We report on the development of an adaptive wavelet image coder based on stack-run representation of the quantized coefficients. The coder works by selecting an optimal wavelet packet basis for the given image and encoding the quantization indices for significant coefficients and zero runs between coefficients using a 4-ary arithmetic coder. Due to the fact that our coder exploits the redundancies present within individual subbands, its addressing complexity is much lower than that of the wavelet zerotree coding algorithms. Experimental results show coding gains of up to 1:4dB over the benchmark wavelet coding algorithm

Awan, A. Majid

Husain, S. Afaq

Rajpoot, Nasir M. (Nasir Mahmood)

English

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http://wrap.warwick.ac.uk           Original citation: Awan, A., Rajpoot, Nasir M. (Nasir Mahmood) and Husain, S. (2003) Stack-run adaptive wavelet image compression. In: IEEE Data Compression Conference (DCC 2003), Utah, US, 25-27 Mar 2003. Published in: Data Compression Conference, 2003. Proceedings. DCC 2003 p. 414.  Permanent WRAP url: http://wrap.warwick.ac.uk/61273              Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions.  Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners.  To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.  Copies of full items can be used for personal research or study, educational, or not-for  profit purposes without prior permission or charge.  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For more information, please contact the WRAP Team at: publications@warwick.ac.uk  Stack-Run Adaptive Wavelet Image CompressionA. Majid Awan∗ Nasir M. Rajpoot† S. Afaq Husain‡AbstractWe report on the development of an adaptive wavelet image coder basedon stack-run representation of the quantized coefficients. The coder works byselecting an optimal wavelet packet basis for the given image and encoding thequantization indices for significant coefficients and zero runs between coeffi-cients using a 4-ary arithmetic coder. Due to the fact that our coder exploitsthe redundancies present within individual subbands, its addressing complexityis much lower than that of the wavelet zerotree coding algorithms. Experimen-tal results show coding gains of up to 1.4dB over the benchmark wavelet codingalgorithm.1 IntroductionWavelet transform based image coding methods have gained popularity during thelast decade or so [11, 10, 5, 14, 15]. This is mainly due to localization propertiesof wavelets in both time (space) and frequency resulting in a sparse representationwith redundancies in and across subbands that can be exploited. Wavelets, however,are not well-suited [6] to represent oscillatory patterns, a special form of texture.Oscillatory variations of intensity can only be described by the small scale waveletcoefficients. Unfortunately, those small scale coefficients carry very little energy, andare often quantized to zero, even at high bit rates. Fingerprints or seismic signals area few examples of non-wavelet-friendly signals. Wavelet packets [2] were developedin order to adapt the underlying wavelet bases to the contents of a signal. The basicidea is to allow nonoctave subband decomposition to adaptively select the best basisfor a particular signal. Results from various image coding methods based on waveletpackets [6, 8] show that they are particularly good in coding images with oscillatorypatterns. There has, however, been relatively much less research on wavelet packetimage coding.The zerotree quantization, proposed first by Shapiro [11], is an effective way ofexploiting the self-similarities among high-frequency subbands at various resolutions.∗Faculty of Computer Science & Engineering, GIK Institute of Engineering Sciences and Tech-nology, Topi, Pakistan; email: gcs0105@giki.edu.pk†Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK; telephone:+44 (24) 7657-3795; email: nasir@dcs.warwick.ac.uk‡Faculty of Computer Science & Engineering, GIK Institute of Engineering Sciences and Tech-nology, Topi, Pakistan; email: syed a h@giki.edu.pk1The success of wavelet based image coding methods has widely been perceived to bedue to across-scale similarities which could enable encoding of a set of wavelet coef-ficients by a single codeword, called a zerotree. However, it has become increasinglyclear [14, 6] that exploitation of redundancies present within subbands can also resultin coding gains comparable to those of the state-of-the-art in image compression.In this paper, we present an adaptive wavelet transform based image coder whichemploys a stack-run representation [14] for quantized transform coefficients in orderto benefit from the intra-subband redundancies. Our compression algorithm can bedivided into four parts: In the first part, an adaptive wavelet packet basis is selectedfor representing the given image using certain entropy-based cost function. We foundthat the Coifman-Wickerhauser (CW) entropy cost function [3] does not often resultin the best wavelet packet basis yielding the highest compression performance forour coding algorithm. Next, the wavelet packet coefficients are quantized using anoptimal scalar quantizer for Laplacian distribution [13]. In the third part, we representthe quantized coefficients with stack-run coding using 4-symbol set of [14] whichgenerates a redundant symbol stream. Finally, this symbol stream is entropy codedusing a high order arithmetic coder. We use a symbol set for the arithmetic codingof run/level values in which context information is used to enable multiple uses ofa single symbol. Our coder offers the following advantages: (a) a relatively simpleand therefore fast coding algorithm, (b) capable of progressive transmission, and (c)coding performance consistently comparable with and often outperforming two state-of-the-art image coders [10, 6].This paper is organized as follows. In the next section, a brief review of waveletpacket transform is presented. The coder algorithm is described in further detail inSection 3. Experimental results of the new coder are presented in Section 4. Finally,the paper concludes with some remarks on coder’s nature and performance.2 Adaptive Wavelet TransformAdaptive wavelet transform, also known as the wavelet packet transform, for imagescan be regarded as an adaptive wavelet subband decomposition that lifts the limi-tation of only decomposing the lowpass subband. This adaptation of representationbasis to the image contents can be achieved in practical applications if a fast basisselection algorithm is employed to select the best basis from among the huge libraryof possible bases.2.1 Best Basis SelectionCoifman and Wickerhauser [3] suggested to use a fast dynamic programming algo-rithm with a time complexity of O(N logN), where N is the number of pixels in theimage, to select the best basis according to a given cost function C. A key criterionmust be met in order to invoke a dynamic programming strategy [4]: the cost-functionshould be separable or additive; ie, C(x) =∑k µ(xk), where xk, k = 0, 1, . . . , N − 1,are the elements of wavelet packet decomposed image x, and µ is a positive function2such that µ(0) = 0.2.2 Role of a Cost FunctionThe notion of the best wavelet packet basis is limited to the cost function basedon which the full wavelet packet tree is pruned to obtain an optimal tree. It is,therefore, crucial that an appropriate cost function is chosen taking into considerationthe quantization strategy employed by the coder. Initially, Coifman andWickerhauser[3] usedh(x) = −∑k( |xk|2||x||2 log|xk|2||x||2)(1)as a cost function. Ramchandran and Vetterli [9] used the optimal bit allocationalgorithm of Shoham and Gersho [12] to distribute the budget across the nodes ofthe wavelet packet tree, and they select the best basis according to the followingrate-distortion criterionC(x, Q, λ) = D(x, Q) + λR(x, Q). (2)Given a set of quantizers Q, the rate R(x, Q) is estimated with the first-order ap-proximation of the entropy H(Q(x)), and the distortion D(x, Q) is defined as themean square error (MSE). The selection of the best basis involves three embeddednonlinear optimization problems. The overall complexity of the approach in [9] is,therefore, quite high.Since a scalar quantizer is used to quantize the transform coefficients, we choseto select the best wavelet packet basis by simply using entropy as a criterion for thesplit/merge decisions. This also enables to reduce substantially the computationalcomplexity of the algorithm. We experimented with three additive entropy-based costfunctions: the CW-entropy h(x), the lp-norm (where p = 1) given byC1(x) =∑k|xk| (3)and the energy entropy given byC2(x) =∑klog(x2k). (4)3 The Coder Algorithm3.1 QuantizationWithin each subband, distribution of the wavelet packet coefficients can be approx-imated by a Laplacian distribution [1]. We used the optimal entropy constrainedscalar quantizer for Laplacian distribution presented in [13]. It is a uniform scalarquantizer with a symmetric dead-zone [−∆+δ,∆−δ] and a step size ∆. The optimalreconstruction offset in the sense of MSE is: δ = 1−∆(e−∆/(1−e−∆)). A dichotomicsearch is employed to find the optimal value of ∆ that uses up the bit budget.33.2 Stack-Run RepresentationThe stack-run representation of [14] partitions the quantized transform coefficientsinto two groups containing zero valued and nonzero valued, referred to as significant,coefficients. Each significant coefficient is represented in binary notation as a stackor a column of bits with the MSB at the top and the LSB at the bottom. A subbandis encoded by starting in one corner of the subband and performing a raster scandescribed in terms of stack and run representations, where ‘stack’ is the magnitudeand sign of the significant coefficient and ‘run’ is the number of zerovalued coefficientsencountered before the next significant coefficient.The symbol set S consists of four symbols; S = {0, 1,+,−}, with the symbolshaving following meanings: ‘0’ is used to signify a binary bit value of 0 in encoding ofsignificant coefficients, ‘1’ is used for binary 1 in significant coefficients, but it is notused for the MSB, ‘+’ is used to represent the MSB of a significant positive coefficient;in addition ‘+’ is used for binary 1 in representing run lengths, and ‘−’ is used for theMSB of negative significant coefficients, and for binary 0 in representing run lengths.Since all binary run lengths start with 1, one can omit the final (eg, MSB) ‘+’ frommost runlength representations without loss of information. A potential problemoccurs in representation of a run of length of one which would not be representableif all MSB ‘+’ symbols were eliminated. For tackling this, it is necessary to retainthe MSB ‘+’ symbol for all runs of length 2j − 1, where j is an integer. Becausethe symbols ‘+’ and ‘−’ are used to code both runs and levels, a level of magnitudeone, which would be represented only by the ‘+’ or ‘−’ that is its MSB, would beindistinguishable from a run. This is handled by simply incrementing by one theabsolute value of all levels prior to performing the symbol mapping.3.3 Entropy CodingBoth the geometry of the best wavelet packet basis selected and the quantized coeffi-cients are converted into a stream of symbols from S using stack-run representationdescribed above. This symbol stream is entropy coded using an order-3 arithmeticcoder [7] to give actual bitstream.4 Experimental ResultsWe implemented the stack-run wavelet packet (SRWP) image coder, generating anactual bitstream, and decoder. Nine different variations of the SRWP coder weretested on six standard 256-level greyscale images each having a resolution of 512×512:Barbara, Houses, Lighthouse, Fingerprints, Goldhill, and Lena. The coder variationswere based on three different filters: symmlet-8, biorthogonal 17/11, and biorthogonal9/7 filters. Each of these variations employed three cost functions given in equations(1), (3), and (4) – denoted respectively by CW, log, and lp – for basis selectionpurposes. Compression results in terms of the peak-signal-to-noise-ratio (PSNR) indecibels (dB) are given in Tables 1–6. Comparisons are provided for two state-of-the-art image coders: the wavelet based SPIHT coder [10] and the fast adaptive4wavelet packet (FAWP) coder [6]; results for these algorithms were not available forlast two images used for evaluating SRWP. The experimental results show that thecompression performance of our SRWP coder is comparable to and often better thanthat of SPIHT and FAWP. Coding gains achieved by the SRWP coder over bothother coders are noticeable, especially for Barbara (1.2dB over SPIHT and 0.4dBover FAWP) and Fingerprints (1.1dB over SPIHT and nearly 0.2dB over FAWP).Empirical rate-distortion curves for the best SRWP coder variation, SPIHT, andFAWP are given in Figure 1. Visual results for four test images compressed at 0.25bits per pixel (bpp.) are provided in Figure 2.5 ConclusionsAn adaptive wavelet packet image coding algorithm based on stack-run representationof the quantized transform coefficients is presented. Our coder is relatively simpleand does not need to maintain any list of coefficients, as is the case with EZW[11] and SPIHT. Decent coding gains over SPIHT and FAWP coders demonstratethe successful exploitation of intra-subband redundancies. Experimental results alsoshow that the l1-norm and energy cost functions typically result in the best waveletpacket basis whose coefficients are more efficiently encoded by our algorithm. Anotherinteresting result is the consistently better performance of the symmlet-8 filters overfilters in popular use. Future work on this coder may focus on the development ofa cost function that takes into account the Laplacian distribution of wavelet packetcoefficients.6 AcknowledgementsWe gratefully acknowledge extensive use of WaveLab, a MATLAB r© toolbox devel-oped by Professor David Donoho of Stanford and his colleagues, in this work.References[1] K. A. Birney and T. R. Fischer. On the modeling of DCT and subband imagedata for compression. IEEE Transactions on Image Processing, 4(2):186–193,1995.[2] R.R. Coifman and Y. Meyer. Orthonormal wave packet bases. Technical Report,Department of Mathematics, Yale University, New Haven, 1990.[3] R.R. Coifman and M.V. Wickerhauser. Entropy-based algorithms for best ba-sis selection. IEEE Transactions on Information Theory, 38(2):713–718, March1992.[4] L. Cooper and M.W. Cooper. Introduction to Dynamic Programming. Pergamon,New York, 1988.5Bit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7) SPIHT FAWP(bpp.) CW log lp CW log lp CW log lp1.0 37.13 37.65 37.53 37.00 37.39 37.28 36.04 36.41 36.26 36.41 37.240.5 32.43 32.87 32.81 32.28 32.59 32.51 31.34 31.53 31.49 31.39 32.820.25 28.80 28.97 28.98 28.60 28.73 28.72 27.84 27.78 27.87 27.58 29.12Table 1: Compression results (PSNR values in dB) for 512× 512 Barbara imageBit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7) SPIHT FAWP(bpp.) CW log lp CW log lp CW log lp1.0 30.61 30.88 30.89 30.39 30.61 30.62 30.14 30.23 30.22 30.84 30.640.5 26.41 26.56 26.58 26.22 26.19 26.26 25.84 25.87 25.85 26.15 26.480.25 23.37 23.31 23.31 23.16 22.94 23.03 22.71 22.71 22.70 23.17 23.41Table 2: Compression results (PSNR values in dB) for 512× 512 Houses imageBit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7) SPIHT FAWP(bpp.) CW log lp CW log lp CW log lp1.0 33.98 34.16 34.07 33.60 33.93 33.74 33.34 33.51 33.42 34.03 34.060.5 30.27 30.40 30.35 30.01 30.15 30.05 29.68 29.80 29.77 30.25 30.440.25 27.70 27.75 27.78 27.50 27.55 27.54 27.17 27.20 27.27 27.43 27.98Table 3: Compression results (PSNR values in dB) for 512× 512 Lighthouse imageBit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7) SPIHT FAWP(bpp.) CW log lp CW log lp CW log lp1.0 37.11 36.48 37.12 36.57 36.01 36.64 34.82 33.57 35.02 36.01 36.850.50 31.92 31.70 31.95 31.51 31.36 31.58 30.25 29.14 30.26 31.27 31.790.25 27.84 27.90 27.86 27.54 27.60 27.49 26.51 25.48 26.52 27.12 27.59Table 4: Compression results (PSNR values in dB) for 512× 512 Fingerprints imageBit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7)(bpp.) CW log lp CW log lp CW log lp1.0 36.29 36.35 36.34 35.98 36.12 36.10 35.72 35.77 35.770.50 32.83 32.86 32.87 32.57 32.65 32.64 32.28 32.33 32.330.25 30.16 30.16 30.21 29.95 29.94 29.96 29.61 29.64 29.65Table 5: Compression results (PSNR values in dB) for 512× 512 Goldhill imageBit rate SRWP (symm-8) SRWP (17/11) SRWP (9/7)(bpp.) CW log lp CW log lp CW log lp1.0 39.83 39.85 39.84 39.52 39.66 39.66 39.10 39.13 39.120.50 36.42 36.42 36.43 36.06 36.18 36.19 35.41 35.43 35.450.25 32.98 33.01 32.98 32.75 32.71 32.70 31.94 31.96 31.98Table 6: Compression results (PSNR values in dB) for 512× 512 Lena image6(a) (b)(c) (d)Figure 1: Comparative rate-distortion curves(a) Barbara, (b) Houses, (c) Lighthouse, and (d) Fingerprints7(a) (b)(c) (d)Figure 2: Decoded images compressed by SRWP at 0.25 bpp.(a) Barbara, (b) Houses, (c) Lighthouse, and (d) Fingerprints8[5] M. Effros. Zerotree design for image compression: Toward weighted universalzerotree coding. In IEEE Intl. Conference on Image Processing, November 1997.[6] F.G. Meyer, A.Z. Averbuch, and J-O. Stro¨mberg. Fast adaptive wavelet packetimage compression. IEEE Transactions on Image Processing, 9(5):792–800, May2000.[7] M. Nelson and J.-L. Gailly. The Data Compression Book. M&T, New York,1996.[8] N.M. Rajpoot, R.G. Wilson, F.G. Meyer, and R.R. Coifman. A new basis selec-tion paradigm for wavelet packet image coding. In Proceedings IEEE Interna-tional Conference on Image Processing (ICIP’01), October 2001.[9] K. Ramchandran and M. Vetterli. Best wavelet packet bases in a rate-distortionsense. IEEE Transactions on Image Processing, pages 160–175, April 1993.[10] A. Said and W.A. Pearlman. A new fast and efficient image codec based on setpartitioning in hierarchical trees. IEEE Transactions on Circuits & Systems forVideo Technology, 6:243–250, June 1996.[11] J.M. Shapiro. Embedded image coding using zerotrees of wavelet coefficients.IEEE Transactions on Signal Processing, pages 3445–3462, Dec. 1993.[12] Y. Shoham and A. Gersho. Efficient bit allocation for an arbitrary set of quan-tizers. IEEE Transactions on Acoustics, Speech, and Signal Processing, 36:1445–1453, Sept. 1988.[13] G.J. Sullivan. Efficient scalar quantization of exponential and Laplacian randomvariables. IEEE Transactions on Information Theory, 42(5):1365–1374, Septem-ber 1996.[14] M.J. Tsai, J.D. Villasenor, and F. Chen. Stack-run image coding. IEEE Trans-actions on Circuits and Systems for Video Technology, 6:519–521, October 1996.[15] Z. Xiong, O. Guleryuz, and M.T. Orchard. A DCT-based embedded image coder.IEEE Signal Processing Letters, 3(11):289–290, 1996.9

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