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Block-based graph-cut rate allocation for subband image compression and transmission over wireless networks

Block-based graph-cut rate allocation for subband image compression and transmission over wireless networks
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  Block-based graph-cut rate allocation for subband imagecompression and transmission over wireless networks Maria Trocan Institut Supérieurd’Electronique de Paris21 rue d’Assas, 75006 Paris maria.trocan@isep.frBeatricePesquet-Popescu Telecom ParisTech37-39 rue Dareau, 75013Paris beatrice.pesquet@telecom-paristech.frJames E. Fowler Mississippi State UniversityU.S.A. fowler@ece.msstate.eduCharles Yaacoub USEKLiban ABSTRACT The compression of natural images and their transmissionover multi-hop wireless networks still presents many chal-lenges for the researchers and industry. In this paper wepresent a new block-based rate-distortion optimization al-gorithm that can encode efficiently the coefficients of a criti-cally sampled, non-orthogonal or even redundant transform.The basic idea is to construct a specialized graph such thatits minimum cut minimizes the energy functional. We pro-pose to apply this technique for rate-distortion Lagrangianoptimization in block-based subband image coding. Themethod yields good compression results compared to thestate-of-art JPEG2000 codec, as well as a general improve-ment in visual quality. 1. INTRODUCTION Nowadays, the majority of image-compression algorithmsuse wavelet transforms, attempting to exploit all the sig-nal redundancy that can appear within and across the dif-ferent subbands of a spatial decomposition. The wavelettransform has been succesfully used for image representation[1], due to its energy compaction capacities and compres-sion efficiency [2]. However, efficiency of a coding schemehighly depends also on bit allocation. In order to maximizethe compression efficiency, high-complexity subband-basedimage-compression schemes, as the state-of-the-art compres-sion standard, JPEG2000 [1], may be used in wireless net-works.In this paper we present a rate-distortion optimization basedon graph cuts, which can compress efficiently the coefficientsof a critically sampled or even redundant, non-orthogonal Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, orrepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee. Mobimedia’09, September 7-9, 2009, London,UK. Copyright 2009 ICST 978-963-9799-62-2/00/0004 ... $5.00 transform. As described in [3, 4, 5], problems that arise incomputer vision can be naturally expressed in terms of en-ergy minimization. Each of these methods consists in mod-elling a graph for an energy type, such that the minimumcut minimizes globally or locally that functional. Usually,these graph constructions are dense and complex, design-ing the energy function at pixel level. For example, in [6]the graph cut provides a clean, flexible formulation for im-age segmentation. With a grid design, the graph provides aconvenient manner to encode simple local segmentation de-cisions and presents a set of powerful computational mecha-nisms to extract global segmentation from these simple local(pairwise) pixel similarities. Good energy-optimization re-sults based on graph cuts were obtained in image restoration[7], as well as in motion segmentation [8], texture synthesisin image and video [9], etc. As it will be shown by the exper-imental results, the method gives good compression resultscompared to the state-of-the-art JPEG2000 codec. The pa-per is organized as follows: Section 2 describes the solutionfor rate-distortion optimization using graph-cuts, by model-ing distortion energy interactions at block level. Some ex-perimental results obtained with the proposed methods forboth wavelet-based and edge-oriented contourlet-based im-age coding are presented in Section 3. Finally, conclusionsand future work directions are drawn in Section 4. 2. IMAGECOMPRESSIONUSINGGRAPHCUTS As mentioned in the introduction, we propose to use thegraph-cut mechanism for the minimization of the rate- dis-tortion Lagrangian function and thus find the optimal set of quantizers satisfying the imposed constraints. To this aim,we have designed a specialized graph able to represent asubband decomposition taking into consideration the corre-lations between subbands in a multiresolution approach.In the following, we express the Lagrangian functional asa discrete sum accumulating the contribution of each cod-ing unit (subband or block) in terms of rate and distortioninduced by the quantization. Moreover, the graph model isplanar, and the energy function we intend to optimize is con-  vex, so the minimum graph cut can be found in polynomialtime. 2.1 Graph design Consider the weighted graph G = ( V,E,W  ), with V   -vertices, E  -edges and positive edge weights  W  , which have not onlytwo, but a set of terminal nodes,  Q  ∈  V   . Recall that asubset of edges  E  C   ∈  E   is called a  multiway cut   if the ter-minal nodes are completely separated in the induced graph G ( E  C  ) = ( V,E   − E  C  ,W  ) and no proper subset of   E  C   sep-arates the terminals in  E  C  . If   C   is the cost of the multi-way cut, then the multi-terminal min-cut problem is equiv-alent to finding the minimum-cost multiway cut. For ouroptimization problem, the terminals are given by a set of quantizers  Q , and the coding units give the rest of the ver-tices  V    − Q . The edges and their weights/capacities will bedefined in the following depending on the coding strategy(subband or block coding) and the distortion functional.In [7], Y. Boykov  et al.  find the minimal multiway cut bysuccesively finding the min-cut between a given terminal andthe other terminals. This approximation guarantees a lo-cal minimization of the energy function that is close to theoptimal solution for both concave and convex energy func-tionals. As the rate-distortion Lagrangian lies on a convexcurve (i.e.  D ( R )), we propose to use the method in [7] forits optimization. 2.2 Lagrangian rate-distortion functional Consider the problem of coding an image at a maximal rate R max  with a minimal distortion D . Each image consists of afixed number of coding units (spatial subbands or blocks of coefficients), each of them coded with a different quantizer q  i ,  q  i  ∈  Q  ( Q  being the quantizers set). Let  D i ( q  i ) be thedistortion of the coding unit  i  when quantized with  q  i , andlet  R i ( q  i ) be the number of bits required for its encoding.The problem can now be formulated as: find min  i D i ( q  i ) , such that  i R i ( q  i ) = R ≤ R max .In the Lagrange-multiplier framework, this constrained op-timization is written as the equivalent problem:min  i ( D i ( q  i ) + λR i ( q  i )) , R ≤ R max  (1)where the choice of the Lagrangian parameter  λ >  0 mea-sures the relative importance between distortion and ratefor the optimization and which can be determined using abinary search. The advantage of problem formulation inEq. (1) is that the sum and the minimum operator can beexchanged to:  i min( D i ( q  i ) + λR i ( q  i )) , R ≤ R max  (2)This formulation obviously reveals that the global optimiza-tion can now be carried out independently for each codingunit, making an efficient implementation feasible. 2.2.1 Rate estimation For the rate estimation of the quantized coding units weconsider a non-contextual arithmetic coder [10], which usesa zero-order entropy model, where the  M   quantized coef-ficients of a given coding unit are random i.i.d. variablesfollowing a Gaussian distribution. Thus, the zero-order en-tropy ( H  ) estimation in bits/variable (i.e., coefficient) is ob-tained as: H   =  − M   i =1  p i  log 2  p i ,  (3)where  p i  is the probability of the  i th coefficient. The result-ing entropy estimate per coding unit is weighted by the sizeof the coding unit in order to obtain the total entropy of thequantized image. 2.2.2 Distortion estimation The distortion D between the srcinal image x and the quan-tized one,   x , is estimated in the following as the  L 2 norm,i.e. : D  =   x −   x  2 .  (4)This model will be futher developed, in order to obtain agood distortion estimate in the spatial domain, rather thanin the transform domain, as is usually done for orthonormaltransforms. 2.3 Graph design with cross-correlation dis-tortion at the block level Recall that we have written the distortion  D  between thesrcinal image,  x , and the quantized one,   x , as the L 2 norm,i.e.  D  =   x −   x  2 . In a first approximation [11], we haveconsidered only the diagonal terms, i.e.: D I   ∼ =  i  x i  −   x i  2 (5)which amounts to estimating the distortion between the con-tribution to the image and to the quantized image of onlythe  i th subband.In a second approximation, we have also considered the cross-correlation   terms, i.e.: D  ∼ = D I   +  i  i ′ ∈N  ( i )    x i  − x i ,   x i ′  − x i ′   (6)where  N  ( i ) is a neighborhood of   i , containing closely cor-related subbands. Indeed, given the limited support of thewavelets, the closer in scale and frequency are the subbands,the higher the correlation among them. In practice, thisneighborhood could be described by the geometrical positionof the subbands in a multiresolution decomposition (whereonly the vertical and horizontal directions are considered),or by simply linking the subbands in a chain-manner, oneafter another (for example, in Fig. 1, the neighborhood rela-tions are indicated by the black edges in the graph). Thus,Eq. (6) can be written as: D  =  i  x i  −   x i  2      D i +  i  i ′ ∈N  ( i )    x i  − x i ,   x i ′  − x i ′       D i,i ′ (7)We have shown in [12] that in this case, the function to be  Figure 1: Contourlet decomposition with three levels (left) and three-way graph-cut repartition (right) ( q  1 partition in red,  q  2  partition in green,  q  3  partition in blue, where the regular edges are with full black lines,terminal links in colors and the cut-edges in gray lines). minimized is:min  i   x i  −   x i  2 + λR ( i )      E data ( i ) +  i ′ ∈N  ( i )    x i  − x i ,   x i ′  − x i ′       E smooth ( i )  (8)In the following, we propose to extend the subband leveldistortion estimation presented in [12] to the block level(Fig. 2). This extension comes naturally, as the smallerthe coding unit, the more correlated in amplitude are thecoefficients within it. At block level, Eq. (8) becomes:min X  i =1 N  b  j =1  x i,j  −   x i,j  2 + λR ( i,j )      E data ( i,j ) +  ( i ′ ,j ′ ) ∈N  ( i,j ) |   x i,j  − x i,j ,   x i ′ ,j ′  − x i ′ ,j ′ |      E s mooth ( i,j ) (9)where X  , respectively N  b  represent the number of subbands,respectively blocks in each subband,  x i,j  denotes the imagereconstucted only from the  j th block of the  i th subband and    x i,j  − x i,j ,   x i ′ ,j  − x i ′ ,j ′   measures the correlation betweenthe neighbour blocks.The minimization of the energy function defined above isequivalent to the best partition of quantizers per subbandsblocks. Note that for  E  smooth  we have used the sum of absolute values of cross-correlation terms, in order to en-sure that our regular vertices will have associated positiveweights. Our graph will have therefore  B  =  X   ×  N  b  −  1regular vertices. The neighbourhood system,  N  , contains Figure 2: Block graph design: two-level wavelet de-composition with four-blocks subband division andchain network design for the regular vertices. now only position correlation links  E  N  (i.e., edges betweenneighbour blocks, as described in Fig.2). The geometricalmodel can be described as:  G  = ( V,E  ) where  V    =  B  ∪ Q , E   =  E  N   ∪ E  Q  and  Q / E  Q  represent the quantizers set/thelinks between block nodes and quantizers. For the terminallinks,  E  Q , the weights are given by the direct costs in termsof distortion and rate induced by the quantization (i.e., theedge between block  b  and quantizer  q  , ( b,q  ), has the associ-ated weight  w b,q  =  D b ( q  ) + R b ( q  )). The capacity betweentwo regular neighbour blocks (( b i ,b i ′ )  ∈  E  N  ) is defined asthe absolute value of the cross-correlation distortion inducedby the current quantization of these blocks. 3. APPLICATION TO SUBBAND IMAGECOMPRESSION In the following, we propose to apply the proposed graph-cutminimization model to subband image compression. Someresults are drawn in the framework of classical separable  wavelet image coding, as well as for a geometrical transform,namely the contourlet decomposition [13]. Note that themethod can be applied to almost any existing decomposition(wavelets, Xlets, subbands, blocks, may them be criticallysampled / redundant etc.). 3.1 Wavelet image compression with graph-cuts Due to their energy compaction efficiency, the biorthogo-nal filter banks are the most used in image compression [1].This is the reason for which we consider in our simulationframework both the 5/3 and 9/7 filter banks for the spatialdecomposition. 3.1.1 Experimental results For our simulations, we have considered two representa-tive test images: Barbara (512x512 pixels) and Mandrill(512x512 pixels), which have been selected for the difficultyto encode their texture characteristics.We have used dead-zone scalar quantization, with  q   ∈ { 2 0 ,..., 2 10 } . The dead-zone has twice the width of the otherquantization intervals. All the images have been decom-posed over five spatial levels with the floating-point 5/3 and9/7 filter banks. Note that for rate estimation in the alloca-tion algorithm we have used a simple (non-contextual) arith-metic coder [10], while JPEG2000 codec [1] uses a highly op-timized contextual coder. The JPEG2000 results have beenobtained with the Kakadu framework. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.42020.52121.52222.52323.52424.525 Mandrill (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   ) GCC−Simple distortionGCC−Cross−correlated distortionGCC−Block cross−correlated distortion9/7 JPEG2000 Figure 3: Rate-distortion comparison for Mandrillimage with 9/7 wavelet subband decomposition As it can be remarked from Fig. 3 and Fig. 4, the resultsobtained with the 9/7 wavelet subband decompostion of JPEG2000 are between 0.5 and 1.5 dB higher than thoseobtained with the proposed graph-cut rate-distortion algo-rithm. This situation can be explained by the fact that the9/7 filter bank is very close, from an energy partition pointof view, to an orthonormal decomposition. As illustratedin Fig. 5 and Fig. 6, our method seems to better cope withnon-orthogonal decompositions at very low bitrates ( ≤  0 . 1bpp). Barbara (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   ) GCC−Simple distortionGCC−Cross−correlated distortionGCC−Block cross−correlated distortion9/7 JPEG2000 Figure 4: Rate-distortion comparison for Barbaraimage with 9/7 wavelet subband decomposition Mandrill (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   )   GCC−First order subband distortionGCC−Cross−correlated subband distortionGCC−Cross−correlated block distortion5/3 JPEG2000 Figure 5: Rate-distortion comparison for Mandrillimage with 5/3 wavelet subband decomposition One can remark that distortion approximation at subbandlevel taking into account the cross-correlation among sub-bands always leads to better results than the simple modelwithout cross-correlation terms, by using a more realisticcorrelation model. Moreover, the finer level of represen-tion for the coding units, the higher the correlation amongthese units, as it can be remarked from the presented re-sults, having an average gain of 0.25 dB over the preceedingrate-distortion curve obtained with a subband-level cross-correlated distortion model. Barbara (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   )   GCC−First order subband distortionGCC−Cross−correlated subband distortionGCC−Cross−correlated block distortion5/3 JPEG2000 Figure 6: Rate-distortion comparison for Barbaraimage with 5/3 wavelet subband decomposition 3.2 Contourletimagecompressionwithgraph-cuts The drawback of separable wavelets is the limited orienta-tion selectivity, as they fail to capture the geometry of theimage edges. In order to overcome the problem of edge rep-resentation, Minh N. Do and Martin Vetterli have defined anew family of geometrical wavelets, called contourlets [13].With contourlets, one can represent the class of smooth im-ages with discontinuities along smooth curves in a very effi-cient and sparse way. These decompositions have been suc-cessfully applied in image segmentation and noise removal,as well as in image compression: as shown in [14], the codecbased on wedgelets gives better performance in image com-pression than the JPEG2000 standard at very low rate. 3.2.1 Experimental results For a better comparison, we have considered the same testimages: Barbara (512x512 pixels) and Mandrill (512x512pixels). We have used dead-zone scalar quantization, with q   ∈ { 2 1 ,..., 2 10 }  and a 5-level contourlet decomposition,where the coarsest three decomposition levels consist of a9/7 separable wavelet transform (i.e., 3 directions), and thefinest two levels are represented with a 16- and 32-bandbiorthogonal directional filter. The efficiency of this hybridscheme has been proved in [15].As shown in Figs. 7 and 8, our method surpasses JPEG2000at low bitrates, even though it employs a redundant trans-form. Note that for the rate estimation in the allocation al-gorithm we have used a simple (non-contextual) arithmeticcoder [10], while JPEG2000 codec uses highly optimized con-textual coder. 4. CONCLUSION In this paper we have presented a block-based graph-cutmethod for rate-distortion optimization in image coding.Its great advantage is that it can be applied to decompo-sitions which are not necessarily orthonormal. As shown byexperimental results, it can efficiently encode both wavelet Mandrill (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   ) GCC−First order subband distortion approx.GCC−Cross−correlated subband distortion approx.GCC−Cross−correlated block distortion approx.JPEG2000 Figure 7: Rate-distortion comparison for Mandrillimage with contourlet subband decomposition Barbara (512x512) Bitrate (bpp)    P   S   N   R    (   d   B   ) GCC−First order subband distortionGCC−Cross−correlated subband distortionGCC−Cross−correlated block distortionWavelets J2K Figure 8: Rate-distortion comparison for Barbaraimage with contourlet subband decomposition and contourlet coefficients compared to standard RD cod-ing tools, enhancing thus the wireless transmission efficiency.Moreover, the proposed method could be further used withvector quantizers. 5. REFERENCES [1] “Information technology – JPEG 2000 image codingsystem,”Tech. Rep., ISO/IEC 15444-1, 2000.[2] S. G. Mallat,“A theory for multiresolution signaldecomposition: The wavelet representation,”  IEEE Transactions on Pattern Analysis and Machine Intelligence  , vol. 11, pp. 674–693, 1989.[3] V. Kolmogorov and R Zabin,“What energy functionscan be minimized via graph cuts,”  IEEE Transactions on Pattern Analysis and Machine Intelligence  , vol. 26,pp. 147 – 159, 2004.[4] Y. Boykov and V. Kolmogorov,“An experimental
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