fast fractal image compression using spatial correlation-2004

Fast fractal image compression using spatial correlationT.K.Truong,C.M.Kung,J.H.Jeng *,M.L.HsiehDepartment of Information Engineering,I-Shou University,Kaohsiung County 840,TaiwanAccepted 8March 2004Communicated by Prof.I.AizawaAbstractFractal image compression is time consuming in the encoding process.The time is essentially spent on the search for the best-match block in a large domain pool.In this paper,the spatial correlations in both the domain pool and the range pool are utilized to reduce the searching space.With this technique,the encoding speed is 2.6times faster than that of the full search method while the quality of the retrieved image is almost the same.Moreover,since the searching space is limited to the matched blocks of the previous range blocks,fewer bits are required to represent the transform.The bit rate is thus improved by about 20%.Ó2004Elsevier Ltd.All rights reserved.1.IntroductionFractal image compression was original proposed by Barnsley and coworkers [1–3]and first realized by Jacquin in 1990[4].The underlying premise of fractal image compression is based on the partitioned iteration function system (PIFS)which utilizes the self-similarity property in the image to achieve the purpose of compression.To encode an image according to the self-similarity property,each block must find the most similar domain block in a large domain pool.For baseline method,the encoding process is time consuming since a large amount of compu-tations of similarity measurement are required to find the best match.Also,in order to achieve the global optimization,global offsets have to be recorded,which increase the storage spaces.Therefore,focal aims of fractal image compression are to speed up the encoder and to increase the compression ratio.In the Fisher’s classification method [5],a given image block was divided into the four quadrants.For each quadrant,the average and the variance were computed.According to certain combination of these values,72classes were constructed.This method reduced the searching space efficiently.However,it required large amount of compu-tations and moreover,the arrangement of these 72classes was complicated.In Wang et al.[6],four types of range block were defined base on the edge of the decoded image.They used a hybrid type of coding mechanism to achieve higher compression ratio while maintaining a reasonable image quality.Their method does provide speedup ratio of 1.6–5times,but it still requires the same amount of storage space as that of the baseline method.In this paper,a new search strategy based on image correlation is used to improve the encoding speed and reduce the storage space,while the quality of retrieved image is preserved.The spatial correlation reveals that neighbor blocks usually have some similar properties such as edge and shade,etc.Moreover,the characteristics of the spatial correlation depend on the orientations of the edge and shade.For instance,consider the 8nearest neighbor blocks of a given block.If the given block possesses a horizontal edge,it’s left and right neighbors usually posses the similar horizontal edges,*Corresponding author.Tel.:+886-7-657-7251;fax:+886-7-657-8944.E-mail address:jjeng@.tw (J.H.Jeng).0960-0779/$-see front matter Ó2004Elsevier Ltd.All rights reserved.doi:10.1016/j.chaos.2004.03.015Chaos,Solitons and Fractals 22(2004)1071–1076/locate/chaosbut not the other 6neighbors.Similarly,if the block possesses a diagonal edge,then it’s left-up and right-down neighbors usually posses a diagonal edge,but not the others.Based on this property,one can limit searching space of the current block to the matched domain blocks of the neighbor range blocks.Since the searching space is much smaller than that of the full search method,the compression speed is improved.On the other hand,in order to avoid poor matches using this mechanism,one also pre-defines a threshold to determine if a full search process for this range block should be invoked.Thus the quality of the retrieved image can be maintained.This algorithm can also improve the compression ratio.Since the searching space is limited relative to the previous matches,fewer bits are required to record the offset of the domain block instead of the absolution position.2.Mathematical backgroundThe idea of fractal image compression is based on the Iteration Function System (IFS)in which the governing theorem is the Contractive Mapping Fixed-Point Theorem [5]given as follows.Theorem.Let x be a complete metric space and f :X !X be a contractive mapping.Then there exists a unique point x f 2X such that x f ¼f ðx f Þ¼lim n !1f ðn Þðx 0Þ,for any point x 02X .The point x f is called the fixed point or the attractor of the mapping f .For a collection of functions,the Collage Theorem [5]says that if w 1;w 2;...:;w n ,w i :X !X ,are contractive transforms then the map W ¼[w i is contractive.Thus,by Contractive Mapping Fixed-Point Theorem,there exist a unique attractor S 2X ,and S ¼W ðS Þ.IFS may be regarded as a collection of contractive transforms which has a unique attractor.Fractal image compression is an inverse problem,i.e.,if some set S is given,how to find the IFS which has S as its attractor.Another difficulty is that for natural images,only local self-similarities exist.Thus,there may not exist a transform that can make a natural image to be its own contractive fixed-point.Therefore,the idea of local self-similarity is adopted to form the Partitioned Iterated Function System (PIFS)in which is a collection of contractive maps w i :D i !X where D i &X for i ¼1;...;n .3.Baseline methodFor simplicity,let f be a given 256Â256gray level image.The domain pool ‘D’is defined as the set of all possible blocks of size 16·16of the image f ,which makes up ð256À16þ1ÞÂð256À16þ1Þ¼58081blocks.The range pool ‘R’is defined to be the set of all non-overlapping blocks of size 8·8,which makes up ð256=8ÞÂð256=8Þ¼1024blocks.For each block v from the range pool,the fractal transformation is constructed by searching all of the elements in the domain pool to find the most similar block.Let u denote a sub-sampled domain block which is of the same size as v .The similarity of u and v is measured using mean square error (MSE),which is defined by MSE ¼164P 7j ¼0P 7i ¼0ðu ði ;j ÞÀv ði ;j ÞÞ2.The fractal transformation allows the Dihedral transforms of the domain blocks,i.e.,the 8orientations of the blocks generated by rotating the blocks counter clockwise at angles 0°,90°,180°and 270°and flipping with respect to the line y ¼x ,respectively.Rotate 0°,Rotate 90°,Rotate 180°,Rotate 270°1001 ;01À10 ;À100À1 ;0110 Flip with the line X ¼Y from above0110;100À1;0À1À10 ;À1001Thus for a given block from the range pool,one needs to compute 58081·8¼464,648MSE to obtain the most similar block from the domain pool.Thus,one needs 1024·464,648¼475,799,552MSE computations in total to encode the whole image using this base line compression method.The fractal transformation also allows the adjustment of the contrast p and the brightness q on the block u .Thus the similarity is to minimize the quantity d ¼k p Áu k þq Àv k ,where u k ,06k 67are the 8orientations of u .By calculus,p and q can be computed directly by1072T.K.Truong et al./Ch aos,Solitons andFractals 22(2004)1071–1076p¼½N h u;v iÀh u;1ih v;1i ½N h u;u iÀh u;1iq¼1N½h vÁ1iÀp h uÁ1iwhere N is the number of pixels of the range pool‘R’.Finally,the position of the domain block(after sub-sampled,it is denoted by u),the contrast p,the brightness q,and the orientation k constitute the fractal code of the given range block v.For256Â256image,16bits are required to represent the position of the domain block.4.Fast encoding algorithmTo speed up the encoder time,one makes use of the spatial correlation to reduce the searching space.Let r j be the range block to be encoded,06j<1024.Denote the neighbor range blocks of r j,as depicted in Fig.1,by r H,r V,r D1and r D2which have been encoded.These neighbors are the same as those utilized to improve the vector quantization(VQ)image coding[7,8].Assume d H1,d V1,d D11and d D21are the corresponding matched domain blocks,respectively.Now,onewill restrict the searching space of r j to d H1,d V1,d D11,d D21including some domain blocks in the relative directions.Forexample,d H1is the mapped domain block of r H which is in the horizontal direction of r j.Thus one expands the searchingspace in the horizontal direction to d H0,d H1,d H2and d H3as depicted in Fig.1.Similarly,d V1,d D11and d D21are expandedaccording to their corresponding directions.Thus,the searching space of r j is limited toS¼f d H0;d H1;d H2;d H3;d V;d V1;d V2;d V3;d D1;d D11;d D12;d D13;d D2;d D21;d D22;d D23gIn this case,the expansion width is said to be4,which can be set to other values according to the trade offbetween the encoding speed and the bit rate.It should be noted that some of these neighbors and their extended domain blocks might not exist.They are considered whenever they are applicable.To avoid large gaps between this local minimum and the global minimum obtained through the baseline method, one pre-defines a threshold T.If the local minimum exceeds this threshold a full search will be invoked.The detail steps of the modified encoding algorithm are given as follows:1.j¼0.2.Perform the full search for r j.Let dðjÞbe the matched domain block.Record the fractal transformation.3.j¼jþ1,if j¼1024then stop.4.Define the searching space S byS¼f d H0;d H1;d H2;d H3;d V;d V1;d V2;d V3;d D1;d D11;d D12;d D13;d D2;d D21;d D22;d D23gas given in Fig.1.5.Search the best match of r j from S.Let dðjÞbe the best matched block.If MSEðdðjÞ;r jÞ<T,then record the fractaltransformation and go to step3.Otherwise go to step2.T.K.Truong et al./Ch aos,Solitons andFractals22(2004)1071–10761073In step2,since the full search is performed,the absolute position is recorded.In step5,when the‘‘If’’condition holds,the range block r j is called a‘‘hit’’block.It stands for the local minimum being acceptable.For such a hit block, only4bits are required to record the offset of the domain block instead of the16-bits absolute position.Two bits are used to record the range correlation and the other2to record the domain correlation as depicted in Fig.1.Bigger expansion width will produce more hit blocks and save more encoding time,but the bit rate will be higher,since more bits are required to record the relative offset.Let N R and N H denote the number of range blocks and hit blocks, respectively.For hit blocks,2þB W bits are required to record the relative positions,where B W denotes the number of bits to represent the expansion width.For non-hit blocks(N RÀN H in total),the B A bits are required to record the absolute positions.Let B k,B p and B q denote the number of bits required to represent the orientation,the contrast and the brightness,respectively.Then the bit rate(bit per pixel,bpp)can be computed directly in terms of the number of hit blocks asbpp¼N Hð1þð2þB WÞþB kþB pþB qÞþðN RÀN HÞð1þB AþB kþB pþB qÞTPwhere,N TP is the total number of pixels in the image.Note that,one bit is required to indicate if the block is a hit block or not.5.Experiment resultsThe images Lena,Baboon,F16and Pepper are tested to demonstrate the speed-up rate bit rate and quality of the proposed algorithm in comparison to the baseline method.For a given image of size256·256,the size of range blockare chosen to be8·8and4·4.The software simulation is done using C++on a Pentium IV1800Windows XPPC.Fig.2.(a)Original image,Lena of size256Ã256.(b)The initial image for the decoder of the fractal compression.(c)Baseline method, time used¼12.01min,PSNR¼28.15,bit rate¼0.4844.(d)Proposed method,time used¼4.63min,PSNR¼27.87,bit rate¼0.3824, T¼100.1074T.K.Truong et al./Ch aos,Solitons andFractals22(2004)1071–1076As an illustrative example,Fig.2shows the results of the proposed method in comparison to the baseline method. Fig.2(a)is the original Lena image and Fig.2(b)is the initial image used to retrieve the fractal-compressed image.Fig. 2(c)and(d)show the retrieved images using the baseline method and the proposed method,respectively.With threshold being set as T¼100,the proposed method is2.59times faster and the bit rate is also improved while there is only0.3dB decay.The relation between the number of hit blocks and the image qualities at various thresholds are depicted in Fig.3, which is obtained using the image Lena.It is observed that higher value of threshold produces more hit blocks and higher compression ratio,but the quality will decrease.Different sizes of range blocks will change the number of hit blocks.Since smaller blocks have simpler properties than bigger blocks,it will be easier for them tofind the best matches.Therefore,more hit blocks will be encountered in the case of smaller range block size.Such a factor will affect the speed-up rate,bit rate and PSNR.The PSNR is a measurement of the distortions between two images f and g of sizes mÂn,which is defined as:PSNR¼10Âlog2552 MSEwhereMSE¼P nÀ1j¼0P mÀ1i¼0ðfði;jÞÀgði;jÞÞ2Tables1and2show the results of the proposed method and the baseline method using range block sizes of8·8and 4·4,respectively.The threshold is100and the expansion width is4,i.e.2-bit width.According to the tables,it can be easily seen that the proposed method produces very little influence on the quality of the decoded image,while it produces lower bit rates and higher speed-up rates for bothcases.Table1The comparison of baseline method and proposed method of size8·8Method PSNR Time bpp Hit blocks Speed-up rate Pepper Baseline28.8612.120.484401Proposed28.44 4.350.3782665 2.78Lena Baseline28.1512.010.484401Proposed27.87 4.630.3824642 2.59F16Baseline24.8712.120.484401Proposed24.82 5.300.3925587 2.29Baboon Baseline19.9112.120.484401Proposed19.9110.940.4778121 1.11T.K.Truong et al./Ch aos,Solitons andFractals22(2004)1071–107610751076T.K.Truong et al./Ch aos,Solitons andFractals22(2004)1071–1076Table2The comparison of baseline method and proposed method of size4·4Method PSNR Time bpp Hit blocks Speed-up rate Pepper Baseline34.1217.02 1.937501Proposed32.82 2.24 1.345835737.58 Lena Baseline34.2116.78 1.937501Proposed32.68 2.67 1.36393474 6.28 F16Baseline30.9217.01 1.937501Proposed30.38 4.43 1.43933062 3.84 Baboon Baseline23.6117.01 1.937501Proposed23.5212.72 1.79351128 1.346.ConclusionIn this paper,the spatial correlations in both of the domain and range blocks are utilized to speed up the encoding process and improve the bit rate for fractal image compression.The rationale is the correlation between the neighbor blocks in an image.The matched domain blocks and the expanded blocks of the four neighbors are collected as the searching space.As a consequence,the searching space is much smaller than that of the baseline method.Because the proposed algorithm employs the characteristics of spatial correlations of nature image to compute local minimum which is different from the global minimum obtained by the baseline method such that for the images having less apparent spatial correlation,such as Baboon,the proposed algorithm might result in the large gaps between this local minimum and the global minimum.For this reason,the proposed algorithm uses the threshold T to avoid the large gaps between this local minimum and the global minimum obtained through the baseline method.In comparison to the baseline method,the proposed algorithm spends less encoding time and achieves higher compression ratio,while the quality of the retrieved image is almost the same.References[1]Barnsley MF,Demko S.Iterated function systems and the global construction of fractals.Proc Roy Soc1985;A399:243–75.[2]Barnsley MF.Fractal everywhere.New York:Academic;1988.[3]Barnsley MF,Elton JH,Hardin DP.Recurrent iterated function systems.In:Constructive approximation.1989.p.3–31.[4]Jacquin AE.Image coding based on a fractal theory of iterated contractive image transformations.IEEE Trans Image Process1992;1:18–30.[5]Fisher Y.Fractal image compression,theory and application.New York:Springer-Verlag;1994.[6]Wang Z,Zhang D,Yu Y.Hybrid image coding based on partial fractal mapping.Signal Process:Image Commun2000;15:767–79.[7]Tsai JC,Hsieh CH.Predictive vector quantization for image compression.Electron Lett1998;32:2325–6.[8]Hsieh CH,Tsai JC.Lossless compression of VQ index with search-order coding.IEEE Trans COM-38.1990.p.2166–73.。