H. Alt, K. Mehlhorn, H. Wagner, E. Welzl. Congruence, Similarity, And Symmetries Of Geometric Objects. Discrete and Computational Geometry, vol. 3, pages 237--256, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and good performance in many practical situations. Cass, 1990] has proposed an algorithm which is essentially a sweep of the arrangement (subdivision into cells, see [Edelsbrunner, 1987] in transformation space generated by the constraints that arise from individual correspondences. [Alt et al. 1988], have proposed a similar algorithm. Such sweep based algorithms, while polynomial time in principle, have disappointing performance and cannot incorporate many of the known heuristics used for speeding up search based methods. This paper describes the pruned correspondence search (PCS) ....

Alt H., Mehlhorn K., Wagener H., Welzl E., 1988, Congruence, Similarity, and Symmetries of Geometric Objects., Discrete and Computational Geometry.

....our basic algorithm without much effort. Such restrictions of transformation space can result in major speedups. Geometric pattern matching, can be seen as the process of finding a mapping between two patterns in such a way that some similarity measure is optimised. Exact pattern matching methods [3, 20, 12] decide whether or not transformations exist which map a pattern precisely onto another pattern. Unfortunately, in most practical applications measurement errors and limited numerical precision render exact matching algorithms ineffective. Flexibility in case of perturbations and roundoff errors ....

....minimum deviation matching and bottleneck matching [2, 1] A major disadvantage of these methods is that, in principle, only point sets having equal sizes can be matched. Much work has been done on matching point set patterns without establishing correspondences by means of the Hausdorff distance [12, 11, 16, 9, 8, 22, 19]. Although robust against perturbations, the Hausdorff distance is very sensitive to occlusion and absent or outlying parts. The partial Hausdorff distance [7, 21] is a pattern similarity measure which overcomes this drawback by matching only parts of patterns. As a result, the partial Hausdorff ....

Helmut Alt, Kurt Mehlhorn, Hubert Wagener, and Emo Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237--256, 1988. 33

....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 ....

....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 determine whether two shapes are congruent for some 0. The answer to such a query can be yes or no, or, if is very close to the ....

H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237--256, 1988.

.... common point set problem, or, LCP [ATT97, AH] An easier problem is that of determining whether two equal cardinality point sets are congruent, i.e. does there exist a rigid transformation of one set such that the other point set is superimposed on to the transformed point set [Atk87, Aku92, AMWW88] A generalization of this, called the congruent copy detection, asks, given two point sets A and B, whether there exists a rigid transformation under which A becomes congruent to a subset of B [CGH 93, GMO94, dRL95] The LCP as studied in computational geometry [ATT97] does not, however, ....

....of our problem is that of finding the LCP of two 3 D point sets with the exact congruence replaced by ffl congruence. 1.1 Previous Work Here we shall briefly review related work. Given two 2 D point sets, an O(n 8 ) decision algorithm for ffl congruence under general isometry was given in [AMWW88] in contrast to the O(n log n) algorithms [AMWW88, Ata84] for testing exact congruence) Alt et al. AMWW88] and Iwanowski [Iwa90] gave O(n log n) algorithms for some restricted decision problems and Imai et al. ISI89] gave O(n 3 log n) algorithms for Euclidean metric and O(n) algorithm for ....

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237--256, 1988.

....other set, and the bottleneck matching metric [14] seeks a perfect bipartite matching between two equal cardinality point sets such that the maximum distance between any two matched points is minimized, and it returns this distance. A systematic study of these problems was initiated by Alt et al. [5]. They presented algorithms for several versions of the problem for planar point sets under the bottleneck matching metric. In particular, they proposed an O(n 8 ) decision algorithm to determine if there exists an isometric transformation using which two equal cardinality planar point sets can ....

....of the sets, the existence of a bottleneck matching is a global property of the point sets and therefore complicates the problem. As a result, it is not apparent how the algorithms concerned with the Hausdor metric can be adapted for computing the bottleneck matching. Neither do the algorithms of [5] extend from the planar case to work in three or higher dimensions in any simple way. Very recently, a new paradigm for point set pattern matching based on algebraic convolutions was proposed in [9] and [19] This reduced the complexity of the problem under Hausdor metric to nearly quadratic ....

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237{ 256, 1988.

....2 [fit b2 b4] goal 3 [fit b3 b4] KA3 starts its first stage: finding possible mating feature pairs. The process starts from the body that has the smallest number of surfaces. From bodies b1; b2 and b3, the same type of compound features multi Gamma insertor are found: b1 : allinsertors [[1 3] [1 4] 3] 4] allmultiinsertors [ 1 3 4] 1 4 3] comp1 : 1 3 4] b2 : allinsertors [ 1 3] 1 4] 3] 4] allmultiinsertors [ 1 3 4] 1 4 3] comp1 : 1 3 4] b3 : allinsertors [ 1 2] 1 3] 2] 3] allmultiinsertors [ 1 3 2] comp1 : 1 3 2] The compound features that ....

....b2 b4] goal 3 [fit b3 b4] KA3 starts its first stage: finding possible mating feature pairs. The process starts from the body that has the smallest number of surfaces. From bodies b1; b2 and b3, the same type of compound features multi Gamma insertor are found: b1 : allinsertors [ 1 3] [1 4] [3] 4] allmultiinsertors [ 1 3 4] 1 4 3] comp1 : 1 3 4] b2 : allinsertors [ 1 3] 1 4] 3] 4] allmultiinsertors [ 1 3 4] 1 4 3] comp1 : 1 3 4] b3 : allinsertors [ 1 2] 1 3] 2] 3] allmultiinsertors [ 1 3 2] comp1 : 1 3 2] The compound features that have ....

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3(3):237--256, 1988.

....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 6 log n) AMWW88] For translations only, it can be computed in time O(n 5 log 2 n) EI96] An approximation translation within factor two, d(A ; B) 2d(A ; B) can be obtained by translating A such that the lower left corner of the axis parallel bounding box (called reference point) coincides with ....

Helmut Alt, Kurt Mehlhorn, Hubert Wagener, and Emo Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237-256, 1988.

.... of minimizing the resemblance under some rigid motion Bottleneck Matching 17 08 1997 10:24 Introduction 3 or other transformation of one image relative to the other, has been investigated mainly from a practical point of view, and the best known algorithms are either computationally inefficient [7], or significantly restrict the inputs [8] For the case where the sets A and B are points in the plane, Vaidya [50] explored the geometric structure of the problem to obtain an algorithm for finding a matching between A and B, for which the sum of distances between the matched points is minimal ....

....show in Section 6 an application of our technique for the translation problem: Let A and B be two n point sets in the plane, and ae a fixed number. The problem is to find a translation B 0 of B such that Match(A; B 0 ) is at most ae, or determine that no such translation exists. Alt et al. [7] gave an O(n 6 ) time algorithm for this problem. We improve this bound to O(n 5 log n) and show how to 1 Throughout the paper, stands for a positive constant which can be chosen arbitrarily small with an appropriate choice of other constants of the algorithms. Bottleneck Matching ....

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H. Alt, K. Mehlhorn, H. Wagener and E. Welzl, Congruence, similarity and symmetries of geometric objects, Discrete and Computational Geometry 3 (1988), 237--256.

....and translation so that each point lies within exactly one of the noise regions R. Related Work. The exact point matching problem has been solved in time O(n d Gamma2 log n) for d = d 0 and T the set of congruences (translations and rotations, and possibly a reflection) see Alt et al. [1] , earlier work by Atallah [3] and Atkinson [4] and the recent work of Sprinzak and Werman [19] Baird [6] did some of the pioneering work on the inexact point matching problem but left open the question of obtaining polynomial time algorithms. Alt et. al [1] applied techniques of ....

....see Alt et al. 1] earlier work by Atallah [3] and Atkinson [4] and the recent work of Sprinzak and Werman [19] Baird [6] did some of the pioneering work on the inexact point matching problem but left open the question of obtaining polynomial time algorithms. Alt et. al [1] applied techniques of computational geometry to give polynomial time algorithms for a wide variety of inexact point matching problems. Several other papers have also obtained efficient algorithms for instances of the inexact point matching problem [2; 9; 10; 14; 17; 18] Closely related to ....

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H. Alt, K. Mehlhorn, H. Wagener and E. Welzl, 1988. Congruence, Similarity, and Symmetries of Geometric Objects, Discrete and Computational Geometry 3, 237--256.

....paper shows that if d is not bounded, the problem is at least as hard as the graph isomorphism problem in the sense of the polynomiality. Several related results are described too. 1 Introduction Recently, geometric pattern matching problems have been studied extensively in computational geometry [3, 4, 11]. Most of such studies have been done for approximate matchings in two or three dimensions. Few studies for exact matchings in higher dimensions have been done. This paper studies the problem of determining the exact congruity in higher dimensions. Several studies have been done for exact ....

.... the congruity of two polyhedra in three dimensions [14] Atkinson developed an O(n log n) time algorithm for determining the congruity of two point sets in three dimensions [6] Alt et al. developed an O(n d02 log n) time algorithm for determining the congruity of two point sets in d dimensions [4]. However, to my knowledge, no improvement on their result has been done. In this paper, we present an O(n d01 2 (log n) 2 ) time randomized algorithm for determining the congruity of two point sets A and B (jAj = jBj = n) in d dimensional Euclidean space (d 3) This improves the previous ....

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H. Alt, K. Melhorn, H. Wagener and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, Vol. 3, pp. 237-256, 1988.

....nature. Furthermore, the problem of minimizing the resemblance under some rigid motion or other transformation of one image relative to the other, has been investigated mainly from a practical point of view, and the best known algorithms are either computationally inefficient (see Alt et al. [5]) or significantly restrict the inputs (see Arkin et al. 6] Bottleneck Matching March 16, Introduction 3 Dim A B Norm Oracle Thm. Match Thm. R 2 points points L p 8p O(n 1:5 log n) 5.4 O(n 1:5 log n) 5.10 additive O(n 1:5 ) 6.7 O(n 1:5 ) 6.8 weights segments L p 8p O(n 1:5 ....

....show in Section 8 an application of our technique for the translation problem: Let A and B be two n point sets in the plane, and ae a fixed number. The problem is to find a translation B 0 of B such that Match(A; B 0 ) is at most ae, or determine that no such translation exists. Alt et al. [5] gave an O(n 6 ) time algorithm for this problem. We improve this bound to O(n 5 log n) and show how to find in O(n 5 log 2 n) time a translation B of B that minimizes Match(A; B 0 ) over all translations B 0 . We also present a scheme to find a translation that approximates ....

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H. Alt, K. Mehlhorn, H. Wagener and E. Welzl, Congruence, similarity and symmetries of geometric objects, Discrete and Computational Geometry 3 (1988), 237--256.

....For Exact Translations on the real line, Rote gave an O(n log n) time algorithm. For Exact Rigid Motions, with k = N , Atallah [3] provided an O(N log N ) time algorithm in dimension 2 for planar figures. Atkinson [4] obtained the same runtime in dimension 3. Alt, Mehlhorn, Wagener and Welzl [2] obtained runtime O(N d Gamma2 log N ) in dimensions d 3. The undirected Hausdorff distance between two sets is the maximum of each of the directed distances. Computation of the least such distance obtainable under a given group of transformations, is quite a different problem from ....

H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237256, 1988.

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H. Alt, K. Mehlhorn, H. Wagner, and E. Welzl. Congruence, similarity and symmetries of geometric objects. Discrete and Computational Geometry, 3:237-256, 1988.

.... geometry parlance the above problem is called the largest common point set (LCP) problem and has been studied both under # congruence (as described above) and the exact matching metric (where # = 0) For the two dimensional version of the problem under # congruence, an exact algorithm was given in [4] which runs in O(n 8 ) time. However, it could not be extended to the three dimensional case in any straight forward way. An approximation algorithm for the three dimensional case was given in [1] which returned a set S # A of cardinality at least as large as the LCP between A and B under ....

H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237--256, 1988.

....method for externalizing plane sweep algorithms; ffl persistent B trees: an off line method for constructing an optimal space persistent version of the B tree data structure. For batched problems this gives a factor of B improvement over the generic persistence techniques of Driscoll et al. [1]; ffl batch filtering : a general method for performing K simultaneous external memory searches in data structures that can be modeled as planar layered dags and in certain fractional cascaded data structures; ffl on line filtering : A technique based on the work of Tamassia and Vitter [3] that ....

....National Science Foundation grant DMR 9217290. Author is currently visiting Duke University, email: dev cs.duke.edu 4 This research was supported in part by National Science Foundation grant CCR 9007851 and by Army Research Office grant DAAL03 91 G 0035, email: jsv cs.duke.edu References [1] J. R. Driscoll, N. Sarnak, D. D. Sleator, and R. E. Tarjan, Making Data Structures Persistent, J. Comput. System Sci. 38 (1989) 86 124. 2] D. G. Kirkpatrick and R. Seidel, The Ultimate Planar Convex Hull Algorithm , SIAM J. Comput. 15 (1986) 287 299. 3] R. Tamassia and J. S. Vitter, ....

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3(3):237--256, 1988.

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H. Alt, K. Mehlhorn, H. Wagner, E. Welzl. Congruence, Similarity, And Symmetries Of Geometric Objects. Discrete and Computational Geometry, vol. 3, pages 237--256, 1988.

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity, and symmetries of geometric objects. Discrete and Computational Geometry, 3:237256, 1988.

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Helmut Alt, Kurt Mehlhorn, Huber Wagener, and Emo Welzl. Congruence, Similarity, and Symmetries of Geometric Objects. Discrete and Computational Geometry, 1988.

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H. Alt, K. Mehlhorn, H. Wagener, and E. Welzl. Congruence, similarity and symmetries of geometric objects. Discrete and Computational Geometry, 3:237--256, 1988.

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