fast fractal image block coding based on local variance-1998

near edges,the whole coding scheme is applied on an angiograph (Fig.7)at a low bit rate(0.2b/pixel).Edge detection is a particu-larly important image analysis tool in angiography.If there is any difference in quantization error,especially in the neighborhood of the contours,we might expect a better localization of the contours with nearly linear phase than with the minimal one.Thus,we have implemented an edge detection algorithm[14]as a postprocessing for the reconstructed images.The results presented in Fig.7(b)–(d) shows that the reconstructed images remain robust to edge detection whatever thefilters phase characteristic,even if a low bit rate(0.2 b/pixel)is used.Visually,when zooming the neighborhood of arteries, the perceptual quality is slightly better with the approximately linear phase32-tapfilter than with the minimal phase one[Fig.7(e)and (f)].This effect is clearly invisible with8-tapfilters.IV.C ONCLUSIONIn this work,we have investigated the influence of the phase of CQF’s on the performances of a subband coding scheme.We have compared group delays,and illustrated the phase distortions. We have observed that onlyfilters with minimal phase introduce some distortions.Concerning the edges observed in detail subbands, the dispersion of energy is also perceptible for long minimal phase filters.But it must be noticed that these distortions disappear in a realistic subband coding scheme:short lengthfilters,subsamplings, and moreover quantization noise which generally is greater than the residual phase distortions.An adaptive subband coding scheme has been implemented using CQFfilters and DPCM and pyramidal VQ spatial quantizers.PSNR versus bit rate curves show that the phase characteristic does not influence the coding performances for short lengthfilters(L<32). Even at a low bit rate(0.2b/pixel),the quality of the reconstructed images remains almost equivalent for approximately linear and min-imal phasefilters.Finally,CQF phase characteristics were found to have no influence on edge-detection postprocessing in an angiography application.A CKNOWLEDGMENTThe authors thank the Associate Editor,Prof.M.T.Orchard,for his valuable suggestions,and the three anonymous reviewers for their constructive comments.This work is in the scope of the scientific topics of the PRC-GdR ISIS Research Group of the French National Center for Scientific Research(CNRS).R EFERENCES[1]J.W.Woods and S. D.O’Neil,“Subband coding of images,”IEEE Trans.Acoust.,Speech,Signal Processing,vol.ASSP-34,pp.1278–1288,Oct.1986.[2] D.Esteban and C.Galand,“Application of quadrature mirrorfilters tosplit band voice coding schemes,”in Proc.ICASSP-77,pp.191–195.[3]J.D.Johnston,“Afilter family designed for use in quadrature mirrorfilters bank,”in Proc.ICASSP-80,pp.291–294.[4]M.J.T.Smith and T.P.Barnwell III,“Exact reconstruction techniquesfor tree structured subband coders,”IEEE Trans.Acoust.,Speech,Signal Processing,vol.ASSP-34,pp.434–441,June1986.[5]M.V etterli,“Multi-dimensional subband coding:Some theory andalgorithms,”Signal Process.,vol.6,pp.97–112,Apr.1984.[6]L.Vandendorpe,“CQFfilter banks matched to signal statistics,”SignalProcess.,vol.29,pp.237–249,Dec.1992.[7]P.P.Vaidyanathan,“Quadrature mirrorfilter banks,M-band extensionsand perfect reconstruction technique,”IEEE Acoust.,Speech,Signal Processing Mag.,vol.4,pp.4–20,July1987.[8]I.Daubechies,Ten Lectures on Wavelets.Philadelphia,PA:SIAM,1992.[9]O.Rioul and P.Duhamel,“A Remez exchange algorithm for orthonor-mal wavelets,”IEEE Trans.Circuits Syst.II,vol.41,pp.550–560,Aug.1994.[10]T.W.Parks and J.H.McClellan,“Chebyshev approximation for nonrecursive digitalfilters with linear phase,”IEEE Trans.Circuit Theory, vol.CT-19,pp.189–194,Mar.1972.[11]M.J.T.Smith and S.L.Eddins,“Analysis/synthesis techniques forsubband image coding,”IEEE Trans.Acoust.,Speech,Signal Processing, vol.38,pp.1446–1456,Aug.1990.[12]K.Ramchandran and M.Vetterli,“Best wavelet packet bases in a rate-distorsion sense,”IEEE Trans.Image Processing,vol.2,pp.160–175, Apr.1993.[13]T.R.Fischer,“A pyramid vector quantizer,”IEEE rm.Theory,vol.32,pp.568–583,July1986.[14]R.Deriche,“Using Canny’s criteria to derive a recursively implementedoptimal edge detector,”put.Vis.,pp.167–187,1987.Fast Fractal Image Block Coding Based on Local VariancesC.K.Lee and W.K.LeeAbstract—In fractal image block coding,most of the time is spent on finding a close match between a range block and a large pool of domain blocks.For a large image,this effect becomes aggravated as the domain pool increases exponentially.Here,we propose using the local variances of domain blocks to reduce the search space.By sorting the contracted domain pool according to their local variances and defining an acceptance criterion for a close match,we can confine all the potential close matches to a relatively small sized window to limit the search space.The encoding time can hence be shortened with the decoded image quality as good as that using the full search method.The speedup can be over ten times depending on the complexity of encoded images.Index Terms—Fractal coding,image coding,matching method.I.I NTRODUCTIONIn recent years,fractal image block coding,based on the con-tractive image transformation,has received much attention due to its ability to generate high-resolution reconstructed images at very high compression ratios.However,it suffers from very high encoding complexity.This high complexity is due to the extensive number of matchings among range blocks with a large pool of domain blocks. In order to address this problem,several researchers have presented methods[4],[6]–[7],[10]that reduce the search space of the domain pool.However,all the proposed methods either reduce the image fidelity or require complex analysis on contours and textures of image subblocks.Our method is based on the fact that two equal-sized image blocks cannot be closely matched unless their local means and variances are closely matched.This implies that blocks whose local means and vari-Manuscript received September1,1995;revised January13,1997.The associate editor coordinating the review of this manuscript and approving it for publication was Prof.Michael T.Orchard.The authors are with the Department of Electronic Engineering,Hong Kong Polytechnic,Kowloon,Hong Kong.Publisher Item Identifier S1057-7149(98)03991-8.1057–7149/98$10.00©1998IEEEances differ greatly cannot constitute a close match.However,since fractal coding methods allow the local mean to be shifted by an arbi-trary amount,differences in mean cannot be used to exclude matches. Differences in the local variance,though,can be used to eliminate many elements of the domain pool as candidate matches.By sorting the domain blocks according to their local variances and defining an acceptance criterion,we can limit the search space to a relatively small sized window.Though our method is not guaranteed to always find the best match,experimental results show that it provides a good tradeoff between matching accuracy and complexity.For appropriate choice of search window size,the method gives an insignificant increase in decoded image distortion,while speeding up the encoding by a factor of ten or more.Experiments also reveal that a window size of about one-tenth of the domain pool is required for a relatively com-plex image,while about one-twentieth for relatively simple images. Similar methods in[5]and[8]have been proposed in vector quan-tization to reduce the search space within the predefined codebook. They use geometric properties between the input vectors and the codevectors in the codebook.Those codevectors not satisfying the geometric requirements are excluded from the search.These proposed algorithms successfully reduce the number of mathematical opera-tions to only3–20%of that required by a full search with no loss of image quality.However,it is not feasible to apply the exact approach to fractal image compression,since the size of the domain pool is much larger than the size of the codebook.Therefore,the overhead for the precomputed energy measures of the domain pool would be large.Hence,we propose a more straightforward but efficient approach based on sorting the domain pool according to variances. This correspondence is arranged as follows.A concise review of fractal-based image block coding is presented in Section II.Our algorithm is detailed in Section III.Wefirst discuss the relationship between the local variances of a transformed domain block and a range block.Then,based on this result,we explain the steps for fast encoding.Experimental results are given in Section IV to substantiate our claim and further analysis is performed based on several comparison criteria.II.O VERVIEW OF F RACTAL I MAGE C OMPRESSIONFractal image coding technique is based on the mathematical theory of iterated contractive transformations in metric spaces.The original image I,of size W2H,isfirst partitioned into nonoverlapping regions called range blocks,R i,with size w r2h r.Domain blocks are extracted from the original image.Each domain block is of size w d2h d,which should be larger than that of the range block to fulfill the contractive requirement.The pool of domain blocks,f D i g(i=1;2;111;n),is obtained by sliding a window,of size equal to w d2h d,in a single pixel step across the original image from the left to the right and from the bottom to the top.The affine transformation f i g for a grey-scale image can be considered as a three-dimensional(3-D)entity with two spatial dimensions and one intensity dimension,which isi a i b i0c id i000si+b ei t s l o c a l v a r i a n c e;a nw a n dh b e t h e w i d t h a nT h e d o m a i n b l o c k a n d ts a m e n o t a t i o n b u t w i t hv i o u s l y i n S e c t i o n I I,m k(i;j)= k3D(i;j)i s t h e c o n t r a c t e d d o m a i n b l o c k a f t e r t h ek t h s e l f-s y m m e t r i c a l t r a n s f o r m a t i o n a n dm i s t h e i n t e n s i t y s h i f t o f t h e w h od o m a i n b l o c k.He n c e,t h e m e a n of t h e t r a n s f o r m e d d o mm k= k3D:(5)This shows that the variance of the transformed domain block will remain the same after an intensity shift m and one of the k self-symmetrical transformations.Moreover,as the purpose of intensity shift is to match the mean intensity of the range block withthe transformed domain block,i.e., m = Rn 0 k 3D j being small does not imply that mkmatches R n ,we can say that mk does not match R n if j 20 2Rh=2:(6)B.Procedures of the Fast Encoding AlgorithmThe steps of the fast encoding algorithm are listed as follows.Step 1:Partition the original image into n subimages denotedby R n .Step 2:Extract the domain blocks from the original image anddenote them by f D m g .Step 3:Contract each domain block to the size of a range block.Step 4:Calculate the mean and variance of the contracted domainblocks and sort them in a descending order with their local variances.Step 5:Select one range block and calculate its mean and vari-ance.Step 6:Search the sorted domain pool until a contracted domainblock with a variance closest to that of the range block.Step 7:Define a search window size by selecting the percentage,w %,of domain blocks from f D i g to be included in the search.The search window consists of the (w=2)%of domain blocks with lower variances and (w=2)%of blocks with higher variances than the domain block identified in Step 6.Step 8:Employ (6)to search within the search window a do-main block,which has the closest match to the range block,after applying one of the eight self-symmetrical transformations and the intensity shift.Step 9:Store this mapped transformation coefficients.Step 10:Repeat Steps 5–9for all other range blocks.Variable size image partitioning methods in [2],[6],and [9]are usually used in fractal image compression to further improve the image quality.In our approach,we employ quadtree image partition method of [2]and [3]because it assigns fewer bits to the low infor-mation block but more to the high.This gives a better encoded image quality,especially in regions containing edges and texture.However,in those regions we have to pay a penalty:additional storage size.In order to implement this quadtree partitioning method,we have the two variations:i)additional sets of domain blocks for a fixed contractivity factor of 0.5;and ii)an early exit method whenever a match within the error tolerance for the quadtree splitting process occurs.TABLE IC OMPARISON OF THED ECODED I MAGES U SING i)E XHAUSTIVE S EARCH AND ii)S EARCH W ITHIN A W INDOW S IZE :4.7%OF THE M AXIMAL D OMAIN P OOL WITH AN E RROR T HRESHOLD OF36TABLE IIA VERAGE PSNR’s FOR D IFFERENT W INDOW S IZES .A VERAGE PSNR AT A G IVEN W INDOW S IZE I S THE A VERAGE OF THE F IVE PSNR’s O BTAINED BY S ETTING THE E RROR T HRESHOLD TO 36,49,64,81,AND100IV.E XPERIMENTAL R ESULTSThree images of different complexities are used to evaluate this proposed algorithm.Unlike the traditional way of classifying image complexities,we use the principles of fractal image compression.That is,simple images are rich of self-symmetries while complex ones are those with no self-symmetries.We include the images “apple,”“Bab,”and “Lena.”All these images are of resolution 2562256with 256gray levels.Besides image complexity,there are several parameters affecting encoded image quality and encoding speed,including:the size of the search window,the error thresh-old for quadtree partitioning of range blocks,and the acceptance criterion for a close match.We vary these parameters to provide a performance analysis.We decode the encoded images from affine transformation codes as given in [6]and [7].To measure the im-age fidelity,we use the peak-signal-to-noise ratio (PSNR),defined asPSNR =20log10(7)where n is the number of b/pixel,and rmse is the root mean square error between the original image and the reconstructed image.Table I tabulates the different parameters obtained from this pro-posed algorithm when compared to exhaustive search.It reveals that the algorithm produces satisfactory decoded images with a relative speedup of about ten times,where the relative speedup is defined as shown at the bottom of the page.We also note that all the differences in PSNR obtained from exhaustive search and our method are less than 0.3dB,with matching compression ratios.Table II shows the effect of various window sizes on the average PSNR of the selected images.The average PSNR at a given window size is the average of the five PSNR’s obtained by changing the error threshold in an exponential step (i.e.,thresholds 36,49,64,81,and 100).Relative Speedup =Time taken to compress an image using the exhaustive search method Time taken to compress the same image using the new encoding methodFig.2.Relative speedup versus the search window size of the image with an error threshold at100.Fig.3.Relative speedup versus the error threshold.For the apple and Lena images,the average PSNR’s are quite steady,with a maximal difference of only 0.03dB and 0.06dB,respectively,when the window size is increased from 4.7%to 10.9%of the maximal domain pool.For the Bab images,the average PSNR is slightly increased by 0.21dB when the window size is increased from 4.7%to 9.4%and tends to converge when the window size is further increased to 10.9%.From these results,we can generalize that,for simple to moderately complex images,a window size of around 5%is good enough to have decoded images with insignificant differences to those from exhaustive search.However,for very complex images,it needs around 10%.A larger search window is required for Bab because the domain pool consists of a lot of blocks with similar variances,which are extracted from the hair portion.We also perform similar analysis to the compression ratios of the selected images.The average compression ratios are increased by less than 1.5%when the window size is increased from 4.7%to 6.3%of the maximal domain pool and converge when the window size is further increased to 10.9%.The slight increase in the compression ratio comes from the quadtree partition with the early exit method,because it ends the search whenever the error threshold requirement is met.Hence,with a larger window size,the required number of transformations is slightly reduced.Fig.2shows that the relative speedup at different error thresholds is inversely proportional to the size of the search window.In fact,the time for matching takes most of the compression time.Meanwhile,it raises the difference between the ideal and actual speedup.Ideally,a search window size of 5%can achieve a speedup of 20,but the actual one is only around 11,because of an over-head in sorting the domain pool and locating the close domain block.TABLE IIIA RITHMETIC O PERATIONS PER P IXEL WITH THE S EARCH W INDOW E QUAL TO4.7%OF THE M AXIMAL D OMAIN P OOL .T HE R ANGE B LOCK S IZE I S 828.T HE V ALUES IN P ARENTHESES D ENOTE THE O VERHEAD FOR C ALCULATING AND S ORTING THE L OCAL V ARIANCES OF D OMAIN BLOCKSFig.3displays the relative speedup versus the error threshold at a window size of 4.7%.It shows a speedup of eight to 11times on different images.Besides,with a 5%search window size,the relative speedup is increased slightly with an increase in the error threshold,because more range blocks can meet the error threshold requirement before splitting takes place.Finally,a computational analysis is summarized in Table III for comparison.The figures in the table show that this algorithm needs only 4%of the number of mathematical operations required by a full search.V.C ONCLUSIONIn this work,we have proposed a fast encoding algorithm based on local variances.The rationale is that closely matching blocks have closely matching means and variances.Hence,the search space can be limited to a relatively small sized window within a domain pool sorted according to their local variances.Moreover,it has several advantages.First,it does not need to classify the domain pool using complex analysis.Instead,only a simple parameter,the local variance is required.Second,all potential domain blocks are effectively confined to a relatively small region for comparison.Third,it introduces no additional burden on the decoding process.Experimental results also demonstrate its feasibility.For the se-lected images,the search space can be reduced to around 5%,com-pared with exhaustive search,with a relative speedup of ten or more.Also,the quality of the decoded images remains almost the same.R EFERENCES[1]M.F.Barnsley and L.P.Hurd,Fractal Image Compression .Welles-ley,MA:A.K.Peters,1993.[2]Y.Fisher,E.W.Jacobs,and R.D.Boss,“Image compression:A study ofthe iterated transform method,”Signal Process.,vol.29,pp.251–263,1992.[3]Y.Fisher,Ed.,Fractal Image Compression:Theory and Application .New York:Springer-Verlag,1995.[4]M.Gharavi-Alkhansari and T.Huang,“A fractal-based image block-coding algorithm,”in Proc.IEEE ICASSP’93,vol.5,pp.345–348.[5] C.-M.Huang,Q.Bi,G.S.Stiles,and R.W.Harris,“Fast full searchequivalent encoding algorithms for image compression using vector quantization,”IEEE Trans.Image Processing ,vol.1,pp.413–416,July 1992.[6] A.Jacquin,“Image coding based on a fractal theory of iterated contrac-tive image transformations,”IEEE Trans.Image Processing ,vol.1,pp.18–30,Jan.1992.[7] A.Jacquin,“Fractal image coding:A review,”in Proc.IEEE ,vol.81,pp.1451–1465,Oct.1993.[8]M.T.Orchard,“A fast nearest-neighbor search algorithm,”in Proc.IEEE ICASSP’91,pp.2297–2300.[9]L.Thomas and F.Deravi,“Pruning of the transform space in block-based fractal image compression,”in Proc.IEEE ICASSP’93,vol.5,pp.341–344.[10]G.Vines and M.H.Hayes III,“Adaptive IFS image coding withproximity maps,”in Proc.IEEE ICASSP’93,vol.5,pp.349–352.。