L. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, J.M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. ####### ##### ###### ### #####, 7:113-124, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....does not apply to the case of polygons. If we measure distance by the L1 metric, faster algorithms, based on parametric searching, are developed in [54, 64, 66] If we allow P to translate and rotate, then computing the minimum Hausdorff distance becomes significantly harder. Chew et al. [65] have given an O(m when both P and Q are finite point sets, and an O(m when P and Q are polygons. Another way of measuring the resemblance between two polygons P and Q is by computing the area of their intersection (or, rather, of their symmetric difference) Suppose we wish to ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets, Geometric pattern matching under Euclidean motion, Comput. Geom. Theory Appl., 7 (1997), 113--124.

....of times A s is called considerably. The technique is rather complicated, as it requires the design of an efficient parallel algorithm. Fortunately, the generic algorithm does not necessarily have to solve the same problem as the concrete decision; in several cases, sorting can be used instead [3, 8, 11, 13, 16]. However, the existing parallel sorting algorithms that have good worst case time bounds are not easily implemented, and in some cases the hidden constants in the asymptotic running times are enormous [2] Cole [9] shows how sorting based parametric search can be optimized even further, but the ....

.... tree and scheduling problems studied by Megiddo [16] the slope selection problem [11] the Frechet distance problem [3] the bottleneck distance problem [13] and the problem of finding the minimum Hausdorff distance, under rotation and rotation, between two sets of points, lines, or polygons [8]. In several other cases, the generic algorithm consists of several steps, sorting being one of them [17, 19] For sorting based parametric search we have several parallel sorting algorithms at our disposal. The first sorting algorithm that sorts in O(logn) parallel steps using O(n) processors is ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Comput. Geom. Theory Appl., 7:113--124, 1997.

....determine how much these shapes resemble each other. More precisely, one wants to find a rigid motion of one shape that maximizes the resemblance with the other shape. There are several ways to measure resemblance. For example, for point sets or polygonal chains one can use the Hausdorff distance [2, 1, 9, 15, 16]; for polygonal chains one can also use the Frchet distance [3] The resemblance of two convex polygons can also be measured by looking at the Hausdorff or rchet distance between their boundaries. For an application in computer vision, however, it seems more appropriate to look at the area of ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. In Proc. 5th Canad. Conf. Cornput. Geom., pages 151-156, Waterloo, Canada, 1993.

....about the x axis which can be handled by computing the metric separately for each rotation and taking the minimum value instead of incorporating rotation into the alignment process. A great deal of work has been done on point set matchings; see Alt, Aichholzer, and Rote [1] Chew et al. [7], and Goodrich, Mitchell, and Orletsky [14] for several methods of obtaining both optimal and approximate matchings. Different methods can be applied when the correspondence between points is known as it is here; Imai, Sumino, and Imai [15] provide an algorithm that minimizes the maximum distance ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Keriera, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Cornput. Ceom. Theory Appl., 7:113 124, 1997.

....for boundary curves are based on turning angle, see Cohen and Guibas [5] or normalised ane arc length, see Huttenlocher and Kedem [10] The Hausdor metric is de ned on the collection of all (non empty) closed, bounded subsets of a metric space. Some algorithms are based on this metric, see [4], 11] 1] However, the Hausdor metric is not robust with respect to certain types of noise. For example, outliers, i.e. isolated points lying far away from the other points, can cause a dramatic increase in the Hausdor distance. The Hausdor metric is invariant for the group of isometries. ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, Jon M. Kleinberg, and Dina Kravets. Geometric pattern matching under Euclidean motion. In Fifth Canadian Conference on Computational Geometry, pages 151-156, 1993.

....to determine how much these shapes resemble each other. More precisely, one wants to nd a rigid motion of one shape that maximizes the resemblance with the other shape. There are several ways to measure resemblance. For example, for point sets or polygonal chains one can use the Hausdoroe distance [2, 1, 9, 15, 16]; for polygonal chains one can also use the Fr#chet distance [3] The resemblance of two convex polygons can also be measured by looking at the Hausdoroe or Fr#chet distance between their boundaries. For an application in computer vision, however, it seems more appropriate to look at the area of ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. In Proc. 5th Canad. Conf. Comput. Geom., pages 151156, 1993.

....metrics for curves are based on turning angle, see Cohen and Guibas [5] or normalised ane arc length, see Huttenlocher and Kedem [11] The Hausdor metric is de ned on the collection of all (non empty) closed, bounded subsets of a metric space. Some algorithms are based on this metric, see [4], 12] 1] However, the Hausdor metric is not robust with respect to certain types of noise. For example, outliers, i.e. isolated points lying far away from the other points, can cause a dramatic increase in the Hausdor distance. The Hausdor metric is invariant for the group of isometries. ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, Jon M. Kleinberg, and Dina Kravets. Geometric pattern matching under Euclidean motion. In Fifth Canadian Conference on Computational Geometry, pages 151-156, 1993.

....scheme can be found in [8, 9] Related Prior Work. Chew et al. 6] have considered the problem of matching point sets in a ddimensional space using the Hausdorff distance under translation only. For the case d = 3, they provide exact solutions in O(n 3 log 2 n) time. Recent extensions [5] to the more general case of Euclidean motion and of sets of segments have obtained exact solutions in O(n 6 log 2 n) time. However they are limited to the case of planar sets. Due to the high computational complexity of the exact algorithms, efficient but approximate solutions for the case ....

Chew, L. P., Goodrich, M. T., Huttenlocher, D. P., Kedem, K., Kleinberg, J. M., and Kravets, D. Geometric pattern matching under Euclidean motion. Comput. Geom. 7, 1-2 (

....rigid motion # (translation plus rotation) minimizing ######### can be done in #### # ## # ######## time [28] This is done using dynamic Voronoi diagrams. Given a real value #, deciding if there is a rigid motion such that ######### # # can be done in time #### # ### # # # ### ### [13]. Given the high complexities of these problems, it makes sense to look at approximations. Computing an approximate optimal Hausdorff distance under translation and rigid motion is discussed in [1] The Hausdorff distance is very sensitive to noise: a single outlier can determine the distance ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Computational Geometry, Theory and Applications, 7:113-- 124, 1997.

.... A recent attack on this problem is given in [2] See also [13] ffl The segment center problem is actually a special case of the more general problem of computing the one directional Hausdorff distance, under euclidean motion, between two sets of objects, which can be stated as follows (see [16]) Given two sets, S 1 ; S 2 , of objects in the plane, find the smallest value of d such that the objects of S 1 can be translated and rotated (rigidly) into a placement where every object of S 2 is within distance d from some object of S 1 . Hence an obvious open problem is whether our ....

L.P. Chew, M. Goodrich, D. Huttenlocher, K. Kedem, J. Kleinberg and D. Kravetz, Geometric pattern matching under euclidean motion, Proc. 5th Canadian Conf. on Computatoonal Geometry, 1993, 151--156.

.... Voronoi diagrams [2] Given two nite point sets A and B, computing the translation that minimizes the Hausdor distance H(A ; B) is discussed in [9] and [20] Given a real value , deciding if there is a rigid motion m (translation plus rotation) such that H(m(A) B) is discussed in [8]. Computing the optimal rigid motion, minimizing H(m(A) B) is treated in [19] using dynamic Voronoi diagrams. Given the high complexities of these problems, it makes sense to look at approximations. Computing an approximate optimal Hausdor distance under translation and rigid motion is ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Computational Geometry, Theory and Applications, 7:113-124, 1997.

....of Point Pattern Matching in Astronautics 3 structure D which can be used to find a quick answer to the matching problem for any query pattern P . Since P represents data from a camera with bounded resolution, the data contain errors and the match will only be approximate. Therefore, following [5] we define the approximate matching problem: Given point sets C and P on the unit sphere, determine the smallest such that there is a rigid motion mapping each point in P within distance of some point in C. We can consider the query problem for this version, as well. Moreover many ....

....representatives, one from each set, which have the correct distances between each other. 4 C P Figure 2: The mapping between the two pairs defines the whole mapping Within the Computational Geometry community point pattern matching usually was investigated for sets of points in the plane (see [1, 2, 5, 7, 8, 13]) but it seems that the bounds and algorithms described in the following can be transferred to sets of points on the unit sphere, as well. In case of the exact matching problem with no additional information about the pattern P matching candidates are found by determining all occurrences of one ....

[Article contains additional citation context not shown here]

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg and D. Kravets, "Geometric Pattern Matching under Euclidean Motion", Proceedings Can. Conf. on Comp. Geom., 1993, pp. 151-155.

....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 ....

....any two arbitrary patterns. This means that must be capable of detecting similar patterns even if parts in one pattern may be absent from the other or lying outside of it. Examples of pattern similarity measures are the the uniform metric [23] the Fr echet distance [13] the Hausdorff distance [16], the partial Hausdorff distance [21, 10] and the symmetric difference [14] In exact pattern matching algorithms, the discrete metric, defined by A 6= B ) A; B) 1, is used implicitly. Each pattern similarity measure, has a specific pattern space P on which it is defined. Many similarity ....

L. Paul Chew, Michael T. Goodrich, Daniel P. Huttenlocher, Klara Kedem, Jon M. Kleinberg, and Dina Kravets. Geometric pattern matching under Euclidean motion. In Fifth Canadian Conference on Computational Geometry, pages 151--156, 1993.

....) of an illustration P we record which basic shapes appear where. In other words, for each basic shape, we record in the index the translation, rotation, and scale transformations which cause this basic shape to match well some of the shapes present in P , according to the Hausdorff distance [ Chew et al. 1993 ] Thus we can think of the index as a list of colored points in R 4 , where the four coordinates are the four parameters defining the transformation, and the color is the label of the basic shape involved. We actually store the logarithm of the scale parameter, so as to make variations in ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. In Proceedings of the Fifth Canadian Conference on Computational Geometry, pages 151--156, 1993.

....to produce a reasonably small static query data structure D which can be used to find a quick answer to the matching problem for any query pattern P . Since P represents data from a camera of bounded resolution, the data contain errors and the match will only be approximate. Therefore, following Chew et al. 1993) we define the approximate matching problem: Given point sets C and P on the unit sphere, determine the smallest such that there exists a rigid motion mapping each point in P within distance of some point in C. We can consider the query problem for this version, as well. Notice that ....

....for our application. In addition, it is not particularly designed for solving the query problem, in fact it seems that this variant of the problem has not been considered theoretically before. An algorithm solving the approximate matching problem in time O(m 3 n 2 log 2 mn) is presented in Chew et al. 1993). Again, no particular solutions for the corresponding query problem are known. 3. Our Approach We present a solution which solves the query versions of the exact and the approximate matching problem using additional information about the pattern. First, some preliminary deliberations: The ....

Chew, L. P., Goodrich, M. T., Huttenlocher, D. P., Kedem, K., Kleinberg, J. M., Kravets, D. (1993). Geometric Pattern Matching under Euclidean Motion. Proceedings Can. Conf. on Comp. Geom., 151-155.

....Before we present the new metric, we evaluate a number of pattern metrics known from literature. For each, we examine the domain of de nition (type of patterns) the maximal invariance group, and satisfaction of the axioms. Many pattern matching algorithms are based on the Hausdor metric, see [4, 11, 1]. This metric is de ned on the collection of closed, bounded (nonempty) subsets of any underlying metric space. The Hausdor metric is not robust for outliers. In fact, one might even say that the Hausdor metric is de ned in terms of outliers. In addition, the Hausdor metric is only invariant ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, Jon M. Kleinberg, and Dina Kravets. Geometric pattern matching under Euclidean motion. In Fifth Canadian Conference on Computational Geometry, pages 151-156, 1993.

....however, does not apply to the case of polygons. If we measure distance by the L1 metric, faster algorithms, based on parametric searching, are developed in [55, 57] If we allow P to translate and rotate, then computing the minimum Hausdorff distance becomes significantly harder. Chew et al. [56] have given an O(m 2 n 2 log 3 mn) time algorithm when both P and Q are finite point sets, and an O(m 3 n 2 log 3 mn) time algorithm when P and Q are polygons. Another way of measuring the resemblance between two polygons P and Q is by computing the area of their intersection (or, ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets, Geometric pattern matching under Euclidean motion, Proc. 5th Canad. Conf. Comput. Geom., 1993, pp. 151--156.

....one can improve the running times in the approximate case if the noise regions are disjoint. Even so, the methods in these papers are relatively sophisticated, with rather high running times for all but the most simple motions. In work more directly related to this paper, several researchers [9, 10, 25, 26] have studied methods for nding rigid motions that minimize either the directed or undirected Hausdor distance between 2 the two point sets. All of these methods are based on intersecting higher degree curves and or surfaces, which are then searched (sometimes parametrically [1, 11, 12, 13, ....

....running time. 2 Our Algorithms The input to our algorithms is a set B of n points in IR d and a set P of m points in IR d , where dimension d is considered constant. 3 Motion Optimal Match Our Method Factor T in IR 2 nm 2 log n [25] nm log n 2 T R in IR 2 n 2 m 3 log n [9] n 2 m log n 4 T in IR 3 n 3 m 2 log 2 n [25] nm log n 2 T in IR d nm log n 2 R in IR 3 n 2 m log n 4 T R in IR 3 n 3 m log n 8 Table 1: The asymptotic running times for an optimal match and for our approximately optimal match, having a worst case ....

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Comput. Geom. Theory Appl., 7:113-124, 1997.

No context found.

L. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, J.M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. ####### ##### ###### ### #####, 7:113-124, 1997.

No context found.

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Compututational Geometry: Theory and Applications, 7:113--124, 1997.

No context found.

L. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, J.M. Kleinberg, and D. Kravets. Geometric pattern matching under euclidean motion. Proceedings of the Fifth Canadian Conference on Computational Geometry, pages 151--156, 1993.

No context found.

L. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, J.M. Kleinberg, and D. Kravets. Geometric pattern matching under euclidean motion. In Proceedings of the Fifth Canadian Conference on Computational Geometry, pages 151--156, 1993.

No context found.

L. P. Chew, M. T. Goodrich, D. P. Huttenlocher, K. Kedem, J. M. Kleinberg, and D. Dravets. Geometric pattern matching under euclidean motion. Comput. Geom. Theory Appl., 7:113-124, 1997.

No context found.

L. P. Chew, M. Goodrich, D. Huttenlocher, K. Kedem, J. Kleinberg, and D. Kravets. Geometric pattern matching under Euclidean motion. Proc. 5'th Canadian Conf. Comp. Geom., 151-156, 1993.

No context found.

L. Chew, M.T. Goodrich, D.P. Huttenlocher, K. Kedem, J.M. Kleinberg, and D. Kravets. Geometric pattern matching under euclidean motion. In Proceedings of the Fifth Canadian Conference on Computational Geometry, pages 151--156, 1993.

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