D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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. The ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267-291, 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 ....

D. P. Huttenlocher, K. Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267291, 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. The ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267-291, 1993.

....areas belong to the closest resp. sites. becomes the farthest site Voronoi diagram, while for k =1wehavetheusual Voronoi diagram of n points. The FCVD is identical to the orthogonal projection on the plane of the upper hull of the one colored Voronoi surfaces as described by Huttenlocher et al. [9] and Sharir and Aggarwal [15, Section 8.7] they use it to obtain the minimum Hausdor# distance between two point sets. For its computation, they give an O(kn log n) time algorithm. Once the FCVD is given, one can, in time O(nk) determine the smallest color spanning circle: its center is either ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-- 291, 1993.

.... Aplicada II, Universitat Politecnica de Catalunya, Barcelona, Spain Praktische Informatik VI, FernUniversitat Hagen, Germany Institut fur Informatik I, Universitat Bonn, Germany ESCET, Universidad Rey Juan Carlos, Madrid, Spain 1 as described by Huttenlocher et al. [HKS93] and Sharir and Aggarwal [SA95, Section 8.7] their algorithm for computing the solution runs in time O(kn log n) Similarly one can determine the smallest color spanning axis parallel square and other objects with fixed orientation which are unit circles of a convex distance function. In this ....

D. P. Huttenlocher, K. Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267--291, 1993. 15

.... [2] Given two finit point sets # and #, the translation # # that minimizes the Hausdorff distance #### ## ## can be determined in time ######## ### # # when the underlying metric is # # or ## [14] For other # # metrics, # ### ##### it can be computed in time ###### # ####### ##### # ### [29]. #### is the inverse Ackermann function, a very slowly increasing function. This is done using the upper envelopes of Voronoi surfaces. Computing the optimal rigid motion # (translation plus rotation) minimizing ######### can be done in #### # ## # ######## time [28] This is done using ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267--291, 1993.

....is the maximum of h(A; B) and h(B; A) H(A;B) maxf d(A; B) d(B; A)g. For nite point sets, it can be computed using 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 ....

Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267-291, 1993.

....the edges that arise are stable under changes in lighting, and the models are well described as a collection of edges. There are many schemes that represent models and images by collections of edges and define a distance metric between them, Huttenlocher s use of the Hausdorff distance [10] is prominent among them. Some methods use a metric that is proportional to the number of edges that coincide (see the excellent survey articles: 11] 12] A smooth, optimizable version of such a metric can be defined by introducing a penalty both for unmatched edges and for the distance between ....

D.P. Huttenlocher, K. Kedem, K. Sharir, and M. Sharir. The Upper Envelope of Voronoi Surfaces and its Applications. In Proceedings of the Seventh ACM Symposium on Computational Geometry, pages 194--293, 1991.

....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 ....

Daniel P. Huttenlocher, Klara Kedem, and Micha Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267--291, 1993.

....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 ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete and Computational Geometry, 9:267-291, 1993.

....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, ....

D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

....function, as the transformation which gives rise to this minimum value will be the one bringing B into closest correspondence with A. Many approaches to determining this minimising transformation are based on searching the graph of this function (for example, by enumerating the local minima, as in [5]) It is therefore of interest to know what the geometric complexity of this graph may be. Upper bounds have been determined for some transformation groups, but few lower bounds were known. We will also be considering the graph of the function dG (g) h(g(B) A) which is the graph of the ....

.... We now note that ffi can be made as small as desired, thereby narrowing the staircase gap, and the Problem Lower Bound Solution Transformation Set type Norm Complexity Complexity Translation Point Sets L 1 , L1 Omega Gamma n 3 ) O(n 2 log 2 n) 4] L 2 Omega Gamma n 3 ) O(n 3 log n) [5] Points and Segments L 1 , L1 Omega Gamma n 4 ) O(n 4 ff(n) 5] L 2 Omega Gamma n 4 ) O(n 4 log 3 n) 1] Rigid Motion Point Sets L 2 Omega Gamma n 5 ) O(n 5 log 2 n) 3] Point and Segments L 2 Omega Gamma n 6 ) O(n 6 log 2 n) 3] Table 1: Results for the complexity ....

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D.P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. In Proceedings of Seventh ACM Symposium on Computational Geometry, pages 194--203, 1991. To appear in Discrete Computational Geometry.

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D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

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D.P. Huttenlocher, K. Kedem and M. Sharir, The Upper Envelope of Voronoi Surfaces and its Applications, Discrete and Computational Geometry, 9 (1993), 267-- 291.

.... et al. 19] using parametric searching, in O( mn) mn) time, which is significantly faster than the previously best known algorithm by Alt et al. 33] If P and Q are finite sets of points, a more efficient solution, not based on parametric searching, is proposed by Huttenlocher et al. [163]. Their solution, 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 [54, 64, 66] If we allow P to translate and rotate, then computing the minimum Hausdorff distance becomes significantly ....

D. P. Huttenlocher, K. Kedem, and M. Sharir, The upper envelope of Voronoi surfaces and its applications, Discrete Comput. Geom., 9 (1993), 267--291.

....) for any # 0, considerably improving the quadratic time algorithm of [19] Finding the minimum Hausdor# distance under translation between two polygonal regions in the plane under the Euclidean metric. This is a hard instance of a general pattern matching problem. It was left untreated in [26], and solved by a brute force ine#cient method in [8] We solve it in time O( mn) 2 log 3 (mn) where m and n are the number of edges of the given polygons. This is about 3 orders of magnitude faster than the algorithm of [8] Solving the 1 segment center problem given a set of n ....

....recognition, computer vision, etc. Huttenlocher and Kedem [25] showed that if P and Q are sets of points, then D(P,Q) can be computed in O( mn) 2 #(mn) time, where #( is the inverse Ackermann function. This bound has been recently improved to O(mn(m n)#(mn) log mn) by Huttenlocher et al. [26]. They also showed that if the distance between two points is measured in the L 1 or L# metrics, the distance D(P,Q) for sets P, Q each consisting of non intersecting segments, can be computed in time O( mn) 2 log mn) However, their algorithm does not extend to the more useful case of the ....

D. Huttenlocher, K. Kedem and M. Sharir, The upper envelope of Voronoi surfaces and its applications, Proc. 7th ACM Symp. on Computational Geometry, 1991, 194--203.

....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 nonintersecting line segments under Author s address: ....

....known algorithms that find the minimum Hausdorff distance for the four problems stated above are very costly in terms of runtime. If A and B are point sets of cardinalities m and n, respectively, then the minimum Hausdorff distance under translation can be computed in time O(mn(m n) log(mn) [19]. Under Euclidean motion the runtime is O( mn) 2 (m n) log 2 (mn) 12] If A and B are sets of nonintersecting lines segments of cardinalities m and n then the minimum Hausdorff distance under translation can be computed in time O( mn) 2 log 3 (mn) 8] Under Euclidean motion the ....

[Article contains additional citation context not shown here]

D.P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi-surfaces and its applications. In Proc. of the 7th ACM Comp. Geom. Conference, pages 194--203, 1991.

....al. 18] using parametric searching, in O( mn) 2 log 3 (mn) time, which is significantly faster than the previously best known algorithm by Alt et al. 30] If P and Q are finite sets of points, a more efficient solution, not based on parametric searching, is proposed by Huttenlocher et al. [140]. Their solution, 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 ....

D. P. Huttenlocher, K. Kedem, and M. Sharir, The upper envelope of Voronoi surfaces and its applications, Discrete Comput. Geom., 9 (1993), 267--291.

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D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

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K. Kedem D. P. Huttenlocher and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

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D.P. Huttenlocher, K. Kedem and M. Sharir. The upper envelope of Voronoi surfaces and its applications, Discrete Computational Geometry, 9, (1993), 267-291.

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D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

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D. P. Huttenlocher, K. Kedem and M. Sharir. The upper envelope of Voronoi Surfaces and its Applications. Proc. 7'th ACM Symp. Comp. Geom., 194-203, 1991.

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D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom., 9:267-291, 1993.

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D. P. Huttenlocher, K. Kedem, and M. Sharir, "The upper envelope of Voronoi surfaces and its applications," Discrete & Computational Geometry 9 (1993) 267-- 291.

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