IMAGE QUALITY OPTIMIZATION BASED ON WAVELET FILTER DESIGN AND WAVELET DECOMPOSITION IN JPEG2000. Do Quan and Yo-Sung Ho

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1 IMAGE QUALITY OPTIMIZATIO BASED O WAVELET FILTER DESIG AD WAVELET DECOMPOSITIO I JPEG2000 Do Qun nd Yo-Sung Ho School of Informtion & Mechtronics Gwngju Institute of Science nd Technology (GIST) 26 Cheomdn-gwgiro (Oryong-dong), Bu-gu, Gwngju , Kore E-mil: {viequndo,hoyo}@gist.c.r ABSTRACT In JPEG2000, the Cohen-Dubechies-Feuveu (CDF) 9/7-tp wvelet filter dopted in lossy compression is implemented by the lifting scheme or by the convolution scheme while the LeGll 5/3-tp wvelet filter dopted in lossless compression is implemented just by the lifting scheme. However, these filters re not optiml in terms of Pe Signl-to-oise Rtio () vlues, nd irrtionl coefficients of wvelet filters re complicted. In this pper, we proposed method to optimize imge qulity bsed on wvelet filter design nd on wvelet decomposition. First, we propose design of wvelet filters by selecting the most pproprite rtionl coefficients of wvelet filters. These filters re shown to hve better performnce thn previous wvelet ones. Then, we choose the most pproprite wvelet decomposition to get the optiml vlues of imges. Keywords: wvelet filter, lifting, filter design, JPEG2000. ITRODUCTIO Wvelet, proposed by Grossmn nd Morlet [], is nown s the dydic trnsltions nd diltions of prticulr function, clled the m wvelet. It is lso clled the first-genertion wvelets. evertheless, we my use the lifting scheme for genertion of wvelets [2]. In this scheme, wvelets re not necessrily trnsltions nd diltions of function, though they hve ll powerful properties of the first-genertion wvelets. It is the lifting scheme nd is referred to s the second-genertion wvelets in literture. Using the lifting scheme, Every Finite Impulse Response (FIR) wvelet filter or filter bn cn be decomposed into severl lifting steps. Sttement bout perfect reconstruction cn be mde by Lurent polynomil entries. A lifting step then becomes n elementry mtrix tht is tringulr mtrix with ll digonl entries equl to one. In the JPEG2000 stndrd, there re two options for user s choice: the Cohen-Dubechies-Feuveu (CDF) 9/7-tp wvelet filter dopted in lossy compression technique nd the LeGll 5/3-tp wvelet filter dopted in lossless compression [3]. In ddition, there re four types of decompositions, such s mllt, spcl", pcet, nd which is defined by user. Different types of wvelet decompositions hve different performnces in terms of imge qulity nd in terms of vlues. Therefore, in this pper, we exmine ll decompositions in JPEG2000 to find the most pproprite decomposition in order to get the optiml vlues of imges. Then, we concentrte on the design of optiml wvelet filters used in the lossy compression technique s n wy to optimize imge qulity. There re two schemes to construct wvelet filter: the convolution scheme nd the lifting scheme. However, since the lifting scheme cn gretly reduce the number of wvelet decompositions nd reconstructions, the lifting scheme is preferred to the convolution scheme. With the lifting scheme, there hve been severl proposed wvelet filters with different coefficients bsed on the CDF 9/7-tp wvelet filter. In these filters, prmeter is used to find their coefficients. Dubechies first used the lifting scheme nd fctoring method with irrtionl number = to find coefficients of the wvelet filter [4]. This filter is referred to the CDF 9/7-tp wvelet filter. Gungjun proposed simple 9/7-tp wvelet filter bsed on the lifting scheme with = -.5 [5]. Compred with the CDF 9/7-tp wvelet filter, this proposed filter hs the sme performnce in terms of vlue, but the computtionl complexity is much lower. Ling used temporry prmeter t=.25 [6] to generte the filter (for the CDF 9/7-tp wvelet filter, t=.274 ). This filter is lso proved to hve the sme performnce with lower computtionl complexity thn the CDF 9/7- tp wvelet filter. Although the filters proposed by Gungjun nd Ling slightly outperform the CDF 9/7-tp wvelet filter, we rgue tht their performnces re not optiml. The question is whether there is ny vlue of nd the corresponding coefficients of wvelet filters which give the sme or even better performnce with lower computtionl complexity thn the CDF 9/7-tp wvelet filter. In this pper, we ddress this problem. Especilly, we design 9/7-tp wvelet filters by exmining the whole possible rnge for (i.e. from the negtive infinite to the positive infinite) to find the optiml vlue of for filters best performnce. We then compute the optiml coefficients for the 9/7-tp wvelet filters from the optiml. Therefore, bsed on the construction theorem of the biorthogonl wvelet nd on the lifting scheme, we find the optiml wvelet filters to get better performnce thn previous schemes in terms of vlues nd in terms of computtionl complexity. 7

2 2. WAVELET DECOMPOSITIOS AD WAVELET FILTERS I JPEG Wvelet Decompositions Fig. : Structure of different decompositions There re four types of wvelet decompositions in JPEG2000. First, the Mllt decomposition is the most populr scheme in wvelet compression implementtions [7]. Second, the spcl decomposition ws proposed by Signl Processing nd Coding Lb. t the University of Arizon. Third, the pcet decomposition is quite effective for compression of synthetic perture rdr imgery [8]. Finlly, decomposition is similr to the mllt decomposition. 2.2 umericl Anlysis In wvelets, the forwrd trnsform uses two nlysis filters including low-pss filter h, nd high-pss filter g followed by subsmpling. On the hnd, the inverse trnsform first upsmples nd then uses two synthesis filters including low-pss filter h nd high-pss g [4],[6]. In the z-domin, h cn be expressed s [4]: e h z, h z = () = b where b (nd respectively e ) is the smllest (nd respectively the lrgest) integer number for which h is nonzero. The functions of h nd g in the ω -domin re defined s: nd Mllt ( ω) 0 L H = h + 2 h cos nω, (2) ( ω) 0 n n= L2 G = g + 2 g cos nω, (3) n= respectively, where in the design of the CDF 9/7-tp wvelet filter, L is 4 nd L 2 is 3. These functions my be rewritten s [5]: H jω + e = 2, 2 ( ω) P( ω) jω + e G( ω) = 2 P ( ω), (5) 2 P ω nd j P ( ω) re polynomils of e ω ; nd re the number of the vnishing moments of the wvelet where Spcl n Pcet (4) nd its dul, respectively. In the CDF 9/7-tp wvelet, nd re ten by = =4. In our design of 9/7-tp wvelet filters, = 2 nd =4. If H(ω) nd G(ω) construct biorthogonl wvelet, the normlized condition must be stisfied s: H ( 0) = 2; G 0 = 2 (6) Combining Eq. (6) with Eq. (2) nd Eq. (3), we hve: h0 + 2 h + h2 + h3 + h4 = 2. g0 + 2 g + g2 + g3 = 2 Substitutingω = 0 into H(ω) of Eq. (2) nd into G(ω) of Eq. (3) results in: h + 2 h + h h + h = 0. (8) (7) g + 2 g + g g = 0. (9) Then, ting the second derivtive of Eq. (2) results in: 2 g + 4g 9g = 0 (0) 2 3 The polyphse mtrix P (z) which presents the filter pir (h,g) is P ( z) o he z g z =, ho z ge z () where h e contins the even coefficients of h, nd h o contins the odd coefficients s: P h e h2 z = nd ho h2 z +. = (2) Therefore, the polyphse mtrix P (z) cn be written s: ( z) s [4]: P ( ) 2 ( + ) + ( + ) ( + ) h0 + h2 z+ z + h4 z + z g + z + g3 z+ z =. (3) h z h3 z z g0 g2 z z Additionlly, P ( z) nd z cn be fctorized into 5 mtrices ξ 0 δ( z) 0 β( z) 0 = +. (4) 0 0 γ( z ) ( + z ) ξ Compring Eq. (3) nd Eq. (4), we conclude tht ( ) ( 3βγδ β δ ) ξ ( β δ γδ βγδ ) ξ h0 = + β + δ + γδ + βγδ ξ h = + + h2 = , h3 = βγδξ h4 = βγδξ 2βγ + g0 = ξ 3βγ + + γ g = ξ βγ g2 = ξ βγ g3 = ξ (5) (6) Finlly, using Eq. (7), Eq. (8), Eq. (9), nd Eq. (0), we hve: 8

3 β = 2 4 ( + 2 ) 2 ( + 2 ) γ = δ = ( 2) ( 2) ( + 2 ) ξ = + 4 (7) In our proposed method, is exmined in the ( ; + ) rnge to find the optiml vlue for the 9/7-tp wvelet filters in terms of computtionl complexity nd in terms of performnce. With ech, the (β,γ,δ,ξ) set is determined by Eq. (7) nd then the coefficients of h nd g re computed by Eq. (5) nd Eq. (6), respectively. In this wy, we will find the optiml bsed on the imum vlue of (i.e. ) of the corresponding 9/7-tp wvelet filter nd the optiml wvelet filter for ech experimentl imge nd compression rtio. 3. PROPOSED METHOD This prt presents proposed method of wvelet filter design. Since β, γ, δ, nd ξ depend on by Eq. (7), we cn get different vlues for β, γ, δ, nd ξ by chnging in the ( ; ) + rnge. Then, the coefficients of the 9/7-tp wvelet filters re computed using Eq. (5) nd Eq. (6). For ech filter, we find the corresponding vlue. Consequently, hs n optiml vlue when vlue is imum (i.e., ). = new Y Strt = 0 = 0 = + new new opt = = = = = opt 2 opt 2 = 0 = 0 = new new = Fig. 2: Method to find nd optiml. Figure 2 is the flow chrt of our proposed method. We exmine in its whole rnge. First, is exmined in the negtive side, nd then it is exmined in the positive side. In ech side, we find locl nd loclly optiml. After tht, we compre the vlue of in both opt 2 opt 2 = = > 2 Y = = opt End Y = new 9 sides. The higher is considered s the finl nd its corresponding is the finl optiml. With this, we chieve the optiml coefficients of the 9/7-tp wvelet filters. The shpe of for ech 9/7-tp wvelet filter is shown in Fig. 3 s function of. L M optiml Fig. 3: Shpe of s function of. To decrese the serch time for finding optiml, we use the bi-section lgorithm. Suppose we hve in two initil positions Left (L) nd Right (R), we compre the vlue of Middle (M) position with vlue of (L) nd (R). After tht, we updte R or L by M. These steps re iterted until reches its imum vlue. Figure 4 summrizes these min steps of the bisection method. Fig. 4: Bisection method. 4. EXPERIMETAL RESULTS We conducted experiments on the imges of the USC imge dtbse [9] nd some JPEG2000 test imges [0]. In ddition, the proposed lgorithm ws implemented on VM9.0 [] s reference softwre of JPEG Wvelet Filter Design The shpes of nd the improvement of vlue for different imges re quite similr. Therefore, we only described representtive imges in this prt, such s GIRL, HOUSE, PEPPERS, nd FAXBALLS imges in the dtbses. 4.. Computtion with Different R By testing imges in the USC imge dtbse, we get different results. The step size of is 0.. The potentilly optiml hs been drwn in Fig. 5. As it cn be esily seen, the optiml is generlly in the negtive side. By our experiments, in the negtive side of vlue, when decreses from zero to infinitive negtive number, increses. However, t certin vlue of, the stops incresing. After this period, decreses when continues decresing. When reches the negtive infinite, vlue is very smll nd remins constnt. In the positive side of vlue, when increses from zero to infinitive positive number, the first increses. At certin vlue of, does not chnge even continues incresing. After tht, decreses when increses. When goes to the positive infinite, is. s (L). 2. M (L+R)/2. 3. If (M) > s, then L M. Else R M. 4. Return to step 2 unless R L is smll enough.

4 very smll nd remins constnt. Becuse the rnge of is in the ( ; + ) rnge, we only drw potentilly optiml in the negtive side in Fig. 5. [db] [db] :8 :6 25 : :64 :00 : ) b) Fig. 5: survey with different compression rtios for GIRL ) nd HOUSE b) imges Find The Optiml 25.00% :8 :6 : :64 :00 :28 hve n optiml rnge for ll imges of the dtbse, rther thn only one number. Figure 6 represents the optiml distribution for Comprison with Other 9/7-tp Wvelet Filters We lso implemented the previous 9/7-tp wvelet filters. We used = for the CDF 9/7-tp wvelet filter nd =-.5 for Gungjun nd Ling 9/7-tp wvelet filters. Especilly, by our proposed method, we find optiml nd optiml wvelet filter, whose vlue is better thn previous 9/7-tp wvelet filters. Imge GIRL ( opt=-2.203) HOUSE ( opt=-2.2) PEPPERS ( opt=-2.5) FAXBALLS ( opt=-2.3) Bitrte [bpp] Tble : comprison JPEG2000 [db] (=-.5862) Gungjun 2 [db] 2=-.5 Proposed [db] [db] 2 [db] Tble shows the improvement with optiml compred with previous 9/7-tp wvelet filters. With the optiml, the improvement of is presented in nd 2. The chnges re clculted by = ( proposed) ( reference)[ db] (8) Additionlly, vlues of the filters with 2 =-.5 re quite similr to vlues of the CDF 9/7-tp wvelet filter with = [5], [6]. This cn be verified from the result of tble. Beside the improvement of, we cn lso see tht hs quite simpler vlue thn the vlue of in the CDF 9/7-tp wvelet filter, which is n irrtionl number. 4.2 Wvelet Decompositions Percentge [%] 20.00% 5.00% 0.00% 5.00% 0.00% We experimented on five imges from USC dtbse. They re, BABOO,, GRADMA, nd MA. These imges re decomposed through four types of decompositions: mllt, spcl, pcet, nd. Additionlly, these conditions re used for both 5/3-tp wvelet filter in lossless compression nd 9/7-tp wvelet filter in lossy compression. Finlly, we use VM9.0 s reference softwre of JPEG of Different Decompositions Fig. 6: Distribution of optiml for Ech imge in the dtbse is tested with 6 different compression rtios including :8, :6, :, :64, :00, nd :28. It is obvious tht different imges hve different optiml. Even for n imge, the optiml is usully not the sme for different compression rtios. Therefore, we The different decompositions re clculted bsed on Eq. (8). To find pproprite wvelet decomposition for getting the optiml vlues of imges, we choose the worst decomposition s reference. Then we exmine differences of vlues of different decompositions from the decomposition reference s the following tbles. 0

5 Imge BABOO GRADMA MA Imge BABOO GRADMA MA Tble 2: of different decompositions for 5/3-tp wvelet filter. Bitrte Mllt Spcl Other Pcet 2 3 [bpp] [db] 2 [db] 3 [db] 4 [db] [db] [db] [db] Tble 3: of different decompositions for 9/7-tp wvelet filter. Bitrte Mllt Spcl Other Pcet 2 3 [bpp] [db] 2 [db] 3 [db] 4 [db] [db] [db] [db] Tble 2 nd tble 3 show tht the best decomposition is the mllt decomposition while the worst decomposition is the pcet decomposition. These tbles lso present the improvements of the mllt, spcl, nd decompositions compred to the pcet decomposition in terms of nd 2, nd 3, respectively. Additionlly, the imum improvement through choosing wvelet decomposition is significntly up to 0.93dB for 9/7-tp wvelet filter in lossy compression nd up to 3.85dB for 5/3-tp wvelet filter in lossless compression. Bsed on our experiments, we cn choose the most pproprite decomposition for our imge compression ppliction of JPEG Optiml for Different Decompositions We count the numbers of imum vlues for different decompositions nd different bitrtes. As result, we hve sttistic s shown in Fig % 90% 80% 70% 60% 50% % % 20% 0% 0% mllt spcl pcet For 5/3 tp wvelet filter Fig. 7: Optiml for different decompositions Rte-Distortion (RD) Curves For 9/7 tp wvelet filter Figure 8 nd Fig. 9 show the RD curves for different decompositions of nd imges.

6 [db] [db] 28 mllt spcl 26 pcet ) For 5/3-tp wvelet filter b) For 9/7-tp wvelet filter Fig. 8: RD curves of different decompositions () [db] [db] ) For 5/3-tp wvelet filter b) For 9/7-tp wvelet filter mllt spcl pcet mllt spcl pcet mllt spcl pcet Fig. 9: RD curves of different decompositions () 2 5. COCLUSIOS In this pper, we proposed method to optimize imge qulity bsed on wvelet filter design nd on wvelet decomposition. Using the lifting scheme, we proposed method to design the optiml 9/7-tp wvelet filters. We lso found out the rnge of optiml vlue, which ws used to find the coefficients of the filters, nd s function of. Using our proposed method, the complexity of wvelet filters is significntly reduced since their irrtionl coefficients re replced by rtionl coefficients. Additionlly, the re improved. We lso found the most pproprite wvelet decomposition to get the optiml vlues of imges. As result, the imum difference of the most pproprite wvelet decomposition is distinguishble up to 0.93 db for 9/7-tp wvelet filter in lossy compression nd up to 3.85 db for 5/3-tp wvelet filter in lossless compression. ACKOWLEDGEMETS This wor ws supported in prt by ITRC through RBRC t GIST (IITA-2008-C ). REFERECES [] J. Morlet nd A. Grossmnn, "Decomposition of hrdy functions into squre integrble wvelets of constnt shpe," SIAM Journl of Mthemticl Anlysis, Vol. 5, Issue 4, pp , 984. [2] W. Sweldens, "The lifting scheme: construction of second genertion wvelets," SIAM Journl on Mthemticl Anlysis, Vol. 29, pp , 996. [3] D. S. Tubmn, nd M. W. Mrcellin, JPEG2000 Imge Compression Fundmentls, Stndrds nd Prctice, Kluwer Acdemic Publishers, [4] I. Dubechies nd W. Sweldens, "Fctoring wvelet trnsforms into lifting steps," Journl of Fourier Anlysis nd Applictions, Vol. 4, o. 3, 998. [5] Z. Gungjun, C. Lizhi, nd C. Huowng, "A simple 9/7-tp wvelet filter bsed on lifting scheme, Interntionl Conference on Imge Processing, Vol. 2, pp , 200. [6] D. Ling, L. Cheng, Z. Zng, "Generl construction of wvelet filters vi lifting scheme nd its ppliction in imge coding," Opticl Engineering, Vol. 42, o. 7, pp , [7] JPEG 2000 Prt I Finl Drft Interntionl Stndrd FDIS5444-:2000, ISO/IEC JTC/SC29/WG 876, August [8] A. Przelsowsi, "Performnce evlution of JPEG2000-lie dt decomposition schemes in wvelet codec," Interntionl Conference on Imge Processing, Vol. 3, pp , 200. [9] [0] [] "JPEG2000 Verifiction Model 9.0 (VM9.0, reference softwre)," ISO/IEC JTC/SC29/WG 23, April 200.