PROCEEDINGS OF INTERNATIONAL WORKSHOP ON IMAGE PROCESSING: THEORY, METHODOLOGY, SYSTEMS AND APPLICATIONS 2-22 JUNE,1994 BUDAPEST,HUNGARY Adaptve Fractal Image Codng n the Frequency Doman K AI UWE BARTHEL AND THOMAS VOYÉ Insttut für Fernmeldetechnk, Technsche Unverstät Berln Enstenufer 25, D-1587 Berln, Germany Phone: ++ 49 3 31423838, E-mal: barthel@ftsu.ee.tu-berln.de ABSTRACT Fractal mage codng has been used successfully to encode dgtal grey level mages. Especally at very low btrates fractal coders perform better than cosne-transform-based JPEG coders. A block-based fractal mage coder s able to explot the redundancy of grey level mages by descrbng mage blocks through contractvely transformed blocks of the same mage. Prevous fractal coders used affne lnear transformatons n combnaton wth 1 st order lumnance transformatons that change the brghtness and scale the lumnance values of mage blocks. We propose an extenson to hgh order lumnance transformatons that operate n the frequency doman. Wth ths transformaton and an adaptve codng scheme a better approxmaton of mage blocks can be acheved. Btrate reductons are hgher than those acheved by "spatal-doman" fractal codng schemes. An addtonal effect of ths new transformaton s a better convergence at the decoder. I. INTRODUCTION AND OVERVIEW The prncple of fractal mage codng conssts n fndng a constructon rule that produces a fractal mage whch approxmates the orgnal mage. Redundancy reducton s acheved by descrbng the orgnal mage through contracted parts of the same mage (self-transformablty). Fractal mage codng s based on the mathematcal theory of terated functon systems (IFS) developed by Barnsley [1]. Jacqun [2] was the frst to propose a block-based fractal codng scheme for grey level mages. In [3] we have shown that the codng performance can be greatly mproved by applyng a vector quantzaton to the optmal lumnance transformaton and usng a better geometrcal search scheme. In ths paper we descrbe a new lumnance transformaton n the frequency doman. Wth ths transformaton the codng effcency can be further enhanced. At the decoder fewer teratons are needed to reconstruct the mage. In secton II, we brefly present the prncple of a block based fractal mage coder. An mproved codebook desgn and an adaptve geometrcal search scheme are descrbed n secton III. The proposed new lumnance transformaton s presented n secton IV. The descrpton of the new coder can be found n secton V. Fnally, n secton VI, we present some results and dscuss the merts of the new codng scheme. II. THE PRINCIPLE OF A FRACTAL BLOCK-CODER The mage to be encoded s parttoned nto non-overlappng square blocks. R,j s the mage block at the poston (, j) and s called a range block. The task of a fractal coder s to fnd a good approxmaton for all range blocks. Each range block s approxmated by a transformed larger block D l,k of the same mage (doman block) as shown n fgure 1.
τ, j z D l, k R,j z R ',j y y x x orgnal mage transformed approxmaton Fg. 1. Approxmaton of a range block through a transformed doman block The transformaton τ,j combnes a geometrcal transformaton and a lumnance transformaton. The geometrcal transformaton s an affne lnear transformaton that conssts of a spatal contracton and a poston shft that maps the doman block to the poston of the range block. The doman block that has been scaled down to the sze of the range block s referred to as codebook block. Jacqun proposed a 1 st order lumnance transformaton that scales the dynamc range and changes the brghtness of the pxel values of a codebook block. In matrx form τ,j can be expressed as follows: LxO k k x x τ j, M y P = L 11 12 OL O Mk k PMyP + L O 21 M y P NM zqp NM 22 aqp NM zqp NM b QP (1) z denotes the pxel ntensty of an mage at the poston x, y. ( a, b, k m,n R) Only the transformatons of each range block have to be transmtted to the decoder. The set of all transformatons can be seen as the fractal code for the orgnal mage. Ths code, teratvely appled to any ntal mage, generates the reconstructed mage. To ensure the convergence at the decoder the transformatons τ,j have to be contractve. Ths means: det L M N k k k k 11 12 21 22 O < 1 and a < 1 (2) P Q The process of fractal encodng s lossy. The approxmaton error ε, that s determned at the coder ncreases durng the decodng process snce the codebook blocks are generated at the decoder from the fractal reconstructon mage whch s not free of errors. If the scalng factor a s assumed constant, the upper bound for the approxmaton error after the decodng s gven by ε / (1- a ). As the total number of transformatons has to be kept low, herarchcal coders wth varable range block szes are used. If the approxmaton error for a large range block exceeds a gven level, ths block s splt nto up to four smaller range blocks for whch addtonal transformatons are determned. For hgh codng effcency well chosen codng parameters n combnaton wth effcent codng of the fractal transformaton parameters are necessary. III. GEOMETRICAL TRANSFORMATION The search for a geometrcal transformaton can be seen as a search n a codebook that contans the set of contracted doman blocks. Codng effcency strongly depends on the constructon of ths codebook. Another mportant aspect s the order n whch ths codebook s searched. 2
y t l When constructng the codebook, the set of all possble affne-lnear transformatons (equaton 1: k m,n, x, and y) has to be reduced to a sutable subset. As dgtal mages are sampled mages wth a gven spatal resoluton not all affne lnear transformatons are possble. The sze rato of range to doman block s usually chosen to 1:2 n x- and y-drectons. A smaller contracton rato allows a better approxmaton of range blocks but results n a hgher error propagaton at the decoder. Usng hgher contracton ratos leads to decreasng smlartes between range and codebook blocks. To assure contractvty the codebook blocks are generated from the fltered and sub-sampled orgnal mage. Jacqun proposed a smple averagng flter. We obtaned better codng results usng a 1-tap antalasng flter wth a cut-off frequency below π/2. We determned an effcent search path by examnng the dstrbuton of codebook block postons that yeld the best approxmaton for a gven range block. Very often the best codebook block corresponds to the doman block drectly above or close to the poston of the range block to be encoded. Ths fact can be used for an optmzed adaptve search scheme. The codebook blocks beng the most probable are examned frst. The search path has the form of a spral and starts wth the codebook block drectly above the range block (Fg. 3). By ntroducng search regons a varable length of the search path s possble. The search s aborted at the end of each search regon f the approxmaton error s below a threshold value. Ths search scheme reduces the encodng tme and the average search ndex. Fgure 2 shows the probablty densty functon of the search ndces. In the gven example an mage was encoded wth an 8 bt geometrcal codebook as shown n fgure 3. The entropy of the search ndces s reduced f addtonal smaller search regons wth error thresholds are ntroduced. Note that we use a relatve addressng of the codebook blocks and a varable doman block shft. y t s n e d b a b o r p n o t c n u f 1.8.6.4.2 wth 3 search regons and error threshold wthout error threshold 32 64 96 128 16 192 224 256 search ndex Fg. 2. Probablty densty functon of the geometrcal codebook ndces usng a search scheme as shown n fg. 3. The maxmum search regon was 8 bt. The addtonal search regons used a search wdth of and 4 bts. mage border possble postons of doman blocks AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA AA frst doman block range block search path: frst doman block mnmum search maxmum search Fg. 3. Adaptve search scheme usng a mnmum search regon 3
o o IV. LUMINANCE TRANSFORMATION Problems of 1 st order lumnance transformatons The 1 st order lumnance transformaton λ 1 proposed by Jacqun scales the dynamc range (a) and changes the brghtness of the pxel values (b) of a codebook block g: λ 1 agf= a g+ b (3) Ths 1 st order transformaton has two dsadvantages: - Only small and 'smple structured' range blocks can be approxmated well. - The convergence at the decoder s poor. In partcular f a hgh approxmaton error s tolerated at the coder, the error propagaton at the decoder s very hgh. In ths case the number of teratons necessary to decode the reconstructon mage wll rse. Modfed 1 st order lumnance transformaton Fgure 4 shows the dstrbuton of optmal non-quantzed a/b-values obtaned from a fractal coder usng the conventonal 1 st order lumnance transformaton. 4 4 3 3 b t e s f f 2 1 b t e s f f 2 1-1 -.5.5 1-1 -.5.5 1-1 -1 scalng factor a scalng factor a Fg. 4. Dstrbuton of the optmal a/b-coeffcents usng Fg. 5. Dstrbuton of the optmal a/b-coeffcents usng the conventonal 1 st order lumnance transformaton the modfed 1 st order lumnance transformaton From eq. (3) t can be seen that the b-offset serves to adjust the scaled means of the codebook blocks and s dependent on a. Decreasng a-values generally requre ncreasng b-values. Ths leads to a trangular shaped a/b-dstrbuton (Fg. 4). There s a strong accumulaton of the a/b-values n the regon of a-values near 1. Ths ndcates that the dynamc range of most codebook blocks s kept almost constant. Scalng values close to 1 have the dsadvantage that they result n a hgh error propagaton at the decoder. The a/b-dstrbuton and the fact that the means represent the largest energy component of the codebook blocks result n a hgh upper error bound. Avodng the scalng of the codebook means by large a-values reduces the error propagaton at the decoder. We propose a smple modfcaton of the lumnance transformaton. We decorrelate the a/b-values by only scalng the dynamc part of the codebook blocks. Wth ths modfed transformaton a smlar approxmaton of the range blocks s possble. Wth a well-chosen factor a a lower upper error bound at the decoder can be acheved. λ 1mod. g f= a ( g µ ) + a µ + b a g g (4) The constant factor a can be chosen from to 1 and s found as a compromse: For a = 1 the varance of the a/b-coeffcents reaches ts mnmum, but the lumnance transformaton s not contractve anymore. If a s set 4
to we obtan a mnmal error propagaton and a mnmal decodng tme at the decoder. In ths case the varance of the a/b-dstrbuton s maxmum. Our studes have shown that wth quantzed a/b-coeffcents the best codng results are reached for a =.5. Fgure 5 shows the dstrbuton of optmal non-quantzed a/bvalues of the modfed 1 st order lumnance transformaton. Fgure 6 compares the convergence at the decoder for a crtcal part of the "Clown"-mage. We compare the conventonal and the modfed lumnance transformaton wth quantzed and non-quantzed coeffcents. It can be seen that the modfed lumnance transformaton outperforms the conventonal transformaton n the reconstructon error and the number of teratons needed. Decodng examples are shown n fgure 7. The sgnfcance of the artfacts as shown n fgure 7b s mage dependent and they only occur wth quantzed parameters. By usng the modfed lumnance transformaton these artfacts can always be avoded. 1 3 8 1 mean squared error 6 4 2 4 2 5 1 15 2 25 3 Iteraton number Fg. 6. Comparson of the conventonal and the modfed 1 st order lumnance transformaton: conventonal: 1 non-quantzed, 3 quantzed, modfed: 2 non-quantzed, 4 quantzed a) orgnal b) conventonal 1 st order lumnance transformaton (1 th teraton) c) modfed 1 st order lumnance transformaton (1 th teraton) Fg.7. Orgnal and decoded mages usng the conventonal and the modfed 1 st order lumnance transformaton wth quantzed a/b-values. 5
Hgh order lumnance transformatons n the frequency doman Any mprovement n the approxmaton of range blocks wll mprove the mage qualty and can reduce the total number of transformatons needed to descrbe the fractal approxmaton of the mage to be encoded. One approach to do ths s to use addtonal 'basc codebook blocks' [4], such as smple polynomnal blocks. We feel that such an approach s not very promsng because these smple blocks are easy to encode wth the fractal coder tself. Another possblty s the use of squared and cubc scalng of the pxel ntenstes of the codebook blocks. The optmal scalng parameters are dffcult to determne because of the dependency of the parameters on each other. A further problem s to guarantee the contractvty of such a transformaton. A hgh order lumnance transformaton has to fulfll the followng condtons: To enable ther ndvdual adaptaton the transformaton coeffcents should be ndependent of each other. To assure a control of the contractvty, the requrements for the contractvty should be controllable ndependently by the transformaton coeffcents. Our proposal for a hgh order lumnance transformaton s an extenson of our modfed 1 st order lumnance transformaton: Frst we transform all range and codebook blocks va the dscrete cosne transform (DCT). In the frequency doman we obtan the energy compacted spectra of range and codebook blocks. Then by ndvdually settng or scalng the spectral values of the codebook block G(u,v) we can approxmate the spectrum of the range block F(u,v). N 1 N 1 λ( g ) = IDCT UU auv (, ) Guv (, ) + buv (, ) G( uv, ) = DCT( g), Fuv (, ) = DCT( f ) (5) u= v= whereby N denotes the sze of the blocks, the IDCT s the nverse DCT Many codng schemes are possble usng subsets of ths general lumnance transformaton (5). If all spectral values were set or scaled, the number of transformaton parameters to be transmtted would ncrease drastcally. However, many range blocks can be approxmated wth low order lumnance transformatons. In ths paper we propose a codng scheme usng one or more scalng factors for the dynamc part of the codebook spectrum. For a lumnance transformaton of order K we merge subsets of the spectral values to non-overlappng regons R 1 to R K. The mean s approxmated the same way as wth the modfed 1 st order lumnance transformaton. If usng a 1 st order lumnance transformaton, all dynamc coeffcents are scaled wth only one scalng factor (a 1 ). A 2 nd order lumnance transformaton has got three regons, so the dynamc part of the spectrum s scaled wth two coeffcents (a 1, a 2 ). For lumnance transformatons of order 2 and hgher varous frequency doman parttons are possble. Fgure 8 shows some examples of parttons for 2, 3, and 4 regons. These hgh order lumnance transformatons can be expressed the followng way: λ K ( g ) = IDCT N 1 u= N 1 a G(,) u v + b u, v f = = v= auv (, ) Guv (, ) else UU auv (,) = a f (,) uv R = 1,.., K (6) For every regon R the optmal scalng factor a opt can be evaluated : a opt = { Guv (, ) Fuv (, )} ( uv, ) R ( uv, ) R 2 bopt = F(,) a G(,) Guv (, ) (7) To assure contractvty a opt must not exceed a value of 1. a agan s set to.5. 6
If the a /b-parameters are lmted or quantzed, we get an approxmaton error plus a quantzaton error: K 2 2 e( FG, ) = b + F( u, v) = 1 ( uv, ) R ( uv, ) FuvGuv (, ) (, ) R 2 Guv (, ) ( uv, ) R 2 + a 2 2 G( u, v) ( uv, ) R a = aopt a b= bopt b (8) a R 1: a (u,v) = a 1 R 2: a (u,v) = a 2 R 3: a (u,v) = a 3 2 regons ( 1st order ) u 3 regons ( 2nd order ) v 4 regons ( 3rd order ) Fg. 8. Examples of dfferent frequency doman parttons for lumnance transformatons of 1 st, 2 nd, and 3 rd order (block sze 8x8) V. DESCRIPTION OF THE CODER Before descrbng our new codng scheme, we suggest some mportant modfcatons of the conventonal fractal codng scheme: Improvements of the fractal codng scheme Partal approxmaton: To reduce the total number of transformatons generally herarchcal codng schemes wth varable range block szes are used. In a frst step, transformatons for the largest range blocks of the hghest herarchy level are determned. If the approxmaton error s too hgh for any of the four range blocks of the next herarchy level, the large range block contanng these smaller blocks s splt nto sub blocks. For these sub blocks addtonal transformatons are determned. The transformaton for the large range block s kept f the number of addtonal transformatons does not exceed two. The total number of transformatons can be sgnfcantly reduced f new transformaton parameters are determned for the remanng part of the large range block. The codng procedure for large range blocks can then be descrbed as follows: The transformaton of a large range block and the resultng errors n the sub blocks are determned. The sub block that s responsble for the hghest error component s excluded and a new transformaton for the remanng ¾-block s searched. If necessary, ths procedure s repeated for the ¾-block and leads to a ½-block. Many large range blocks that were totally splt usng the conventonal scheme can now be coded as ½- or ¾-blocks. 7
Codebook-update and a/b-update: One problem of fractal mage codng s the error propagaton at the decoder. It results from the fact that at the decoder the codebook s generated from the reconstructed mage whereas at the coder the codebook s generated from the orgnal mage. The error propagaton at the decoder can be reduced f the coder codebook s updated wth the coded versons of the range blocks. The a/b-update s comparable to the codebook-update. At the end of the codng process the best possble approxmaton of the orgnal mage s known. Now the coder could start codng the mage agan and agan gettng a better and better approxmaton of the decoder codebook. As ths ncreases the codng tme we propose to keep the geometrcal transformatons, but to redetermne the best a/b-values. The a/b-update can be repeated. We found that 1 to 2 a/b-updates are useful. Usng the modfed lumnance transformaton and the descrbed update procedures, the error propagaton can be reduced and a slghtly hgher codng effcency s obtaned. The ncrease of the decodng error can be reduced to approxmately 1-4 % of the codng error compared to more than 1 % usng the conventonal scheme. Coder descrpton We use a coder wth a three level herarchy wth range block szes of 16x16, 8x8 and 4x4 pxels. For the quantzaton of the lumnance transformaton we apply a vector quantzaton (VQ) technque. We use an adaptve search algorthm to determne the order of the lumnance transformaton and the search regon that s used for the geometrcal transformaton. For each herarchy level we defne a set of search classes. A search class contans a fxed search regon and a lumnance transformaton wth fxed order and VQ-codebook sze. These search classes are searched successvely. If the approxmaton error after searchng one class fulflls a gven search stop crteron (error threshold) the search s aborted, otherwse the next search class s examned. To obtan good codng effcency the bt costs are ncreased durng the search. Ths assures to encode a range block wth the lowest necessary rate. Smulatons have shown that t s useful to ncrease both the VQ-codebook sze and the search wdth. If even wth the maxmum search class no good approxmaton can be found, then ths transformaton s rejected and addtonal transformatons are determned usng the partal block approxmaton. For smaller range blocks ths scheme s repeated untl the hghest search class of the lowest herarchy level s reached. As the splttng crteron we check all errors of the smallest range block sze. The advantage of ths codng scheme s that we can locally adapt the btrate to the mage contents. No classfcaton of the range blocks s done before the codng process. Fgure 9 shows the search classes of the lowest herarchy level (block sze 4x4 pxels). A complete set of codng parameters s shown n table 1. 3 s o m e try 25 2 lum nance geom etry AAA AAA AAA AAA bts 15 1 5 1 2 3 4 s e a rc h c la s s Fg. 9. Adaptve search scheme combnng geometrcal and lumnance transformaton (search classes 1 to 4) The appled frequency doman parttons of the lumnance transformatons are shown. 8
Transformaton parameters to be transmtted to the decoder are: - splttng partton of range blocks of the hgher herarchy levels, - search class, - geometrcal ndex of the codebook block and the sometry (f used) and - codebook ndex of the lumnance transformaton VQ (scalng factors a 1 to a K and the offset b). (The search class of a block and the splttng partton are entropy-coded.) The mage qualty respectvely the btrate can be controlled over a large range by only adjustng the error thresholds. For very low btrates however, the block szes have to be enlarged to 32x32, 16x16, and 8x8 pxels. herarchy level block sze search class splt error threshold search stop error threshold lumnance transformaton geometrcal transformaton [msq] [msq] order VQ codebook sze [bts] codebook sze (search) [bts] 3 16 1 6 15 1 st 8 3 2 1 st 8 7 2 8 1 9 3 1 st 8 3 2 1 st 8 7 1 4 1-12 1 st 6 2 2 1 st 6 7 3 2 nd 8 1 4 3 rd 9 14 + 3 (som.) Table 1. : Codng parameters used for the codng results shown n table 2. level and search class classfcaton [bts] partton [bts] geometry [bts] lumnance [bts] sum [bts] number of blocks product [bts] 3 x 2 - - - 2 35 7 3_1 2-3 8 13 313 469 3_1x 2 3.13 3 8 16.13 148 2387 3_2 3-7 8 18 68 1224 3_2x 3 3.13 7 8 21.13 145 363 2 x 2 - - - 2 378 756 2_1 2-3 8 13 517 6721 2_1x 3 3.22 3 8 17.22 236 463 2_2 2-7 8 17 394 6698 2_2x 3 3.22 7 8 21.22 34 7214 1_1 1-2 6 9 131 1179 1_2 2-7 6 15 461 6915 1_3 3-1 8 21 371 7791 1_4 3-17 9 29 225 6525 PSNR = 33.45 db total :.266 69835 bpp Table 2. : Codng results for the Lena mage (512 x 512 pxels). k x : level k, block s totally splt; k_j : level k, search class j, block s not splt; k_jx : level k, search class j, block s partally splt (the splttng partton s addtonally coded) 9
VI. SIMULATION RESULTS AND CONCLUSION We have proposed a new block-orented fractal codng scheme usng an adaptve search scheme wth an extended lumnance transformaton n the frequency doman. Ths transformaton s able to better approxmate codebook blocks to range blocks and has a better convergence at the decoder. The btrate s reduced because fewer transformatons are needed to descrbe the fractal approxmaton of the mage to be encoded. The subjectve qualty of mages coded wth our new scheme s superor compared to conventonal fractal coded mages. Blockng artfacts are reduced and detaled structures are better preserved. In our smulatons we used a herarchcal fractal coder wth varable block szes. Our results show that the 'Lena'-mage (512 x 512 pxels) can be coded at the rate of.1 bpp to yeld a peak-to-peak SNR of 3 db. Fgure 1 shows the coder performance compared to JPEG. Due to the hgh number of parameters detaled nvestgatons are needed to acheve optmal codng effcency. Wth optmzed parameters and better codebooks for the lumnance transformaton further mprovements are to be expected. Many dfferent codng schemes are possble usng the general lumnance transformaton expressed n equaton 5. A new effcent codng scheme unfyng fractal and transform codng wll be presented n a further publcaton. 37 36 35 34 fractal coder PSNR 33 32 JPEG 31 3 29.1.2.3.4.5.6 bpp Fg. 1. Codng results of the new fractal codng scheme compared to JPEG. (Image: Lena 512x512 pxels) REFERENCES [1] M. F. Barnsley, Fractals Everywhere. New York: Academc Press, 1988. [2] A. Jacqun, Image Codng Based on a Fractal Theory of Iterated Contractve Image Transforms. SPIE Vol. 136 Vsual Communcatons and Image Processng 9 [3] K. U. Barthel, T. Voyé, P. Noll, Improved Fractal Image Codng, Proceedngs of PCS ' 93, secton 1.5 [4] M. Gharav-Alkhansar and T. S. Huang, A Fractal-Based Image Block-Codng Algorthm, Proceedngs ICASSP 93,V 345-348 [5] E. W. Jacobs, Y. Fsher and R. D. Boss, Image Processng. A study of the terated transform method, Sgnal Processng 29 (1992) 251-263 1