A system approach to disparity estimation using dynamic programming is presented. The four step system can calculate a dense correspondence map between a stereo pair with parallel or
nonparallel camera geometry. Results are presented with CCIR 601 format stereo images

Cosmas, J

Panis, S

Ziegler, M

Electronics Letters

English

Brunel University Research Archive

Fig. 4 Lena.512 image Rate = O.23bit/pixel (i.e. cornpression ratio = 35); PSNR = 30.9dB Fig. 1 Illustration of codingprinciple idea is to expend code-bits at all instances where the marginal gain is greater than p distortion-unithit. For example, assume some part of the image can be encoded in two different ways. Either it costs R bits and the distortion is D ,  or it costs R + AR bits and the distortion is D - AD. Then, we will choose the more expensive alternative if and only if ADlAR 2 p distortion-unitdbit. Flg. 2 Example of three enlarged source-blocks rmd their optimifed fragmentation The fragmentation is described in high resolution only if the two frag- ments have very different statistics. Sites in a fragment do not need to be Connected. As was hinted at above, the fragmentation is encoded by a bin- tree. By this method, the fragmentation can be described in high resolution near sharp edges, and in low resolution at places where the exact shape of a fragment has a minor impact on the distor- tion (see Fig. 2). To find the optimal bintree, we start with a max- imum tree, and prune this tree by cutting off all branches where the marginal gain is less than p distortion-unitdbit. Fig. 3 Fragmentation of L.ena.512 image In each fragment, the six transform components are uniformly quantised and Huffman encoded. It can be shown [l] that the optimised quantisation step is given by Results: Fig. 3 illustrates a coder which is using both fragmented blocks and variable block size. The fragmentation, the variable ELECTRONICS LE7TERS 25th May 1995 Vol. 31 block size, and the quantisation of the transform components are all controlled by the optimality principle. The test image is Lena.512 (Fig. 4). The rate is 0.23bit/pixel, and the peak signal to noise ratio (PSNR) = 30.9dB. As a comparison, JPEG attains PSNR = 29.8dB at the same bit rate. 24 March 1995 0 IEE 1995 Electronics Lefters Online No: I9950583 R. Nohre (Division of Information Theory, Department of Electrical Engineering, Linkoping University, 581 83 Linkoping, Sweden) References NOHRE. R.: ‘Image coding by fragmented blocks’ Technical Report, 1995, Linkdping University, Sweden (LiTH-ISY-R-1734) KUNTM., BENARDM. and LEONARDIR.: ‘Recent results in high- compression image coding’, IEEE Trans., CAS-34, ( I l ) ,  pp. 1306 1336. FORCHHEIMER R. and KRONANDER T.: ‘Image coding - from wave forms to animation’, ZEEE Trans. Acoust. Speech Signal Process., 1989, pp. 2008-2023. image coding: performance comparison with the discrete cosine transform’, IEE Proc. F, 135, (2), pp. 121-132. NAISOPOULOS P., WARD R.K. and MORSE D.J.: ‘Adaptive compression coding’, IEEE Trans., COM-39, (8), pp. 1245-1254. DELP E.J. and METICHELL o.R.: ‘Image compression using block truncation coding’, IEEE Trans., COM-37, (9), pp. 1335-1342. wu H.-s., KING R.A. and KITNEY R.I.: ‘Improving the performance of the quadtree based image approximation via generalised DCT’, Electron. Letf., 29, (lo), pp. 887-888. BIGGAR M.J., MORRIS 0.1. and CONSTANTINIDES A.G.: ‘Segmented- System approach to disparity estimation S. Panis, M. Ziegler and J.P. Cosmas Indexing terms: Stereo imaze processing, Dynamtc prozramming A system approach to disparity estimation using dynamic programming is presented. The four step system can calculate a dense correspondence map between a stereo pair with parallel or nonparallel camera geometry. Results are presented with CCIR 601 format stereo images. Zntroduction: Binocular stereo is the process of obtaining depth information from a pair of left and right camera images [l]. Sur- veys on stereo matching techniques can he found in [2, 31. Many approaches are system oriented like the approach described here. The system consists of four parts (Fig. 1): No. 11 87 1 (i) global modelling of the luminance difference between the stereo images (ii) determination of useful disparity search range along each scan l i e  ( i )  disparity estimation based on 3-D dynamic programming (iv) disparity map postprocessing disparity estirnat io n (dynamic camera difference Progr-.ng) modelling and comemdion h li ‘i Fig. space Di’ determinati disparity map postprocessing Pm , 1 Disparity estimation system (i) Global modelling of luminance dfference between left and right images: In a stereo pair of images, there may be an imbalance in the luminance levels between the left and the right images, cawed by different camera configurations and characteristics. Therefore, statistical modelling and compensation of the luminance difference take place in this system as a preprocessing step, based on a method described in [4]. a 2-D optimisation space b 3-D optimisation space (ii) Determination of useful disparity search range along scan lines: The number of paths that have to be searched by dynamic pro- gramming is (d;d,)’ where n is the column axis, d, is the disparity axis and d, is the y-displacement axis of the optimisation space (Fig. 2b). As the range of disparity and y-displacement increase, the execution time grows and hence the danger of considering paths that may lead to wrong results increases. Therefore, a pre- processing stage calculates the maximum and minimum values of disparity and y-displacement, which are subsequently used to restrict the sue of the dynamic programming optimisation space. The calculation is performaed in three steps. First, a large-block- matching routine gives an estimate of the range of displacement vectors. Secondly, a block-based median filtering smoothens the disparity map and fmally, scanning each scan line, the minimum and maximum vectors are calculated. (iii) Disparity estimation based on 3-0 dynamic programming: In [l, 51, two dynamic programming based algorithms for disparity estimation are described. The algorithms work with ideal stereo images. Therefore, the dynamic programming optimisation space is twodimensional (Fig. 2a) because optimisation is performed only on disparity. In the case of nonparallel geometry or image misalignment, the algorithms fail because the y-displacement is nonzero. In this work, a 3-D optimisation space is used because y- displacement is also considered (Fig. 26). This way, calculation of epipolar geometry is not required and a parameter free system is developed that is able to function with imperfect stereo images. The total cost function for dynamic programming is a combina- tion of a ‘normalised matching cost’ and a ‘disparity jump cost’. 872 The normalised matching cost (NMC) is determined by the quality of matching whereas the disparity jump cost is a reNarisation factor which enforces disparity to be smooth along a iform sur- smooth, the area is classified as occluded. The disparity jump cost is defied, as in [l], by face (monotonicity constraint [I]). Where disparity % ils to be f(.) = p f i +  €1.1 2 2 0 (1) where p = 0.3 and E = 0.15; empirically determined total cost GJj, i) is calculated with the recursive functidn: G,(j,Z) = NMC,(j,i)+f(.)+G,(jb,,t,ibest) R. > 1 Gm(j,i) = NMC,( j , i )  n = l  where x is the disparity jump between the current candidate dis- parity and the disparity of the previous column, n is the current column number, j is the candidate disparity, i is the candidate y- displacement and (jkn iM) is the best predecessor vector. [l]. The ( 2 )  (iv) Disparity map postprocessing: Postprocessing is designed to smoothen the disparity map by correlating each disparity value to its neighbouring values and filling up gaps where disparity failed to be calculated. Postprocessing of disparity is divided into three parts: (a) median filtering with 3 x 3 window (neighbourhood correla- tion) (b) median filtering with 1 x 5 window (vertical dispariry correla- tion) (e) interpolation (gap fdlig) l?s3l Fig. 3 Original left even fzld offirst frame of ‘Aqua’ image sequence rn Fig. 4 Disparity map for stereo pair ‘Aqua’ P Fig. 5 Disparity map after postprocessing for stereo pair ‘Aqua Results: The stereo image pair ‘Aqua’ [6] (Fig. 3), has disparity range in the interval +1 to +34pixels, calculated from the left to the right images. The image size is 720 x 288 and the calculation time was 8min on a Sparc 10 workstation. From the equalised dis- parity map shown in Fig. 4, the depth of each object compared to the other objects can be seen. The objects with a lighter grey value ELECTRONICS LE7TERS 25th May 1995 Vol. 31 No. 11 _ _ ~  __ ~~ (larger disparity) are closer to the camera whereas those with a darker grey value (smaller disparity) are further away from the camera. Black areas have no disparity value because they were classified as occlusion regions. By visual comparison between the original picture and its disparity map, we can recognise the shapes of the objects. Good examples are the fish, plants and the rock. Fig. 5 shows the postprocessed disparity map. It is smoother and the object definition is clearer. The disparity map of Fig. 5 has been used to reconstruct the left image from the right image. The reconstructed image is visually good and has a PSNR = 29dB, which shows that the disparity fidelity is quite high. Block-based disparity gave a PSNR = 23dB. Acknowledgment: This work was camed out within the framework of RACE (Research on Advanced Communications in Europe) under contract R2045 'DISTIMA (DIgital STereoscopic IMaging and Applications). 0 IEE 1995 Electronics Letters Online No: 19950617 S. Panis and J.P. Cosmas (Electronic Engineering Dept.. Queen Mary and Westfiild College, Mile End Rood, London E l  4NS. United Kingdom) M. Ziegler (Siemens AG. ZFE T SN 22, 81730 Munich, Germany) S. Panis: also with Siemens AG, Munich, Germany 20 March 1995 References 1 GEIGER, D., LANDENWRF, D., and YZIILLE, A.: 'Occlusions and binocular stereo', Int. J. Comput. Vis., 1993, 2 DARNARD, s.T., and FISCHLER, M.A.: 'Computational stereo', Computing Surveys, 1982, 14, (4), pp. 553-572 3 DHOND, u.R., and AGGARWAL, J.K.: 'Structure from stereo-A review', IEEE Trans. Syst. Man Cybern., 1989, 19, (6), pp. 1489-1510 4 'Balance compensation for stereoscopic sequences' RACE DISTIMA Doc. 451TUD/WP3.2lDNIC/27.792/1 5 cox, 1.1, HMGORANI, s., MAGGS, D.M., RAO, s.B.: 'Stereo without regularization'. NEC Research Institute Report, 21 October 1992 6 Stereoscopic test sequences (CCIR 601 format) shot within RACE DISTIMA Searching for the best linear approximation of DES-like cryptosystems L. Buttyan and I. Vajda Indexing terms: Cryptography, Graph theory, Viferbi decoding The authors show that the problem of searching for the best characteristic in linear cryptanalysis is equivalent to searching for the maximal weight path in a directed graph. Under certain restrictions, the best characteristic can be easily obtained even on a personal computer. Introduction; The design of information systems should take into account potential security problems. Cryptography provides a strong information processing technique for algorithmic protection of information. Differential cryptanalysis [I] and linear crypt- analysis [2] are the most effective methods to date for attacking DES-like cryptosystems. Differential and linear cryptanalyses are very similar on a structural level as shown in [3, 41. Both methods rely on the existence of an rround characteristic having high prob- ability, where r is the numbez of rounds of the cryptosystem. Therefore, searching for the best characteristic is a very important part of these attacks. However no practical algorithm has yet been published which finds the best characteristic for the full number of rounds without any restrictions. We introduce an abstract model for the searching problem based on searching for the maximal weight path in a directed graph. In this Letter, we focus on linear cryptanalysis, but our results can be adopted for differential crypt- analysis, owing to their duality [4]. ELECTRONICS LETTERS 25th May 1995 Vol. 31 The aim of linear cryptanalysis is to find out the key (or a part of the key) of the cryptosystem with the help of an effective linear approximation $2. P + 0 .  c + 0 , .  K, = 0 (1) ,=l of the cipher and a large number of plaintext-ciphertext pairs. Here, P and C denote the plaintext and the ciphertext blocks, respectively, K, stands for the subkey of the ith round, Q, 0 and Qz are the mask vectors. Furthermore, '.' and '+' stand for the binary scalar product and XOR operation, respectively. The effec- tiveness of eqn. 1 depends on how well it agrees with the probabil- ity Q'. A larger value of IQ' -U21 provides a more effective linear approximation of the cipher. nR) rn Fig. 1 Structure of 1 round and the masking vectors Characteristics; We define an r round characteristic to describe a linear approximation of an r round cipher. Our definition is not exactly the same as that of Biham [3], but it is better suited to our purposes. Definition I :  A 1 round characteristic is a tuple x = (Q, 0, Q, 4'). where Q = (SF, QR), 8 = (O', OR) and QL = OR. q' is the probahil- ity that the linear approximation ( O R  + O L )  .X + 0' .Y + 8 .  K = 0 (2) of round function F holds, where Y = f l X ,  K )  (see Fig. 1). Definition 2: An r round characteristic ( r  > 1) is a series of r 1 round characteristics xI, x2 ,.._, x, ,  where x ,  = (Q, 0, Q,, q'z = 112+qj) such that Q, = Or+, for i = 2, 3,  ..., r .  Thus an r round char- acteristic determines r linear approximations (np + 0:) .xi + $2:. y,  + @* . K ,  = 0 (3) with probability = 1/2 + qt, 0 < i < r + 1. In DES-like cryptosystems, (4) Y, = x,-1 + x,+1 0 < 2 < T + 1 where = f l X j ,  Kj),  and (.Yo, XI) = (P", PR) = P and (X,, X,,,) =(CL, CR) = C are the plaintext and ciphertext blocks, respectively. We omit the swapping of the two halves of C, because it does not have any cryptographic significance. Since Q: = OF, i = 1, 2, ..., r by definition 1 and Qj = eo, i = 2, 3,  ..., r by definition 2, R,lL + Qj+lr + (Q,R + 0:) = 0 for i = 2, 3, ..., r-1. Summing up the r expressions of eqn. 3 we get ni . P +  0,. c+ E@*. K ,  = 0 ( 5 )  i=l The probability of eqn. 5 is Q' = 1/2 + 2"q,q2 ... q, according to the piling-up lemma in [2]. So the problem is to find an r round characteristic for which nlqil is maximal. Graph representation and searching algorithm: We can represent all the 1 round characteristics with a two-stage directed graph. The nodes of the first stage and the second stage of the graph represent the possible values of Q and 0, respectively. An edge leads from a node of the first stage, say from w, to a node of the second stage, say to 8, if there exists a 1 round characteristic x = (w, 8, 1/2+q), for which q z 0. The weight of this edge is loglql. No. 11 873 

Computational stereo', Computing Surveys,

Vol. 31 The aim of linear cryptanalysis is to find out the key (or a part of the key) of the cryptosystem with the help of an effective linear approximation $2. P + 0. c + 0,.

System approach to disparity estimation

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