Novel Pre-Compression Rate-Distortion Optimization Algorithm for JPEG 2000

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1 Novel Pre-Compresson Rate-Dstorton Optmzaton Algorthm for JPEG 2000 Yu-We Chang, Hung-Ch Fang, Chung-Jr Lan, and Lang-Gee Chen DSP/IC Desgn Laboratory, Graduate Insttute of Electroncs Engneerng Natonal Tawan Unversty 1, Sec. 4, Roosevelt Road, Tape 106, Tawan ABSTRACT In ths paper, a novel pre-compresson rate-dstorton optmzaton algorthm s proposed, whch can reduce computaton power and memory requrement of JPEG 2000 encoder. It can reduce the wasted computatonal power of the entropy coder (EBCOT Ter-1) and unnecessary memory requrement for the code-stream. Dstorton and rate of codng passes are calculated and estmated before codng, and therefore truncaton pont s selected before codng. Expermental results show that the computaton tme of EBCOT Ter-1 and memory requrement for the code-stream can be greatly reduced, especally at hgh compresson rato. The qualty of the proposed algorthm s slghtly lower than that of post-compresson rate-dstorton optmzaton algorthm. Keywords: JPEG 2000, rate control, EBCOT, low power, rate dstorton optmzaton. 1. INTRODUCTION JPEG s the new stll magng codng standard n next generaton. The performance of JPEG 2000 s superor to JPEG at all bt-rate. 2 However, the computatonal complexty of JPEG 2000 s much hgher than that of JPEG. There are two major parts n JPEG 2000: Dscrete Wavelet Transform (DWT), and Embedded Block Codng wth Optmzed Truncaton (EBCOT). 3 In general, quantzaton s not used to control the rate of code-stream n JPEG 2000 encoder. It s appled to adjust weghts of dfferent frequency bands based on the flter bank and decomposton level of DWT, and no quantzaton s used at reversble wavelet transform mode. After DWT and quantzaton, the coeffcents are parttoned nto code-blocks, whch are encoded by EBCOT. The most complex part n JPEG 2000 s EBCOT. EBCOT s a two-tered algorthm. Ter-1 s embedded block coder, whch utlzes context-based arthmetc codng to encode each code-block nto ndependent embedded bt-stream. Ter-2 s post-compresson rate-dstorton optmzaton algorthm. EBCOT Ter-1 s the most complex part of JPEG 2000, whch consumes more than 50% of total computaton power 4. 5 Reducng ts computaton tme can sgnfcantly decrease the total run tme of JPEG 2000 encoder. Most lossy stll mage codng standard, ncludng JPEG, use quantzaton scheme to acheve rate control. However, ths scheme cannot provde best qualty at a gven bt-rate and get precse rate n one teraton. Instead of usng quantzaton scheme to perform the rate control, JPEG 2000 uses a better scheme to control the rate by EBCOT Ter-2 processng. It uses lagrange optmzaton to precsely control the bt-rate and guarantees the best qualty at specfc bt-rate. However, n the rate-dstorton optmzaton, all transformed coeffcents must be processed by EBCOT Ter-1 to get the rate and dstorton nformaton. In most cases, most compressed bt-streams generated by EBCOT Ter-1 wll be dscarded through the procedure of EBCOT Ter-2. The memory spent to store the dscarded bt-stream and computatons used are all wasted. Some prevous works, Chang s, 6 Yeung s 7, 8 focus on memory and power reducton for PCRD. Chang et al., 6 use EBCOT Ter-2 feedback control to termnate redundant computaton of EBCOT Ter-1. Computaton tme of EBCOT Ter-1 can be reduced to 40% and 20% at medum to hgh compresson rate. Yeung 78 proposed a scheme based on prorty scannng. It s to encode the truncaton ponts n a dfferent order by prorty nformaton and termnate block codng adequately. The computatonal cost and memory requrement can be reduced by 52% and 71% respectvely n the case of 0.25 bpp. In ths paper, a new dea of qualty control s proposed. Instead of controllng the bt-rate, t mnmzes the bt-rate at gven mage qualty, that s, dstorton constrant. Qualty control has an excellent feature that t s a pre-compresson rate-dstorton optmzaton algorthm. By selectng truncaton pont before codng, wasted computatonal power and unnecessary memory requrement s elmnated. The qualty slghtly degrades comparng wth the post-compresson rate-dstorton algorthm. Vsual Communcatons and Image Processng 2004, edted by Sethuraman Panchanathan, Bhaskaran Vasudev, Proc. of SPIE-IS&T Electronc Imagng, SPIE Vol SPIE and IS&T X/04/$

2 Ths rest of ths paper s organzed as follows. Rate-dstorton optmzaton algorthm s revewed n Secton 2. Secton 3 and 4 wll explan qualty control algorthm n detal. Experment results are shown n Secton 5. Fnally concluson s drawn n Secton RATE-DISTORTION OPTIMIZATION In JPEG 2000, orgnal mage s decomposed as several subbands by DWT and each subband s parttoned nto codeblocks. EBCOT Ter-1 compresses a code-block as an ndependent embedded bt-stream whch s composed of several codng passes. An embedded bt-stream of code-block may truncated at some feasble truncaton pont, n, to yeld rates, R n. The total rate R of the code-stream s R = R n. (1) The dstorton correspondng to n s denoted by D n, whch s measured by the Mean Square Error (MSE). The total dstorton of the code-stream s D = D n. (2) For a Rate-Dstorton (R-D) optmzed code-stream wth rate R and dstorton D, t s mpossble to lower D wthout ncrease of R or vce versa. Taubman 3 has solved the R-D optmzaton problem for rate control. For rate control, the goal s to mnmze the dstorton whle keepng the rate of code-stream smaller than a target bt-rate, R T. The problem has been mapped nto Lagrange optmzaton problem as the mnmzaton of D + λr = An R-D slope for a code-block at each canddate truncaton pont n, whch s defned as S n = Dn R n (D n + λr n ). (3) = Dn 1 D n R n, (4) Rn 1 s used to fnd the optmal λ and the optmal truncaton pont set z. Wth ncrease of n, the canddate truncaton pont wll be from the most bt-plane to least bt-plane. Therefore, the slope value must decrease monotoncally, that s, the value of S n must be larger than S n+1. λ and z s optmal f the followng condtons s satsfed S n λ, n n S n < λ (5), n > n n z and for all code-blocks. In addtonal to rate control, one can mnmze the total rate, R, at the target dstorton, D T. Ths s called qualty control. As n rate control, the R-D optmzaton of qualty control can be acheved by mnmzng R + λ D = (R n + λ D n ) (6) where λ s the Lagrange multpler for qualty control. It can be readly proved that (3) and (6) are equvalent by replacng 1/λ by λ, mn(r + λ D)=mn(λ ( 1 λ R + D)) = λ mn(d + λr). (7) The optmal truncaton pont set of rate control problem that resultng the dstorton D s the same as the qualty control problem wth D T = D. Furthermore, the resultng rate R of qualty control wll be the same as the rate constrant R T of the rate control problem SPIE-IS&T/ Vol. 5308

3 The concept of qualty control s contrary to that of rate control. Rate control mnmzes dstorton of the codestream at gven rate constrant whle qualty control mnmzes rate of the code-stream at gven dstorton constrant. It s beleved that the qualty constrant s more meanngful than rate constrant for mage compresson n general. Besdes, qualty control has an advantage that t can be done at DWT doman,.e. before the EBCOT Ter-1, wthout sgnfcant qualty loss. 3. RATE-DISTORTION CALCULATION IN DWT DOMAIN The dstorton of a code-block s measured as the Mean Square Error (MSE) of the reconstructed pxels to the orgnal ones. A sample coeffcent must be checked whether or not t belongs to magntude refnement pass (P 2 ) snce the reconstructng approach of P 2 s dfferent from that of sgnfcant propagaton pass (P 1 ) and cleanup pass (P 3 ). The checkng of whether a sample coeffcent belongs to P 2 can be done pror to EBCOT. Let µ k be the sample bt at k-th bt-plane of a j-th coeffcent µ, and n be the canddate truncaton pont. Let φ k denotes whether µ k belongs to P 2 or not, whch can be found as φ k 1, 2 = k+1 µ 0, 2 k+1 > µ. (8) Let t k be the value of µ lower than k-th bt-plane, whch s the value of the truncated part. The delta dstorton for coeffcent belongng to canddate truncaton pont n P 2 at bt-plane k s d n j = (t k+1 2 k ) 2 (t k t k ) 2, φ k = 1 0, φ k = 0 The dstorton for coeffcent of canddate truncaton pont n (P 1 orp 3 ) at bt-plane k s 0, φ k = 1 d n j = (t k+1 ) 2 (t k t k ) 2, φ k = 0 u k = 1 (t k ) 2, φ k = 0 u k = 0 (9) (10) where t k s the reconstructed value of t k and s gven by t k = and the the delta dstorton, D n, can be calculated as 2 k 1, k > 0 0, k = 0. (11) D n N 2 1 = j=0 d n j (12) where N s code-block sze. The rate, on the other hand, must be estmated snce the exactly rate can only be known after EBCOT. In order to select the optmzed truncaton pont before EBCOT, the slope of each truncaton pont must be estmated. The randomness property of sample coeffcents n P 2 s utlzed to estmate the rate. Fgure 1 shows the randomness property of P 2. Each pont n Fg. 1 represents one P 2 bt-stream and the x-axs and y-axs s the number of sample coeffcent and the resultng rate of t. It s obvous that the compresson rato of P 2 s almost constant regardless of dfferent mage types, subbands, code-block szes and whch bt-plane t belongs to. Therefore, the rate of P 2 can be accurately estmated. In order to ncrease the number of feasble truncaton ponts, the propagaton property of P 1 s used. In the lowest two bt-planes of an code-block, most of coeffcent has been sgnfcant, and therefore almost all the non-sgnfcant sample coeffcents s propagated n P 1 as shown n Fg. 2. As can be seen, almost all the non-sgnfcant sample coeffcents are propagated n P 1 for the lowest two bt-planes. The delta rate of the truncaton pont at bt-plane k for n P 2 s estmated as r n 1, φ j = k = 1 0, φ k = 0. (13) SPIE-IS&T/ Vol

4 4500 Bts Magntude Refnement Pass lena baboon Jet Pepper Pxel Fgure 1: Randomness property of the magntude refnement pass. The delta rate of the truncaton pont at bt-plane k < 2 for n P 1 s estmated as The total delta rate, R n, can be estmated by R n = and r n j = 1, φ k = 0 0, φ k = 1. (14) ω P2 BC n, n P 2 ω P1 BC n, n P 1 k < 2, otherwse BC n = N 2 j (15) r n j (16) where ω P2 and ω P1 are the emprcally decded compresson rato of P 2 and P 1 of lowest two bt-planes, and BC n s bt-counts of truncaton pont n. 4. QUALITY CONTROL The flow chart of proposed qualty control algorthm s shown n Fg. 3. It has two stages: accumulaton stage and decson stage. Rate and dstorton are calculated and accumulated n the accumulaton state. Optmal truncaton pont set s selected n the decson stage. Let the the depth of magntude bt of DWT coeffcent s H. P 1 and P 2 are truncaton ponts for bt-plane k < 2 and only P 2 s truncaton pont for 2 k H 1. Therefore,the number of truncaton ponts are H + 2. Let the value of n between 0 and H + 1(0 n H + 1) denotes the canddate truncaton pont from hgher bt-plane to lower bt-plane and two sets, A P1 and A P2, represent the collecton of canddate truncaton ponts P 1 and P 2 respectvely,.e. AP1, n = H + 1,H 1 n A P2, 0 n H 2,n = H. (17) 4.1. Accumulaton stage Ths stage ncludng Pass Detecton, R-D Calculaton and R-D Accumulaton. Pass Detecton step detects whether or not the bt belongs to P 2. R-D Calculaton step estmates rate and computes dstorton for each bt. R-D Accumulaton step accumulates rate and dstorton to obtan D n and R n for each truncaton pont. Accumulaton stage s repeated untl coeffcents of a tle are fnshed SPIE-IS&T/ Vol. 5308

5 Btcount(%) Lena Btcount(%) Baboon P1 P2 60 P1 P2 40 P3 40 P Btplane(b) (a) 0 Btplane(b) (b) Fgure 2: Propagaton property of P 1 for (a) lena and (b) baboon Pass Detecton When one DWT coeffcent µ at poston j n code-block s fetched n, pass of each bt at each bt-plane k s detected by usng equaton (8). Then the φ k (0 k H 1) nformaton are obtaned R-D Calculaton The φ k nformaton are exploted to calculate dstorton and estmate rate. The delta rate and dstorton for each truncaton pont n s calculated accordng to the equaton (9) and (13) f n A 2 and (10) and (14) f n A 1. Then d n j and r n j for 0 n H + 1 are obtaned R-D Accumulaton D n and BC n are set to zero ntally for j = 0. To get R n and D n, dn j and cn j are accumulated wth D n = D n + dn j BC n = BC n + rn j j = j + 1 (18) for n=0 to n=h+1. If j = N 2 1 at the end of one code-block, the BC n must be multpled wth ω P1 or ω P2 to get R n wth the equaton (15), and R n and D n for code-block are obtaned Decson stage Ths stage ncludng Calculate Slope step and Truncaton Ponts Selecton step. The Calculate Slope step computes the slope value for all code-block wth the nformaton R n and D n generated by stage 1 by usng equaton (4). The target dstorton of mage are averagely dstrbuted nto every tle and the dstorton of every tle s D T. The Truncaton Ponts Selecton step adjusts threshold value λ wth several teratons to satsfy the condton (5) wthout total dstorton of all code-block exceedng D T. Then the optmal truncaton ponts n for all every code-block s found. These nformaton are exploted by EBCOT Ter-1 to compresses code-blocks wthout processng truncated parts. SPIE-IS&T/ Vol

6 DWT coeffcents Pass Detecton Calculate Slope R-D Calculaton Truncaton Ponts Selecton D T R-D Accumulaton Truncaton Ponts No End of coeffcents Yes Fgure 3: Flow chart of proposed qualty control. (a) lena (b) baboon (c) pepper (d) elane Fgure 4: PSNR comparson between proposed method and VM9.0. (Test mage: , 1 tle, 1 layer, 2 decomposton level) 1358 SPIE-IS&T/ Vol. 5308

7 (a) lena (b) baboon (c) pepper (d) elane Fgure 5: Normalzed computaton power and memory usage. (Test mage: 512x512, 1 tle, 1 layer, 2 decomposton level) 5. EXPERIMENT RESULT 5.1. Expermental results The PSNR value of proposed qualty control algorthm, whch s compared wth VM9.0, s shown n Fg. 4 and the detal s lsted n Table 1. For these four test mages, the value of PSNR dfference between proposed method and VM9.0 s almost the same at medan and hgh bt-rate. At low bt-rate, the average PSNR value degradaton below VM9.0 s about db. The worst case may be over 1 db at very low bt-rate. There are two reasons about PSNR degradaton. Frst, the rate of each truncaton pont s estmated. Estmated rate may not be the same as the real rate. Second, truncaton ponts of proposed algorthm are only P 2 at all bt-plane and both P 1 and P 2 at lower two bt-planes. The truncaton ponts selected by proposed algorthm may not be optmal set, because some truncaton ponts can not be chosen for the sake of lackng slope nformaton. The normalzed computaton power and memory usage for bufferng bt-stream s shown n Fg. 5. The computaton power s measured wth the number of contexts processed by EBCOT Ter-1 snce t s hghly correlated to processng tme of EBCOT Ter-1. The memory usage s measured wth the number of bytes buffered n memory. Compared wth conventonal post-compresson rate-dstorton optmzaton, the proposed algorthm can reduce consderable computaton power and memory requrement Comparson The normalzed computaton power and memory usage compared wth Chang s work 6 and Yeung s work 7 s shown n Fg. 6 and Fg. 7 respectvely. In Fg. 6, computaton power s measured wth the number of contexts for our work and Change s. 6 For Yeung s 7 approach, t s measured wth the number of passes needed to be processed. In Fg. 7, Normalzed memory usage s measured smlarly wth bytes stored n memory Compared wth Chang s 6 work and Yeung s 7 work, our proposed algorthm has better computaton reducton and lower memory requrement. The reason why Chang s 6 work SPIE-IS&T/ Vol

8 Table 1: PSNR value compared wth VM9.0 for four test mage. Image Compresson Rato VM9.0 Proposed Lena Baboon Pepper Elane (a) lena (b) baboon Fgure 6: Comparson of normalzed computaton power. and Yeung s 7 work have less power and memory reducton s that these two works focus on rate control. Although there are many passes can be skpped durng encodng process, some passes, whch wll be dscarded fnally for the rate control ssue, stll need to be encoded by EBCOT Ter CONCLUSION A pre-compresson rate-dstorton optmzaton scheme s proposed n ths paper. A novel dea, qualty control, for the coder control of JPEG 2000 s also proposed. The qualty control explots the randomness property of P 2 and the propagaton property of P 1 to estmate the rate n DWT doman. Dstorton calculaton n DWT doman s also presented. Expermental results show that the proposed algorthm can greatly reduce the computaton tme for EBCOT and the memory requrement for bt-stream. The PSNR degrades by only 0.3 db n average comparng to the reference software. REFERENCES 1. ISO/IEC JTC1/SC29/WG1 N1855, JPEG 2000 Part I: Fnal Draft Internatonal Standard (ISO/IEC FDIS ), Aug SPIE-IS&T/ Vol. 5308

9 (a) lena (b) baboon Fgure 7: Comparson of normalzed memory requrement. 2. A. Skodras, C. Chrstopoulos, and T. Ebrahm, The JPEG 2000 stll mage compresson standard, IEEE Sgnal Processng Mag. 18, pp , Sept D. Taubman, Hgh performance scalable mage compresson wth EBCOT, IEEE Trans. Image Processng 9, pp , July K. F. Chen, C. J. Lan, H. H. Chen, and L. G. Chen, Analyss and archtecture desgn of EBCOT for JPEG-2000, n Proc. IEEE Int. Symp. Crcuts and Systems (ISCAS 2001), pp , M. D. Adams and F. Kossentn, Jasper: A software-based JPEG 2000 codec mplementaton, n Proc. IEEE Int. Conf. Image Processng (ICIP 2000), pp , T. H. Chang, L. L. Chen, C. J. Lan, H. H. Chen, and L. G. Chen, Computaton reducton technque for lossy jpeg2000 encodng through ebcot ter-2 feedback processng, n Proc. IEEE Int. Conf. Image Processng (ICIP 2002), 7. Y. M. Yeung, O. C. Au, and A. Chang, An effcent optmal rate control technque for JPEG2000 mage codng usng prorty scannng, n Proc. IEEE Int. Conf. Multmeda Expo. (ICME 2003), pp , Y. M. Yeung, O. C. Au, and A. Chang, An effcent optmal rate control scheme for JPEG2000 mage codng, n Proc. IEEE Int. Conf. Image Processng (ICIP 2003), SPIE-IS&T/ Vol