A Hybid DWT-SVD Image-Coding System (HDWTSVD) fo Colo Images Humbeto Ochoa a,b, K.R. Rao a a Univesity of Texas at Alington, Box 906 46 ates Steet, Neddeman Hall, Rm 530 Alington, TX 7609 hdo594@exchange.uta.edu ao@uta.edu TxTEC b Univesidad Autónoma de Cd. Juáez Heny Dunant 406, Zona Ponaf Cd. Juáez, Chihuahua. México Fulbight/PROMEP ABSTRACT In this pape, we popose the HDWTSVD system to encode colo images. Befoe encoding, the colo components (RB) ae tansfomed into CbC. Cb and C components ae downsampled by a facto of two, both hoizontally and vetically, befoe sending them though the encode. A citeion based on the aveage standad deviation of 8x8 subblocks of the component is used to choose DWT o SVD fo all the components. Standad test images ae compessed based on the poposed algoithm. Keywods: coding colo images using wavelets, singula value decomposition, HC-RIOT, discete wavelets, vecto quantization.. INTRODUCTION The phenomenal incease in the geneation, tansmission, and use of digital images in many applications is placing enomous demands on the stoage space and communication bandwidth. Data compession algoithms ae a viable appoach to alleviate the stoage and bandwidth demands. They ae key enabling components in a wide vaiety of infomation technology applications that equie handling a lage amount of infomation. Fom text and image epesentation in digital libaies to video steaming ove the Intenet, cuent infomation tansmission and stoage capabilities ae made possible by ecent advances in data compession. In ecent yeas, many compession techniques have been developed in diffeent fields, specially in the subband coding (SBC) field - namely wavelets in applied mathematics, subband coding in digital signal pocessing, and multiesolution in compute vision that have conveged to a unified theoy. SBC is a poweful technique, which is efficiently implemented using filte banks to split and mege the image without distotion. In SBC the filteed images ae downsampled to thei espective Nyquist ates. The downsampling opeation, pefomed by intege decimation fo pactical easons, intoduces distotions due to aliasing and filteing. Reconstuction theoy fo filte banks demonstates that alias-fee and distotion-fee solutions exist [], [], [3]. Thee exist vaious techniques to constuct wavelet bases, o to facto the filtes into basic building blocks. One of these is lifting, which is known as the second geneation wavelets. A constuction using lifting, is completely spatial and is used when Fouie techniques ae no longe available. The basic idea of compession using the DWT is to exploit the local coelation that exists in most of the images fo building an apoximation. In the fist geneation wavelets, the Fouie tansfom is used to build the space-fequency localization. Howeve, in the second geneation wavelets, this can be done in the spatial domain and can educe the computational complexity of the wavelet tansfom by a facto of two [4]. Othogonal tansfomations povide a good pefomance fo signals with high coelation. Howeve, they achieve a poo pefomance fo signals with low coelations. Togethe with the DWT, new algoithms to encode the esulting subbands have emeged. Fo example, the Embeded Zeotee Wavelet (EZW), intoduced by Shapio [5], the Set Patitioning in Hieachical Tees (SPIHT) poposed by Said and Pealman [6], and the HC- RIOT developed by Syed [7], belong to this categoy. The fome is a quantization and coding stategy that incopoates some chaacteistics of the wavelet decomposition. It takes advantage of the fact that thee ae wavelet coefficients in diffeent subbands that epesent the same spatial location in the image. The second uses a patitioning of the tees in a manne that tends to cluste insignifficant coefficients togethe in a lage subset. The last algoithm combines techniques of the zeotee 64 SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER
entopy (ZTE) algoithm intoduced by Matucci [8] and a modified SPIHT to impove the quality of the images at low bit ates. The Singula Value Decomposition (SVD) technique povides optimal enegy-packing efficiency fo any given image, but its application is vey limited due to the computational complexity associated with the computation of eigenvalues and eigenvectos [9]. The best esults fo SVD image compession have been obtained by combining SVD and vecto quantization (VQ) of the eigenvectos [0], []. In ode to have a bette pefomance of the tansfomation in the aeas of the image whee the coelation is low, a combination of SBC o DWT and SVD was poposed in []. This was called A Hybid DWT-SVD Image-Coding System (HDWTSVD). In this eseach we popose a modification of this system fo colo images and discuss its pefomance.. SVD CODIN The SVD is a tansfom suitable fo image compession because it povides optimal enegy compaction fo any given image [9]. A good epesentation of the image can be achieved by taking only a few lagest eigenvalues and coesponding eigenvectos. A (NxN) matix A is decomposed to fom two othogonal matices, U and V T, epesenting the eigenvectos, and a diagonal matix epesenting the eigenvalues. A = U V T () whee is the ank of A T = diag( λ, λ... λ,0,0,0,0) = diag( σ, σ,..., σ,0,0,0,0) λ λ... λ > λ + =... = λ N = 0 We can calculate the columns v( of V by solving (A -λ(i) v(=0 n =,.., (3) whee A =A T A. The columns of U ae u(= Av( n =,.., (4) λ( The oiginal block can be estimated by etaining the q lagest eigenvalues and coesponding eigenvectos () A ˆ λ ( u( v T ( q (5) = q The squae eo is equal to the sum of the discaded eigenvalues m= X ˆ ( X ( = λ ( (6) q+ whee X( is the oiginal sample and Xˆ ( is the econstucted sample of a subblock. The enegy contained in q etained eigenvalues is Enegy λ ( (7) = q SVD yields two matices of eigenvectos and one of eigenvalues. VQ techniques have been used successfully to encode the eigenvectos and scala quantization techniques (SQ) to encode the eigenvalues [] []. 3. THE PROPOSED ALORITHM Figue shows the Hybid DWT-SVD system poposed in []. Figue shows the poposed system to encode colo images based on this system. A. Encode Essentially, the HDWTSVD encode emains the same as in []. The modifications ae done befoe and afte encoding. Befoe encoding, the 5x5, 8 bit PCM, RB colo components ae tansfomed to a 5x5 luminance-chominance components (CbC) by using Eq. (8) [3]. The chominance (Cb, C) components ae then downsampled by using the decimation filte of Eq. (9) [4], decimation along ows is followed by decimation along columns. The downsampled colo components ae of size 56 x 56. 0.57 0.504 0.098 R 6 + C = 0.48 0.9 0.439 8 (8) b C 0.439 0.368 0.07 B 8 [ 3 3 ] // 8 ; (9) whee // denotes the intege division with ounding to the neaest intege away fom zeo. The luminance component is divided into 64x64 blocks (tiles) and the chominance components into 3 x 3. In ode to be consistent with the spatial esolution, the maximum numbe of levels of subband decomposition fo each tile is 3, while fo each chominance tile is one level less than that applied to the coesponding luminance tile (see Fig. 3). To decide which tansfom (DWT o SVD) to use, the aveage standad deviation of the luminance tile is SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER 65
calculated. If it is above a theshold [], then all the thee components ae encoded using SVD and othewise DWT. B. Decode Afte decoding all the tiles, the oiginal size of the chominance components is estoed by using the intepolation filte of Eq. (0) [4], upsampling along columns is followed by upsampling along ows. (b) [ 3 3 ] // 8 ; (0) R.64 =.64 B.64 0.000 0.39.07.596 6 0.83 C 8 b 0.000 C 8 () 4. RESULTS Table shows the esults afte pocessing thee images at low (< bpp) and high bitates (> bpp). These esults show the bitate of the aeas compessed using DWT and using SVD. In this case, the impovement in PSNR at high bitate is not much. This is because HC-RIOT [7] is intended fo low bitate high quality images. The images will have the best quality fo low bitate. Bitate DWT (bpp) Bitate SVD (bpp) Total bitate (bpp) PSNR (db) Mandill 0.08 0.836 0.653 6.03 Mandill 0.0 0.836 0.935 6.35 Mandill 0.398 0.836.440 6.38 Lena 0.039 0.647 0.0 3.46 Lena 0.63 0.647 0.560 3.9 Lena 0.43 0.647.30 3.96 Peppes 0.033 0.438 0.36 3.40 Peppes 0.7 0.438 0.4 33.8 Peppes 0.369 0.438.0 33.8 (d) (c) Table.Results fo low and high bitates. Figue 4 shows the ecoveed images fo low and high bitates. (a) (e) 66 SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER
(f) system gives good esults at both low and high bit ates. We can achieve even moe compession at low bitates by implementing an adaptive multistage VQ to encode the eigenvectos. An adaptive multistage VQ will help us to educe the bitate at low esolutions in the aeas compessed by SVD while keeping the same bitate and quality of tiles compessed by DWT. As we can see fom images in figue 4, PSNR is not an indication of the ecoveed image quality. REFERENCES Figue 4. Recoveed images fo (a) Mandill at 0.653 bpp, 6.03 db, (b) Mandill at.6, 6.38 db, (c) Lena at 0. bpp, 3.4 db, (d) Lena at.3 bpp, 3.98 db, (e) Peppes at 0.36 bpp, 3.40 db, (f) Peppes at. bpp, 33.8 db. Table shows that thee is not much impovement of the ecoveed images (a) and (b) as the ate inceases in tiles compessed by DWT because most of the tiles of the image ae high activity pixels o low coelation tiles. In Peppes image thee is a sensible eduction of distotion as the ate of tiles compessed by DWT inceases because most of the high activity pixel tiles ae compessed using SVD. Eq. () and (3) wee used to calculate the MSE and the PSNR espectively. M = MSE m= 0 3xMxN whee + N 0 ([ ( ˆ( ] [ g( gˆ( ] + [ b( bˆ( ] + () 55 PSNR 0 log0 db MSE = (3) (, g(, b( = Oiginal samples of the R,, B components. ˆ(, gˆ(, bˆ( = Reconstucted samples of the R,, B components. M = Numbe of ows. N = Numbe of columns. 5. CONCLUSIONS We have pesented a system fo colo image compession based on the HDWTSVD developed ealie fo monochomatic images. Simulation esults show that this [] J. W. Woods and S.D. O Neil, Subband Coding of Images, IEEE Tans. ASSP, vol. 34, pp.78-88, Oct. 986. [] M. Vetteli and D. Le all, Pefect Reconstuction FIR Filte Banks: Some Popeties and Factoizations, IEEE Tans. ASSP, vol. 37, pp.057-07, July 989. [3]. Stang and T. Nguyen, Wavelets and Filte Banks. Wellesley-Cambidge Pess, 996. [4] I. Daubechies and W. Sweldens, Factoing Wavelet Tansfoms into Lifting Steps, Pinceton Univesity, Sept. 996. [5] J.M. Shapio, Embeded Image Coding Using Zeotees of Wavelets Coefficients, IEEE Tans. SP, vol. 4: 3445-346, Dec. 993. [6] A. Said and W.A. Pealman, A New Fast and Efficient Code Based on Set Patitioning in Hieachical Tees, IEEE Tans. CSVT, vol. 6, pp.43-50, June 996. [7]. F. Syed, A Low Bit Rate Wavelet-Based Image Code fo Tansmission Ove Hybid Netwoks. Doctoal Dissetation, UTA, 999. [8] S.A. Matucci, et al, A Zeotee Wavelet Video Code, IEEE Tans. CSVT, vol. 7, pp.09-8, Feb. 997. [9] A.K. Jain, Fundamentals of Digital Image Pocessing. Englewood Cliffs, NJ: Pentice-Hall, 989. [0] C.M. oldick, W. Dowling, and A. Buy, Image Coding Using the Singula Value Decomposition and Vecto Quantization, in Image Pocessing and its Applications, pp.96-300, IEE, 995. [] A. Dapena and S. Ahalt, A Hybid DCT-SVD Image-Coding Algoith IEEE Tans. CSVT, vol., pp.4-, Feb. 00. [] H. Ochoa, K.R. Rao, A Hybid DWT-SVD Image- Coding System (HDWTSVD) fo Monochomatic Images, IS&T/SPIE s 5 th Annual Symposiu Santa Claa, CA, Jan. 003. [3]. Q. Shi and H. Sun, Image and Video Compession fo Multimedia Engineeing, Fundamentals, Algoithms, and Standads. CRC Pess, 000. [4] K. R. Rao and J. J. Hwang, Techniques and Standads fo Image, Video and Audio Coding. Pentice-Hall, 996. SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER 67
Calculation of the aveage std dev. of component tiles Analysis bank Daub 9/7 Subtact mean and calculate SVD of 8x8 subblocks HC-RIOT encode Encoded tile HC-RIOT decode Decision VQ - Synthesis bank Daub 9/7 Add tile mean Input component tiles (CbC). Subtact tile mean Adaptive econstuction and compaison stage Decision VQ Code books OR S Q Code books SQ - OR SQ - Output component tiles (CbC). Reconstuction stage Encode SQ Decode Figue. A Hybid DWT-SVD Algoithm (HDWTSVD). 64x64 Input image R B Colo tansfomation. C Cb 3x3 3x3 HDWTSVD (Encode) HDWTSVD (Decode) R Downsampling Cb,C and tiling,cb,c C C(3x3) Cb(3x3) Reconstucted image B Colo tansfomation. Cb Upsampling Cb, C Figue. The HDWTSVD system fo colo images. Tile 68 SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER
8 6 8 6 8 6 3 8 6 3 3 64 3 Cb, C 64 Figue 3. Subband decomposition of luminance () and chominance (Cb, C) components. SSTEMICS, CBERNETICS AND INFORMATICS VOLUME - NUMBER 69