S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett.,Vol. 43, 1992, pp. 29-34.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....but takes considerably more time [17] Because of the high degree in the computational complexity, it is interesting to look at approximations with a factor #: # ## # ## ## # ## # ### ## # # # ###,where# # is the optimal translation. Finding such a translation can be done in ### ### # time [48]. Variations on the bottleneck distance are the minimum weight distance, the most uniform distance, and the minimum deviation distance. 4.4 Hausdorff Distance In many applications, for example stereo matching, not all points from # need to have a corresponding point in #, due to occlusion and ....

S. Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29--34, 1992.

....of S; a translation of P ; a rotation of P ; congruent, or similar, to P ; approximately congruent to P ; a maximal cardinality subset of P that is congruent or similar to a subset of S; etc. Among the papers in which such variants of the Point Set Pattern Matching problem have been studied are [2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 16, 17, 19, 20]. For the exact matching (i.e. nding all congruent copies of) version of the PSPM Problem, the fastest sequential algorithms in the literature are the following. Department of Computer and Information Sciences, Niagara University, Niagara University, NY 14109, USA. e mail: boxer niagara.edu. ....

S. Schirra, \Approximate decision algorithms for approximate congruence," Information Processing Letters 43, pp. 29-34, 1992.

.... deciding whether there exists a translation such that F (A ; B) can also be solved, but takes considerably more time [13] Because of the high degree in the computational complexity, it is interesting to look 5 at approximations with a factor : F (A ; B) 1 )F (A ; T ) [29], where is the optimal translation. Variations on the bottleneck distance are the minimum weight distance, the most uniform distance, and the minimum deviation distance. 3.4 Hausdor Distance In many applications, for example stereo matching, not all points from A need to have a ....

Stefan Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29-34, 1992.

....by Akutsu, without incurring any increase in running time. Next, instead of approximating the constraint imposed by ffl, we propose algorithms which approximate the size of the largest common point set, and give upper and lower bounds on its size. Our algorithms are based on the work of Schirra [Sch92] They are however nontrivial generalizations of Schirra s algorithms which were for solving the ffl congruence decision problem for two equal cardinality point sets in 2 D, making use of the centroids of the point sets. Our approximation algorithms differ significantly these because they involve ....

....general structural properties of proteins. In the next section we formally state our abstract geometric problem and outline the algorithm due to Akutsu [Aku92] which approximates the ffl constraint. We show then how this algorithm, in combination with the decision algorithm due to Schirra [Sch92] leads to an algorithm for approximating the size of the LCP of two point sets. Following this, we state an exact algorithm for finding the LCP of two point sets when the underlying isometry is a pure rotation. In Section 4 we make use of this exact algorithm to improve the approximation ratio ....

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Stefan Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29--34, 1992.

....the size of the largest common point set, and give upper and lower bounds on its size. Our algorithms are based on non trivial generalizations of the notion of partial decision algorithms for solving the ffl congruence decision problem of two equal cardinality point sets in 2 D, due to Schirra [17]. We next suggest various modifications of the basic algorithms, resulting in an improvement of their run time. The first involves the use of an approximate graph matching due to Efrat and Itai [8] and the second is through the use of random sampling. The time complexity of our algorithms can be ....

....in this paper and only the result is mentioned. In the next section we formally state our abstract geometric problem and outline the algorithm due to Akutsu [3] which approximates the ffl constraint. We then show how this algorithm, in combination with the decision algorithm due to Schirra [17], leads to an algorithm for approximating the size of the LCP of two point sets. Following this, we state an exact algorithm for finding the LCP of two point sets when the underlying isometry is a pure rotation. In Section 4 we make use of this exact algorithm to improve the approximation ratio of ....

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S. Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29--34, 1992.

.... and d dimensional point sets, and the underlying metrics being L1 , L 1 and L 2 [11, 12, 18] In an e ort to improve the running time, various approximation algorithms for either the Hausdor or the bottleneck metric for point sets in two, three, and in general d dimensions have been presented in [9, 10, 15 17, 19, 20, 22]. Pattern matching using bottleneck metric It should be noted that most of the known exact algorithms, especially those involving three and higher dimensional point sets, are restricted to either the exact or the Hausdor metric. While the exact metric is ill posed for many practical ....

S. Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29-34, 1992.

....exists a translation such that d(Q ; B) can be done in O(n 5 log n) time. Because of the high degree in the complexity, it is interesting to look at approximations with a factor : d(Q ; B) 1 )d(Q ; T ) Finding such a translation can be done in O(n 2:5 ) time [Sch92] The optimization problem considers the computation of the minimum distance under a group of transformations. It nds the optimal transformation f such that d(f(A) B) is minimized. For rigid motions (translations plus rotations, sometimes called congruences) this can be found in time O(n ....

Stefan Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29-34, 1992.

....the ffi approximate (k; ffl) stereo matching problem in time O( ffl ffi ) k n 2 2k ) and space O( ffl ffi n 2 ) Again, for jffl 0 Gamma fflj ffi , our solution is without any loss in accuracy. The special case k = 1, called approximate point set congruence, was introduced in [17] and subsequently also studied in [8] O( ffl ffi ) 2 n 2:5 ) and O( ffl ffi ) 6 n 3 ) time algorithms, respectively, were presented. Our methods for the general (k; ffl) stereo matching problem are a non trivial generalization of [8] The algorithms in [17] and [8] are based on ....

....was introduced in [17] and subsequently also studied in [8] O( ffl ffi ) 2 n 2:5 ) and O( ffl ffi ) 6 n 3 ) time algorithms, respectively, were presented. Our methods for the general (k; ffl) stereo matching problem are a non trivial generalization of [8] The algorithms in [17] and [8] are based on using the centroids and lower left corners, respectively, of the two sets A and B. This is not possible for the general (k; ffl) stereo matching problem, as the partitioning into subsets A 1 ; A k , and B 1 ; B k is not given a priori (while for A B Figure ....

S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett., Vol. 43, 1992, pp. 29-34.

....= jM 0 j since d(TPP;QQ (P 0 ) Q 0 ) 8ffi. Since jM 1 j = jM j and m = n, the theorem holds. 2 Note that the constant factor 8 can be reduced to an arbitrary constant c 0 1 with only increasing the time complexity by a constant factor, using a technique described in Ref. [16]. Here, we briefly describe how to apply the technique. For each point q i in Q, we construct a set of points fq 1 i ; 1 1 1 ; q t i g so that a ball of radius 8ffi centered at q i is covered by a union of balls of radius fl = c 0 0 1) 8)ffi centered at q h i . We modify the procedure ....

S. Schirra, "Approximate decision algorithms for approximate congruence," Information Processing Letters, vol.43, pp.29--34, 1992.

....shapes are similar is an important problem in pattern recognition and computer vision. Various measures of shape similarity have been investigated, e.g. Fr echet distance for shapes given as polygonal curves [4, 5] approximate congruence for shapes given as equal cardinality sets of points [4, 6, 7, 10, 11, 15, 16, 20, 23, 24], and Hausdorff distance [2, 3, 4, 9, 12, 13, 14, 17, 18, 19, 22] In this paper we consider the Hausdorff distance between (a) two sets of points under translation, b) two sets of points under Euclidean motion, c) two sets of nonintersecting line segments under translation, and (d) two sets of ....

....is O( mn) 3 log 2 (mn) 12] These large runtimes, as well as the difficulty of actually implementing some of these algorithms, provide motivation for finding simple and efficient approximation algorithms for these problems. The approximation approach that we take here extends ideas from [15, 16, 23, 24] where approximation algorithms are developed for the problem of approximate congruence between point sets. See, for instance, 6] or one of the other references mentioned above for definitions and examples of approximate congruence. These algorithms are approximate decision procedures that ....

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S. Schirra. Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29--34, 1992. 10

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S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett.,Vol. 43, 1992, pp. 29-34.

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S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett.,Vol. 43, 1992, pp. 29-34.

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S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett.,Vol. 43, 1992, pp. 29-34.

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S. Schirra, "Approximate decision algorithms for approximate congruence," Inform. Process. Lett.,Vol. 43, 1992, pp. 29-34.

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S. Schirra, Approximate decision algorithms for approximate congruence. Information Processing Letters, 43:29--34, 1992.

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