A New Hybrid DWT-SVD Image-Coding System for Color Images

Transcription

1 Poceedings of the 5th WSEAS Int. Conf. on Signal Pocessing, Computational eomety & Atificial Vision, Malta, Septembe 5-7, 005 (pp0-06) A New Hybid DWT-SVD Image-Coding System fo Colo Images Humbeto Ochoa a,b, K.R. Rao a a Univesity of Texas at Alington, Box ates Steet, Neddeman Hall, Rm 530 Alington, TX USA b Univesidad Autónoma de Cd. Juáez/PROMEP Heny Dunant 406, Zona Ponaf Cd. Juáez, Chihuahua. MEXICO Abstact: In this pape, we popose the HDWTSVD system to encode colo images. Befoe encoding, the colo s (RB) ae tansfomed into CbC. Cb and C s 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 x subblocks of the is used to choose DWT o SVD fo all the s. Standad test images ae compessed based on the poposed algoithm. Keywods: quantization. coding colo images using wavelets, singula value decomposition, HC-RIOT, discete wavelets, vecto. Intoduction 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 s 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 entopy (ZTE) algoithm intoduced by Matucci [] 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

2 Poceedings of the 5th WSEAS Int. Conf. on Signal Pocessing, Computational eomety & Atificial Vision, Malta, Septembe 5-7, 005 (pp0-06) 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 Coding 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 = 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 N () A ˆ λ ( u( v T ( q (5) = q The squae eo is equal to the sum of the discaded eigenvalues X ( Xˆ ( = λ ( (6) m= q whee X( is the oiginal sample and X ˆ ( is the econstucted sample of a subblock. The enegy contained in q etained eigenvalues is E = q λ ( (7) 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 poposed algoithm Figue shows the Hybid DWT-SVD system poposed in []. Figue shows the poposed system to encode colo images based on this system. A. Encode The HDWTSVD encode emains the same as in []. The only diffeence is that befoe encoding, the 5x5, bit PCM, RB colo s ae tansfomed to a 5x5 luminance-chominance s (CbC) by using Eq. () [3]. The chominance (Cb, C) s ae downsampled by using the decimation filte of Eq. (9) [4], decimation along ows is followed by decimation along columns. The downsampled colo s ae of size 56 x R 6 C = () b C B [ 3 3 ] // ; (9) whee // denotes the intege division with ounding to the neaest intege away fom zeo. The luminance is divided into 64x64 blocks () and the chominance s into 3 x 3. The maximum numbe of levels of subband decomposition fo is 3 fo and fo Cb and C (see Fig. 3). The decision on which tansfom to use depends on the aveage standad deviation (ASTD) of the luminance. If it is above a theshold [], then all the thee s ae encoded using SVD and othewise DWT. B. Decode Afte decoding all the, the oiginal size of the chominance s is estoed by using the intepolation filte of Eq. (0) [4], upsampling along columns is followed by upsampling along ows. [ 3 3 ] // R = B ; (0) C b C ()

3 Poceedings of the 5th WSEAS Int. Conf. on Signal Pocessing, Computational eomety & Atificial Vision, Malta, Septembe 5-7, 005 (pp0-06) 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 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 good quality fo hight and low bitates. (c) Bitate DWT Bitate SVD Total bitate PSNR (db) Lena Lena Lena Peppes Peppes Peppes Table.Results fo low and high bitates. Figue 4 shows the ecoveed images fo low and high bitates. (d) (a) Figue 4. Recoveed images fo (a) Lena at 0. bpp, 3.4 db, (b) Lena at.3 bpp, 3.9 db, (c) Peppes at 0.36 bpp, 3.40 db, (d) Peppes at. bpp, 33. db. Table shows that thee is not much impovement of the ecoveed images (a) and (b) as the ate inceases in compessed by DWT because most of the of the image ae high activity pixels o low coelation. In Peppes image thee is a sensible eduction of distotion as the ate of compessed by DWT inceases because most of the high activity pixel ae compessed using SVD. Eq. () and (3) wee used to calculate the MSE and the PSNR espectively. (b) M = MSE m= 0 3xMxN N 0 ([ ( ˆ( ] [ g( gˆ( ] [ b( bˆ( ] () 55 PSNR 0 log0 db MSE = (3) (, g(, b( = Oiginal samples of the R,, B s.

4 Poceedings of the 5th WSEAS Int. Conf. on Signal Pocessing, Computational eomety & Atificial Vision, Malta, Septembe 5-7, 005 (pp0-06) ˆ(, gˆ(, bˆ( = Reconstucted samples of the R,, B s. 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 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 compessed by DWT. As we can see fom images in figue 4, PSNR is not an indication of the ecoveed image quality. Refeences: [] J. W. Woods and S.D. O Neil, Subband Coding of Images, IEEE Tans. ASSP, vol. 34, pp.7-, Oct. 96. [] M. Vetteli and D. Le all, Pefect Reconstuction FIR Filte Banks: Some Popeties and Factoizations, IEEE Tans. ASSP, vol. 37, pp , July 99. [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 [5] J.M. Shapio, Embeded Image Coding Using Zeotees of Wavelets Coefficients, IEEE Tans. SP, vol. 4: , Dec [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. [] S.A. Matucci, et al, A Zeotee Wavelet Video Code, IEEE Tans. CSVT, vol. 7, pp.09-, Feb [9] A.K. Jain, Fundamentals of Digital Image Pocessing. Englewood Cliffs, NJ: Pentice-Hall, 99. [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 , 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 [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. Calculation of the aveage std dev. of Analysis bank Daub 9/7 Subtact mean and calculate SVD of x subblocks HC-RIOT encode Encoded tile HC-RIOT decode Decision VQ - Synthesis bank Daub 9/7 Add tile mean Input (CbC). Subtact tile mean Adaptive econstuction and compaison stage Encode Decision VQ Code books OR S Q Code books SQ - OR SQ - Output (CbC). Reconstuction stage SQ Decode Figue. A Hybid DWT-SVD Algoithm (HDWTSVD).

5 Poceedings of the 5th WSEAS Int. Conf. on Signal Pocessing, Computational eomety & Atificial Vision, Malta, Septembe 5-7, 005 (pp0-06) 64x6 Input imag 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 Tile Figue. The HDWTSVD system fo colo images Cb, C 64 Figue 3. Subband decomposition of luminance () and chominance (Cb, C) s.