 M 2 M wavelet coefficients if M levels of wavelet decomposition is performed. The traversal of the set (represented by a tree) is performed in a fashion similar to BFS (Breadth-First- Search) algorithm. The level (root corresponds to the lowest resolution) of the tree is according to the resolution of wavelet domain, where the tree is quadtree or 4-ary for 2 dimensional wavelet transform. The list L contains the sets to be coded. The set s 0,0 rooted in LL subband has three subsets {s 0,1, s 1,0, s 1,1 }, corresponding to subbands HLM,LHM,HHM. Every leaf set in highest frequency subbands HL1 or LH1 or HH1 is not partitioned and so has no subsets. Except these two cases, every set s i,j has four subsets, {s 2i,2j, s 2i,2j+1, s 2i+1,2j, s 2i+1,2j+1 }. Every dynamic range is represented as discussed in section 3.1, which is derived from a log 2 scale of the maximum magnitude in the range. Each sign bit is coded just after the magnitude of a coefficient is coded. // indicates the comments in corresponding statement.

4.1 Encoding 1. Find the maximum dynamic range r parent,max ; 2. if r parent,max = 0 return; // no coefficients to encode? 3. Encode r parent,max ; 4. Initialize a list L a set in the lowest resolution (i.e. LL subband); 5. for each resolution level (from the lowest to the highest) (a) for each set in current resolution level i. Enumerate subsets of the current set; ii. r max maximum dynamic range of subsets in current set; iii. r parent,max maximum dynamic range of current set; iv. d base r parent,max r max ; v. Unary encode d base ; vi. if r max = 0 continue // means, nothing to encode (i.e. zerotree), goto (a) vii. for each subset i A. if subset i is not in the highest resolution; r subset maximum dynamic range of the subset i; d local r max r subset ; Unary encode d local ; if r subset = 0 continue; // means, no more descendants, goto vii. Binary encode the coefficient of subset i using r subset bits; Append subset i to the list L for next resolution coding; B. else // i.e. subset i is in the highest resolution Binary encode the coefficient using r max bits; viii. Remove the current set from the list L; 5 Random Access Decoding The purpose of random access decoding is to enable fast access to interesting area with minimal decoding work. For this, an input image is encoded as independent 2D blocks, so that each block is random accessed (Figure 2). Each image block i has M resolution scales. Each resolution bitstream is identified the length information as shown in Figure 3. The target image block can be accessed by following the length information with minimal decoding. The length information stored in each block causes the inevitable overhead and loss of coding efficiency. The

the lowest resolution the highest resolution b 0,0 b 0,1... b 0,M-1 b 1,0 b 1,1... b 1,M-1... block 0 block 1 Figure 2: Bitstream structure for random access decodable and resolution scalable image. Each resolution γ in block β is notated as b β,γ average block seek time will be Mp N, where M is the number of resolution scales, 2 p is the average skip time, and N is the total number of image blocks. The average n seek time can be improved order O( ) faster by following the method in [10]. log 2 n l i,0 + l i,1 +... + l i,m-1 = length of b i,0,b i,1,...,b i,m-1 b i,0 b i,1... b i,m-1 l i,0 l i,1 l i,m-1 bitstream for block i Figure 3: Bitstream structure for resolution scalability of block i. 6 Experimental Results The compression is performed on the platform of Intel 2.0GHz Xeon processor, MS- Windows System, and Visual C Compiler with no speed optimization. The suggested method is faster than original 2D SPIHT, on average, three times in encoding and six times in decoding. This method naturally gives the resolution scalability. In

addition, with only less than 0.5dB loss of quality, it is easily adapted to random access decoding functionality. Some of this loss comes from the overhead of each block length information, which is used to skip blocks on the bitstream for minimal decoding. The actual elapsed times for compressing and decompressing the grey level image Lenna and Woman, at the rate of 0.5 bpp and 0.25 bpp are shown in Table 1. The 2D SPIHT from RPI is chosen for comparison. Note that wavelet transformation times are not included. The sizes of source images are 512 512 for Lenna and 2048 2048 for Woman. Both are 8 bits/pixel grey-level images. A five level of wavelet decomposition with Daubechies 9/7 filter (non lifting version) is used. Thus, we can obtain five different resolution scales. Rate control is done by a quality factor which pre-quantizes the wavelet coefficients. In PROGRES, the wavelet coefficients are arranged in the way that each 32 32 subimage is coded independently for random access decoding. Both experimented methods do not exploit the entropy coding. Table 1: The comparison of compression and decompression time between original RPI 2D SPIHT and the presented PROGRES Compression (in sec) Decompression (in sec) Image & Bitrate (bpp) 2D SPIHT PROGRES 2D SPIHT PROGRES Lenna 0.5 0.375 0.032 0.140 0.000 Lenna 0.25 0.063 0.032 0.015 0.000 Woman 0.5 1.500 0.515 0.390 0.062 Woman 0.25 1.156 0.485 0.188 0.032 7 Conclusion Non bit-plane coding scheme is applied to compress wavelet transformed images. Two-stage differential coding scheme is presented, which efficiently codes the dynamic range of coefficients magnitudes in the spatial orientation tree. The suggested method is faster than original 2D SPIHT, three times in encoding and six times in decoding. It naturally gives the resolution scalability. In addition, with only less than 0.5dB loss of quality, it is easily adapted to random access decoding functionality. References [1] A. Said and W.A. Pearlman, Low-Complexity Waveform Coding via Alphabet and Sample-Set Partitioning, SPIE Visual Communications and Image Processing, San

Jose, CA, Feb. 1997 [2] J. M. Shapiro, Embedded Image Coding Using Zerotrees of Wavelet Coefficient, IEEE Trans. on Signal Processing, vol. 41, pp. 3445-3462, Dec. 1993. [3] Amir Said and William A. Pearlman, A New Fast and Efficient Image Codec Based on Set Partitioning in Hierarchical Trees, IEEE Trans. on Circuits and Systems for Video Technology, vol. 6., pp. 243-250, June 1996. [4] M. Marcellin, M. Gormish, A. Bilgin, M. Boliek, An Overview of JPEG-2000, Proc. Data Compression Conference, J. A. Storer and M. Cohn, eds., Snowbird, Utah, pp. 523-541, Mar. 28-30, 2000. [5] William A. Pearlman, Trends of Tree-Based, Set Partitioning Compression Techniques in Still and Moving Image Systems, Proceedings Picture Coding Symposium 2001 (PCS-2001), Seoul, Korea, 25-27 April, 2001, pp. 1-8. (Invited, keynote paper) [6] William A. Pearlman, Asad Islam, Nithin Nagaraj, and Amir Said, Efficient, Low- Complexity Image Coding With a Set-Partitioning Embedded Block Coder, IEEE Trans. on Circuits and Systems for Video Technology, Vol. 14, No. 11, pp. 1219-1235, Nov. 2004. [7] Yong Sun, Hui Zhang, and Guangshu Hu, Real-Time Implementation of a New Low- Memory SPIHT Image Coding Algorithm Using DSP Chip, IEEE Trans. on Image Processing, Vol. 11, No. 9, Sep. 2002. [8] H. Danyali and A. Mertins, Fully Spatial and SNR Scalable, SPIHT-Based Image Coding for Transmission over Heterogenous Networks, Journal of Telecommunications and Information Technology, vol. 2, pp. 92-98, 2003. [9] S.-T. Hsiang, Embedded Image Coding using Zeroblocks of Subband/Wavelet Coeffcients and Context Modeling, in Proc. 2001 IEEE Data Compression Conference, pp. 83-92, Snowbird, Utah, Mar. 2001. [10] Yushin Cho, Sungdae Cho, and W. Pearlman, Fast and Constant Time Eandom Access Decoding with log 2 n Block Seek Time, to be published on SPIE Electronic Imaging 05, San Jose, CA, Jan. 2005. [11] Jose Oliver and Manuel Prez Malumbres Fast and Efficient Spatial Scalable Image Compression using Wavelet Lower Trees, in Proc. 2003 IEEE Data Compression Conference, pp. 133-142, Snowbird, Utah, Mar. 2003.