Cataloged from PDF version of article.A lossless image compression algorithm based on adaptive subband decomposition is proposed. The subband decomposition is achieved by a two-channel LMS adaptive filter bank. The resulting coefficients are lossy coded first, and then the residual error between the lossy and error-free coefficients is compressed. The locations and the magnitudes of the nonzero coefficients are encoded separately by an hierarchical enumerative coding method. The locations of the nonzero coefficients in children bands are predicted from those in the parent band. The proposed compression algorithm, on the average, provides higher compression ratios than the state-of-the-art methods. (C) 2001 Elsevier Science B.V. All rights reserved

Cetin, A. E.

Egiazarian, K.

Gerek, O. N.

Oktem, L.

Oktem, R.

Signal Processing

Bilkent University Institutional Repository

*Corresponding author. Tel.: #90-312-266-4126; fax: #90-312-266-4126.E-mail address: cetin@ee.bilkent.edu.tr (A.E. C7 etin).Signal Processing 81 (2001) 447}450Fast communicationLossless image compression by LMS adaptive "lter banksRus7 en OG ktem!, A. Enis C7 etin",*, Omer N. Gerek#, Levent OG ktem!,Karen Egiazarian!!Signal Processing Laboratory, Tampere University of Technology, P.O. Box 553, FIN 33101 Tampere, Finland"Department of Electrical and Electronics Engineering, Bilkent University, Bilkent, TR 06533 Ankara, Turkey#Anadolu University, Eskisehir, TurkeyReceived 7 August 2000AbstractA lossless image compression algorithm based on adaptive subband decomposition is proposed. The subbanddecomposition is achieved by a two-channel LMS adaptive "lter bank. The resulting coe$cients are lossy coded "rst, andthen the residual error between the lossy and error-free coe$cients is compressed. The locations and the magnitudes ofthe nonzero coe$cients are encoded separately by an hierarchical enumerative coding method. The locations of thenonzero coe$cients in children bands are predicted from those in the parent band. The proposed compression algorithm,on the average, provides higher compression ratios than the state-of-the-art methods. ( 2001 Elsevier Science B.V. Allrights reserved.Keywords: Adaptive subband decomposition; Lossless image compression; LMS algorithm1. IntroductionLossless compression of images is required inmany practical applications including medical andspace imaging for archiving or transmission. Earlylossless image coders, e.g. JPEG lossless mode, arebased on DPCM. Wu and Memon improvedDPCM schemes by using adaptive prediction andcontext modeling in CALIC [13]. In [9], an e$-cient progressive lossless compression is achievedby introducing S#P transform, a subband de-composition scheme, and an embedded entropycoding. In [2,3], perfect reconstruction "lter banks(PRFB) employing adaptive LMS "lters are intro-duced for subband decomposition, and they areused for lossy image compression in [3,6].In this paper, we use the adaptive PRFB struc-ture in a lossless image compression algorithm andpropose to code the subband coe$cients by themethod developed in [7] which exploits the multi-resolution structure of the subband decomposi-tion/wavelet transform.2. LMS adaptive prediction 5lter banksThe concept of the adaptive "lterbanks is intro-duced in [1,4,12]. Classical adaptive predictionconcepts are combined with the PRFB in [2,3]0165-1684/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 6 8 4 ( 0 0 ) 0 0 2 0 4 - 8Fig. 1. Analysis and synthesis stages of the 2-channel adaptive "lter bank structure.where the key idea is to decorrelate the polyphasecomponents of the multichannel structure by usingan adaptive predictor P (see Fig. 1). The adaptationof the predictor coe$cients is carried out by theleast mean square (LMS) algorithm, and this helpsto cope with the unstationary behavior of the inputdata.In Fig. 1, x1(n) is the downsampled version of theoriginal signal, x(n), thus it consists of the evensamples of x(n). Similarly, the signal x2(n) consistsof the odd samples. An LMS-based FIR predictorof x2(n) from x1(n) can be expressed asx(2(n)"w(n)xT1(n), (1)where x1(n)"[x1(n!N),2, x1(n#M)]T is theobservation vector, and w(n) is the vector of pre-dictor coe$cients which is adapted by the equationw(n#1)"w(n)#k x1(n)e(n)DDx1(n)DD2, (2)where the error signal e(n) is given bye(n)"x2(n)!x(2(n). (3)The subband decomposition structure shown inFig. 1 compacts the most of the energy in the lowestresolution band, and the resulting subsignals areexpected to be decorrelated. A weakness of thestructure shown in Fig. 1 is that the subsignal x1(n)may su!er from aliasing due to downsampling.Aliasing a!ects the quality of prediction especiallywhen further decompositions over x1(n) are carriedout. In order to eliminate this problem an anti-aliasing "ltering stage is introduced in [3], wherex(n) is lowpass "ltered by a halfband "lter of theformH-(z)"12[1#z~1A(z2)]. (4)With the use of the so-called ‘noble identitya [11],the lowpass "ltering operation can be carried outafter downsampling as shown in Fig. 2, and thesubsignal x2(n) is predicted using x-(n) which isa smoothed version of x1(n).The above adaptive PRFB structures areextended to two dimensions in a separablemanner.3. Application to lossless image compressionThe decomposition structure in Fig. 2 is used forlossless image compression as follows: input imageis decomposed into multiresolution bands by con-secutive row-wise and column-wise operations. Inorder to obtain integer-valued coe$cients, both thepredictor output and the low-resolution coe$cientsx-, are rounded to the nearest integer at each stage.This rounding operation makes the perfect recon-struction of Mx1N impossible. It is experimentallyobserved that transmitting the error between Mx1Nand Mx(1N is generally costly. To overcome thisproblem, we apply quantization to Mx-N and trans-mit the quantized coe$cients "rst. The decoderreconstructs Mx(1N by using Q(Mx-N), where Q denotesquantization. Then, we transmit the residual errorbetween Mx1N and Mx(1N. Both the subband de-composition coe$cients and the residual error ex-hibit a multiresolution structure such that locationsof the nonzero coe$cients in a band can be pre-dicted from those in the coarser band. Hence, bothare coded by using the method in [7], which wasdeveloped for lossy coding of the wavelet transformcoe$cients. The method of [7] is an improvedversion of the scheme in [10]. Its two most impor-tant features are scanning the coe$cients of a band448 R. O$ ktem et al. / Signal Processing 81 (2001) 447}450Fig. 2. Adaptive "lter bank structure with an antialiasing "lter.Fig. 3. Prediction neighborhood in columnwise processing.Table 1Bit-rate results for di!erent lossless compression methodsImages JPEG S#P CALICProposedmethodLenna 4.71 4.18 4.12 3.96Mandrill 6.38 5.93 5.88 5.52Harbour 5.23 4.73 4.44 4.64Doex 5.59 4.18 3.79 3.81House 4.95 4.22 4.00 3.57Cameraman 5.51 4.48 4.19 3.83Barbara 5.58 4.69 4.63 4.59Goldhill 5.65 4.75 4.63 4.51by magnitude preference, and employment ofhierarchical enumerative coding [5] instead ofarithmetic coding.4. Experimental resultsWe used six 512]512 images and two 256]256images, House and Cameraman, for testing our algo-rithm. The prediction neighborhood of a pixel isdepicted in Fig. 3. We check the variances of leftdiagonal neighborhood Md1, d2,d3N, right diagonalneighborhood Md4, d5,d6N, and horizontal neigh-borhood Mh1, h2, h3, h4N of the pixel to be predicted(shaded in Fig. 3), and use the one with minimumvariance if its variance does not exceed a threshold.Otherwise, we use the neighborhood Md2, n1, d5,h3,h4, d6N. The bit-rate results in Table 1 show thatthe proposed algorithm, on the average, achievesbetter performance than lossless JPEG [8], S#P[9], and CALIC [13] codecs. In this study,H(3)"(1#z~1)/z in Fig. 2.5. ConclusionsA lossless image compression algorithm usingmultiresolution decomposition by LMS adaptivePRFB is proposed. The algorithm primarily trans-mits lossy coe$cients and then transmits the resid-ual error. The locations of the nonzero coe$cientsin children bands are predicted from those in theparent band through morphological dilation. Boththe location and the magnitude information areentropy coded by hierarchical enumerative coding.The proposed compression algorithm, on the aver-age, achieves a better bit-rate e$ciency than that ofthe state-of-the-art lossless codecs, while proposingprogressive transmission.References[1] O. Egger, W. Li, M. Kunt, High compression image codingusing an adaptive morphological subband decomposition,Proc. IEEE 83 (2) (February 1995) 272}287.[2] OG .N. Gerek, A.E. C7 etin, Linear/nonlinear adaptive poly-phase subband decomposition structures for image com-pression, ICASSP 98, Seattle, WA, Vol. 3, 12}15 May1998, pp. 1345}1348.[3] OG .N. Gerek, A.E. C7 etin, Adaptive polyphase subband de-composition structures for image compression, IEEETrans. Image Process. 9 (2000) 1649}1660.[4] F.J. Hampson, J.C. Pesquet, A nonlinear subband de-composition structure with perfect reconstruction, IEEEInternational Conference on Image Proceeding (1996).R. O$ ktem et al. / Signal Processing 81 (2001) 447}450 449[5] L. OG ktem, J. Astola, Hierarchical enumerative coding oflocally stationary binary data, Electr. Lett. 35 (23) (1999)2003}2005.[6] R. OG ktem, K. Egiazarian, E. C7 etin, Subband decomposi-tion based image compression algorithms with nonlinearadaptive "lter banks, NSIP 99, Antalya, Turkey, Vol. 2,June 1999, pp. 766}769.[7] L. OG ktem, R. OG ktem, K. Egiazarian, J. Astola, E$cientencoding of the signi"cance maps in wavelet based imagecompression, ISCAS 2000, Geneva, Switzerland, 2000, toappear.[8] W.B. Pennebaker, J.L. Mitchell, JPEG: Still Image Com-pression Standard, Van Nostrand Reinhold, New York,1993.[9] A. Said, W.A. Pearlman, An image multiresolution repres-entation for lossless and lossy image compression, IEEETrans. Image Process. 5 (September 1996) 1303}1310.[10] S.D. Servetto, K. Ramchandran, M.T. Orchard, Imagecoding based on a morphological representation ofwavelet data, IEEE Trans. Image Process. 8 (9) (1999)1161}1174.[11] G. Strang, Wavelets and Filter Banks, Wellesley } Cam-bridge Press, Wellesley, MA, 1996.[12] W. Sweldens, The lifting scheme: A new philosophy inbiorthogonal wavelet constructions, Proc. of SPIE 2569(September 1995) 68}79.[13] X. Wu, N. Memon, Context-based, adaptive, lossless im-age coding, IEEE Trans. Commun. 45 (4) (1997) 437}444.450 R. O$ ktem et al. / Signal Processing 81 (2001) 447}450

Lossless image compression by LMS adaptive filter banks

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