Comparing image compression predicors using fracal dimension RADU DOBRESCU, MAEI DOBRESCU, SEFA MOCAU, SEBASIA ARALUGA Faculy of Conrol & Compuers POLIEHICA Universiy of Buchares Splaiul Independenei 313 ROMAIA radud@isis.pub.ro Absrac: - he paper analyzes he abiliy of a fracal dimension esimaor o evaluae he performance of wo ypes of predicors employed in he lossless image compression. framework of he LOCO-I algorihm. he fracal dimension is compued for grayscale images using a box-couning procedure and allows o choose he bes predicor. Key-Words: - lossless compression, predicive compression echniques, fracal dimension, fracal esimaor, residual enropy 1 Inroducion he mos recen lossless image compression echniques are based on predicive echniques [1], [2], [3]. For his ype of echniques boh he coder and he decoder parse he image pixel by pixel in a predefined order and predic he value of he curren pixel using he values of he previously encoded pixels. If we denoe by x he value of he curren pixel and by xˆ he prediced value for x, hen only he residual error ε = x xˆ has o be sen. If he predicion is good hen he residual errors have small values (being concenraed around 0) and so he enropy of he residual error image is smaller, leading o a beer compression raio. A he decoder s side, he original value of he pixel is deermined based on he residual errors received and on he same inference a he coder s side. In his paper he framework is ha of he LOCO-I (Low Complexiy Lossless Compression for Images) algorihm, which is he core of he ISO/IU sandard for lossless and near-lossless compression of coninuous-one images, JPEG-LS. he main seps of he LOCO-I algorihm [4] are he predicion of he value of he curren pixel, he conex-based modeling of he residual error and enropy coding. We have observed ha by replacing he original predicor wih a more complex one, based on he leas squares mehod, he resuls were beer in erms of compression raio when ess were run on a se of medical microscopic images. his poins ou ha i is possible o ge beer compression resuls by combining he wo predicors in he same scheme. his raises he problem of selecing he predicor ha performs bes for he curren pixel, using only informaion ha is also available a he decoder s side. We assumed ha fracal dimension gives informaion abou he smoohness of he region surrounding he curren pixel, and ha he degree of smoohness correlaes wih he performance of he predicors. his paper analyzes he abiliy of a fracal esimaor o selec he predicor mos likely o perform a beer esimaion of he value of he curren pixel. he res of he paper is organized as follows: in he nex secion he fracal esimaor used is presened. In secion 3 he wo predicors are described. he resuls are presened in secion 4, and in he las secion some conclusions are drawn. 2 he fracal esimaor he fracal esimaor ha we used is a modified version of ha devised by Sarkar and Chaudhuri [5] and i was designed o esimae he fracal dimension for grayscale images. We used a boxcouning procedure in which he analyzed image of size M M pixels is divided in grids of size s s, where1 < s M 2, s Ζ. For each s he scale raio is r= s M. he image is considered as a 3D space wih ( x, y) denoing he posiion of he pixel and he hird coordinae( z ) denoing he gray level. On each of he grids here is a column of boxes of size s s s. he value of s is chosen so ha G s = M s where G is he oal number of gray levels. For each grid he boxes are couned
from boom o op saring wih 1. Le l and k be he boxes in which he maximum and he minimum gray levels of he curren grid appear. hen he quaniy n r ( i, j) = l k+ 1, where ( i, j) denoe he curren 1 i M s, grid is compued for every j M s and n ( i j) 1. All hese quaniies are summed up r = i, j r, is compued for all differen values of r. he fracal dimension is esimaed from he leas square linear fi of log ( r ) agains log ( 1 r). In his paper we propose a echnique derived by he box-couning mehod described which allows processing 24-bi color images [6]. he algorihm supposes he nex five seps: 1. he area of ineres is seleced, using a mobile cursor. he size area can be 64X64, 128X128, 256X256 or 512X512. 2. he color image is convered ino 256-gray levels image, using he formula: I = 0,299R + 0,587G + 0,114B where R/G/B are he red/green/blue componens which defines he color of a pixel. 3. he image is binarized using a hreshold beween 1-255 gray level: all pixels whose gray level is greaer or equal o he hreshold will be ransformed in whie, he res will become black. A his poin, he forms inside he image are whie on a black background. 4. racing he conour: once he image is binarized, he nex sep is o race an ouline of he whie areas: all he whie pixels which have a leas one neighbor black will become par of he conour (in our analysis we considered ha one pixel has 8 neighbors:, E, E, SE, S, SV, V, V). he res of pixels will be ransformed in black. 5. he resuled ouline can now be analyzed by esimaing is global fracal dimension, using he box-couning algorihm described earlier. Using he above algorihm we obain a fracal dimension specrum, where a box-couning dimension is relaed o every gray level conained in he image, An imporan limi of his mehod for he deerminaion of he local fracal dimensions is he presence of a supplemenary parameer: he size of he maximum covering square. ess shows ha in mulifracal cases, his value influens significan he resuls. Bu if he image analyzed is a single fracal, hen he global fracal dimension (compued wih box-couning algorihm) is closed o he local dimension wih he higher frequency: In order o appreciae he accuracy of he algorihm, le consider he case of he well known Koch Snowflake. As one can see in he hisogram of local dimension depiced in Fig.1, he mos frequen dimension is 1.25. he box-couning dimension was beween 1.25-1.26, he Koch Snowflake having he analyic dimension of 1.2618. Fig. 1. he mos frequen local dimension for he Koch Snowflake is 1.25 I is imporan o make a clear disincion beween he resul given by a fracal esimaor used for measuring a cerain objec and he fracal dimension owards which he geomery of he objec ends o. I is possible ha he obained resul does no provide an accurae measure of he fracal dimension bu insead offers significan daa ha describe oher characerisics relaed o he srucural qualiy of he objec.. 3 Characerisics of he predicors he predicor of LOCO-I has a fixed par ha esimaes he value of he curren pixel based on a causal emplae of surrounding pixels, and an adapive par ha performs a conex-dependen adjusmen of he value prediced by he fix par. In he following we will focus only on he fixed par of he predicor, and presen briefly he wo predicors we have used. 3.1. MED (Median Edge Deecion) Predicor his predicor is he one used in he LOCO-I algorihm. he causal emplae ha his predicor uses is presened in fig. 2. Rc Ra Rb Fig.2. he causal emplae of he MED predicor he MED predicor performs a simple es in order o deec verical and horizonal edges a x
he lef and a he op of he curren pixel. he esimae is compued as follows: ( Ra, Rb) if Rc max( Ra, Rb) ( Ra, Rb) if Rc min( Ra, Rb) min xˆ MED = max (1) Ra+ Rb Rc oherwise If a verical edge a he lef of he curren pixel is deeced, hen he predicor chooses Rb as he value of he predicion, and if a horizonal edge a he op of he curren pixel is deeced, hen he value of Ra is chosen. If no edge is deeced he predicor esimaes he value of he curren pixel as Ra + Rb Rc. 3.2. LS (Leas Squares) Predicor his predicor uses a linear model o compue he prediced value for he curren pixel. In each conex (deermined by he LOCO-I algorihm see [4] for deails on conex selecion) a 7 parameer vecorθ R is sored and updaed each ime he conex is visied using he leas squares mehod. he prediced value xˆ LS is deermined as follows: xˆ =ϕ θ, (2) where = [ Ra Rb Rc Rd Rw Rn Rne] LS ϕ is deermined from he values of he pixels in he causal emplae presened in fig. 3, θ is he parameer vecor of he curren conex and is he number of imes he conex was visied. Rw Rn Rc Ra Fig.3. he causal emplae of he LS predicor he updae of he parameer vecor follows from he leas squares minimizaion of [ ] 2 ( ) 1 V θ = x( n) ϕ ( n) θ( n), (3) 2 n= 1 ha is 1 ( n) ϕ ( n) ( n) x( n) θ + 1 = ϕ ϕ (4) n= 1 n= 1 Rne Rb x Rd he recursive version of he leas squares mehod was used. oe: Because his mehod for parameer vecor adjusmen implies he reversibiliy of he marix [ ϕ(n) ϕ (n) ], for he implemenaion n= 1 anoher algorihm was chosen working ieraive in he following seps: Iniializaion: P 0 = τ 2 I m, τ is a consan, I m is he uni marix of order m şi θ 0 =[1/p.1/p,..1/p], where p represen he number of he pixels from he image. Run. For = 1,2, p 1. ε +1 = x() - ϕ () θ 2. A = P -1 ϕ() A 3. k +1 = 1 + ϕ ( ) A 4. P = P 1 k +1 A 5. θ +1 =θ + k +1 ε +1 he efficiency of he mehod can be verified by compuing he residual (smaller) enropy relaed o he original image enropy. able 1 offer a comparison on eigh medical images compressed wih a MED predicor and wo LS predicors, LS 3_1 (3 pixels and a single predicion conex) and LS7_365 (7 pixels and 365 predicion conexs) able 1. Compressed images enropy values original MED LS LS 3_1 7_365 cilia.bmp 7.265 5.210 4.927 4.741 dna.bmp 6.933 6.918 6.485 6.448 dyad.bmp 7.446 5.500 5.473 5.410 Micro1.bmp 7.268 6.608 6.571 6.497 Micro2.bmp 7.824 6.644 6.592 6.563 erve.bmp 7.765 6.876 6.629 6.621 Virus.bmp 7.359 5.205 5.104 4.874 virus1.bmp 7.339 5.412 5.261 5.091 he resuls from able 1 are obained wihou condiioning he disribuion of he predicion error from he conex where his error appears. When his condiion operaes, he conexs decompose he original disribuion in several disribuions characerizing he degree of smoohness around he curren pixels [7]. his
effec is visible in fig 4. which depics he disribuion of he predicion error for one image (cilia.bmp), in his case when using he LS predicor. 4 Experimenal resuls We analyzed wo ses of images: one se of 20 medical microscopic images and one se of 14 naural images. Each of he images was analyzed as follows: For each pixel hree quaniies were compued: he residual error of he MED predicor, he residual error of he LS predicor and he esimaion of he fracal dimension of a viciniy of he curren pixel. he viciniy on which he fracal dimension was esimaed is presened in fig. 5. As i can be observed he viciniy was chosen o be causal. 30 pixels a) wihou conex condiioning 30 pixels x 15 pixels Fig.5. he viciniy for fracal dimension esimaion b) wih conex condiioning Fig.4. Error predicion disribuion able 2 presens he (much beer) resuls obained when using conex condiioning. able 2. Enropy values wih conex resricion Origial MED LS LS 3_1 7_365 cilia.bmp 7.265 3.039 3.004 2.904 dna.bmp 6.933 4.145 4.017 4.019 dyad.bmp 7.446 3.351 3.34 3.33 micro1.bmp 7.268 3.954 3.95 3.919 micro2.bmp 7.824 4.187 4.04 4.029 nerve.bmp 7.765 4.024 4.005 4.018 virus.bmp 7.359 3.151 3.14 2.996 virus1.bmp 7.339 3.095 3.085 3.083 Afer he errors and he esimaes were compued for all he images one can observe ha he range [1.8; 2.8] covers all he values compued by he fracal esimaor, and we divided his range in 100 inervals of size 0.01. For each sub-inerval we compued he number of imes he MED predicor performs beer (i.e. yields a smaller error in absolue value) han he LS predicor. his number divided by he number of pixels having he associaed esimaion of fracal dimension inside he analyzed sub-inerval gives he probabiliy ha he MED predicor performs beer han he LS predicor for he analyzed sub-inerval. If we denoe his probabiliy by p MED, hen he same probabiliy associaed o he LS predicor is p LS = 1 p MED. hese probabiliies were accumulaed for each of he wo image ses. In Fig.6. and Fig.7. are represened he probabiliies p MED, and p LS versus he esimaion of he fracal dimension for he medical se of images and respecively he naural se of images. In boh hese figures p MED is figured wih dark dos and p LS wih gray circles. I can be observed ha he esimaion provided by he fracal esimaor disinguishes wo main inervals: one owards smaller values where he MED predicor ends o perform beer, and he oher owards higher values where he LS predicor ends o provide beer predicion. his indicaes ha he esimaion
provided by he fracal esimaor ha was used correlaes wih he performance of he wo predicors. I can also be noiced ha his propery applies o boh ypes of images considered. he reason we considered hese wo ypes of images was he fac ha he LS predicor performed beer on he medical se. Fig.8. Probabiliies for all he images: p MED - dark dos, p LS - gray circles. Fig. 6. Probabiliies for he medical se of images Fig. 9. he scaled disribuions of he esimaes of he fracal dimension associaed wih he pixels from he medical images (gray circles) and naural images (dark dos) Fig.7. Probabiliies for he naural se of images. Figure 8 shows he wo probabiliies p MED, and p LS versus he esimaion of he fracal dimension for all he images (boh he medical and he naural se). he disribuion of he esimaed fracal dimensions associaed o he pixels (shown in fig. 9) indicaes ha here is an imporan number of pixels wih he esimaed fracal dimension in he region where he LS predicor is expeced o perform beer. his leads o he idea ha an improvemen of he compression raio is o be expeced if he LS predicor would be used in his region. 5 Conclusions In his paper we have analyzed he abiliy of an esimaor of fracal dimension o valuae he performance of wo ypes of predicors (MED and LS) employed in he lossless image compression framework of he LOCO-I algorihm. he resuls show ha he esimaor disinguishes wo main regions: one owards lower esimaes of he fracal dimension were he MED predicor performs beer, and he oher one owards higher esimaes of he fracal dimension, where he LS predicor performs beer. his resul indicaes ha he fracal
esimaor could operae as a discriminaor in a new compression algorihm working in he framework of LOCO-I, in which he fixed predicor would be chosen each ime, based on he esimae of he fracal dimension provided by he presened fracal esimaor. Since he emplae used o esimae he fracal dimension was chosen o be causal, his choice would be available a he decoder s side, avoiding he sending of any side informaion. As furher work, we inend o include his discriminaor in a conex-based rerieval procedure, as suggesed in [8] References: [1] I. Avcybas,. Memon, B. Sankur and K. Sayood, A Progressive Lossless/ear-Lossless Image Compression Algorihm, IEEE Signal Processing Leers, 9, no. 10, 2002, pp.312-314 [2] Y. Bai and. Cooklev, An improved mehod for lossless daa compression, Proceedings of he Daa Compression Conference (DCC 05), 2005, pp.451-454 [3] H. Wang, S. Derin Babacan, K. Sayood, Lossless Image Compression Using Conex- Based Condiional Averages, Proceedings of he Daa Compression Conference (DCC 05), 2005, pp. 418-426 [4] M. J. Weinberger, G. Serroussi and G. Sapiro, he LOCO-I Lossless Image Compression Algorihm: Principles and Sandardizaion ino JPEG-LS, IEEE rans. Image Processing 9 (8) 2000, pp. 1309-1324 [5]. Sarkar and B. B. Chaudhuri, An Efficien Approach o Esimae he Fracal Dimension of exural Images, Paern Recogniion, Vol.25, o. 9, 1992, pp. 1035-1041 [6] R.Dobrescu., M. Dobrescu and F. alos, Mulifracal medical image analysis using fracal dimension. In: Dobrescu R, Vasilescu C, eds. Inerdisciplinary Applicaions of Fracal and Chaos heory, Romanian Academy Publ. House, Buchares; 2004, pp. 78-83 [7] R. Dobrescu, Evoluion and rends in image compression, in (Ionescu, F. & Sefanoiu, D. Eds.: Inelligen and Allied Approaches o Hybrid Modeling, Seinbeis-Ediion Sugar/Berlin, 2005 [8] R. Dobrescu, M. Dobrescu, A Compuaional Framework in Modeling Cellular Communicaion, Proceedings of he 5h WSEAS In. Conf. on Applied Informaics and Communicaions, Mala, 2005