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This is page 1 Printer: Opaque this Blo ck Transforms in Progressive Image Co ding Trac D. Tran and Truong Q. Nguyen 1 Intro duction Blo ck transform co ding and subband co ding have b een two dominant techniques in existing image compression standards and implementations. Both metho ds actually exhibit many similarities: relying on a certain transform to convert the input image to a more decorrelated representation, then utilizing the same basic building blo cks such as bit allo cator, quantizer, and entropy co der to achieve compression. Blo ck transform co ders enjoyed success rst due to their low complexity in im- plementation and their reasonable p erformance. The most p opular blo ck transform co der leads to the current image compression standard JPEG [1] which utilizes the 8 8 Discrete Cosine Transform DCT at its transformation stage. At high bit rates 1 bpp and up, JPEG o ers almost visually lossless reconstruction image quality. However, when more compression is needed i.e., at lower bit rates, an- noying blo cking artifacts showup b ecause of two reasons: i the DCT bases are short, non-overlapp ed, and have discontinuities at the ends; ii JPEG pro cesses each image blo ck indep endently. So, inter-blo ck correlation has b een completely abandoned. The development of the lapp ed orthogonal transform [2] and its generalized ver- sion GenLOT [3, 4] helps solve the blo cking problem to a certain extent by b or- rowing pixels from the adjacent blo cks to pro duce the transform co ecients of the current blo ck. Lapp ed transform outp erforms DCT on two counts: i from the anal- ysis viewp oint, it takes into accountinter-blo ck correlation, hence, provides b etter energy compaction that leads to more ecient entropy co ding of the co ecients; ii from the synthesis viewp oint, its basis functions decay asymptotically to zero at the ends, reducing blo cking discontinuities drastically.However, earlier lapp ed- transform-based image co ders [2,3,5]have not utilized global information to their full advantage: the quantization and the entropy co ding of transform co ecients are still indep endent from blo ck to blo ck. Recently, subband co ding has emerged as the leading standardization candidate in future image compression systems thanks to the development of the discrete wavelet transform. Wavelet representation with implicit overlapping and variable- length basis functions pro duces smo other and more p erceptually pleasant recon- structed images. Moreover, wavelet's multiresolution characteristics have created an intuitive foundation on which simple, yet sophisticated, metho ds of enco ding the transform co ecients are develop ed. Exploiting the relationship between the parent and the o spring co ecients in a wavelet tree, progressivewavelet co ders [6, 7, 9] can e ectively order the co ecients by bit planes and transmit more sig- ni cant bits rst. This co ding scheme results in an emb edded bit stream along with many other advantages such as exact bit rate control and near-idemp otency 1. Blo ckTransforms in Progressive Image Co ding 2 p erfect idemp otency is obtained when the transform maps integers to integers. In these subband co ders, global information is taken into account fully. From a frequency domain p oint of view, the wavelet transform simply provides an o ctave-band representation of signals. The dyadic wavelet transform is analogous to a non-uniform-band lapp ed transform. It can suciently decorrelate smo oth images; however, it has problems with images with well-lo calized high frequency comp onents, leading to low energy compaction. In this app endix, we shall demon- strate that the emb edded framework is not only limited to the wavelet transform; it can b e utilized with uniform-band lapp ed transforms as well. In fact, a judicious choice of appropriately-optimized lapp ed transform coupled with several levels of wavelet decomp osition of the DC band can provide much ner frequency sp ectrum partitioning, leading to signi cant improvementover currentwavelet co ders. This app endix also attempts to shed some lightonto a deep er understanding of wavelets, lapp ed transforms, their relation, and their p erformance in image compression from amultirate lter bank p ersp ective. 2 The wavelet transform and progressive image transmission Progressive image transmission is p erfect for the recent explosion of the internet. The wavelet-based progressive co ding approach rst intro duced by Shapiro [6] relies on the fundamental idea that more imp ortant information de ned here as what decreases a certain distortion measure the most should b e transmitted rst. As- sume that the distortion measure is the mean-squared error MSE, the transform is paraunitary, and transform co ecients c are transmitted one by one, it can b e i;j jc j i;j , where N is the total number proven that the mean squared error decreases by N of pixels [16]. Therefore, larger co ecients should b e transmitted rst. If one bit is transmitted at a time, this approach can b e generalized to ranking the co ecients by bit planes and the most signi cant bits are transmitted rst [8]. The progressive transmission scheme results in an emb edded bit stream i.e., it can be truncated at any p ointby the deco der to yield the b est corresp onding reconstructed image. The algorithm can b e thought of as an elegant combination of a scalar quantizer with p ower-of-two stepsizes and an entropy co der to enco de wavelet co ecients. Emb edded algorithm relies on the hierachical co ecients' tree structure that we called a wavelet tree, de ned as a set of wavelet co ecients from di erent scales that b elong in the same spatial lo cality as demonstrated in Figure 1a, where the tree in the vertical direction is circled. All of the co ecients in the lowest frequency band make up the DC band or the reference signal lo cated at the upp er left corner. Besides these DC co ecients, in a wavelet tree of a particular direction, eachlower-frequency parent node has four corresp onding higher-frequency o spring nodes. All co ecients b elow a parentnode in the same spatial lo calityis de ned as its descendents. Also, de ne a co ecient c to be signi cant with resp ect to i;j a given threshold T if jc j T , and insigni cant otherwise. Meaningful image i;j statistics have shown that if a co ecient is insigni cant, it is very likely that its o spring and descendents are insigni cant as well. Exploiting this fact, the most sophisticated emb edded wavelet co der SPIHT can output a single binary marker 1. Blo ckTransforms in Progressive Image Co ding 3 to representvery eciently a large, smo oth image area an insigni cant tree. For more details on the algorithm, the reader is refered to [7]. parent offspring descendents xx xx xxxx xxxxxxxx xx xxxx xxxxxxxx xxxx xxxxxxxx xx xxxx xxxxxxxx xxxxxxxx xxxxxxxx xx xx xxxxxxxx xx xx xxxxxxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx xxxx (a) (b) FIGURE 1. Wavelet and blo ck transform analogy. Although the wavelet tree provides an elegant hierachical data structure which facilitates quantization and entropy co ding of the co ecients, the eciency of the co der still heavily dep ends on the transform's ability in generating insigni cant trees. For non-smo oth images that contain a lot of texture, the wavelet transform is not as ecient in signal decorrelation comparing to transforms with ner frequency selectivity and sup erior energy compaction. Uniform-band lapp ed transforms hold the edge in this area. 3 Wavelet and blo ck transform analogy Instead of obtaining an o ctave-band signal decomp osition, one can have a ner uniform-band partitioning as depicted in Figure 2 drawn for M = 8. The ner frequency partitioning compacts more signal energy into a fewer number of co ef- cients and generates more insigni cant ones, leading to an enhancement in the p erformance of the zerotree algorithm. However, uniform lter bank also has uni- form downsampling all subbands now have the same size. A parent no de do es not have four o spring no des as in the case of the wavelet representation. How would one come up with a new tree structure that still takes full advantage of the inter-scale correlation b etween blo ck-transform co ecients? The ab ove question can be answered by investigating an analogy between the wavelet and blo ck transform as illustrated in Figure 1. The parent, the o spring, and the descendents in a wavelet tree cover the same spatial lo cality, and so are the co ecients of a transform blo ck. In fact, a wavelet tree in an L-level decomp osition L is analogous to a 2 -channel transform's co ecient blo ck. The di erence lies at the bases that generate these co ecients. It can be shown that a 1D L-level wavelet 1. Blo ckTransforms in Progressive Image Co ding 4 |H| DC band |H| (a) (b) π π ω π π π π ω 0 2 o o o π 0 MM 8 4 2 FIGURE 2. Frequency sp ectrum partitioning of a M -channel uniform-band transform b dyadic wavelet transform. decomp osition, if implemented as a lapp ed transform, has the following co ecient matrix: 3 2 n n n ] h [ ] h [ ] h [n] h [ 0 0 0 0 L2 L1 2 2 2 n n n 7 6 ] h [ ] h [ ] h [n] h [ 0 L2 1 L1 0 0 2 2 2 7 6 n n 7 6 h [n] h [ ] ] h [ 0 0 L2 1 2 2 7 6 n n 7 6 h [n] h [ ] h [ ] 0 0 1 L2 2 2 7 6 7 6 .

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