Congruence, similarity, and symmetries of geometric objects

Alt, Helmut; Mehlhorn, Kurt; Wagener, Hubert; Welzl, Emo

doi:10.1007/bf02187910

Cited by 185 publications

(195 citation statements)

References 10 publications

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“…Most of the formulations of the point matching problem in computational geometry are not suitable for noisy, cluttered images either because they require exact matches [12], they require 1-1 matches [13,14] or they assume that every point in one set has a close match in the other set in terms of the (standard) Hausdorff distance [15][16][17][18]. Even under these relatively restrictive assumptions, the computational complexity can be quite high.…”

Section: Prior Workmentioning

confidence: 99%

Efficient algorithms for robust feature matching

Mount

Netanyahu

Moigne

1999

Pattern Recognition

136

103

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One of the basic building blocks in any point-based registration scheme involves matching feature points that are extracted from a sensed image to their counterparts in a reference image. This leads to the fundamental problem of point matching: Given two sets of points, find the (affine) transformation that transforms one point set so that its distance from the other point set is minimized. Because of measurement errors and the presence of outlying data points, it is important that the distance measure between the two point sets be robust to these effects. We measure distances using the partial Hausdorff distance. Point matching can be a computationally intensive task, and a number of theoretical and applied approaches have been proposed for solving this problem. In this paper, we present two algorithmic approaches to the point matching problem, in an attempt to reduce its computational complexity, while still providing a guarantee of the quality of the final match. Our first method is an approximation algorithm, which is loosely based on a branch-andbound approach due to Huttenlocher and Rucklidge, (Technical Report 1321, Dept. of Computer Science, Cornell University, Ithaca, 1992; Proc. IEEE Conf. on Computer vision and Pattern Recognition, New York, 1993, pp. 705-706). We show that by varying the approximation error bounds, it is possible to achieve a tradeoff between the quality of the match and the running time of the algorithm. Our second method involves a Monte Carlo method for accelerating the search process used in the first algorithm. This algorithm operates within the framework of a branch-and-bound procedure, but employs point-to-point alignments to accelerate the search. We show that this combination retains many of the strengths of branch-and-bound search, but provides significantly faster search times by exploiting alignments. With high probability, this method succeeds in finding an approximately optimal match. We demonstrate the algorithms' performances on both synthetically generated data points and actual satellite images.

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Section: Prior Workmentioning

confidence: 99%

Efficient algorithms for robust feature matching

Mount

Netanyahu

Moigne

1999

Pattern Recognition

136

103

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“…Condition (2). If the cardinality |C| of a cluster C is larger than 2k, the cluster is split using Lemma 4.1.…”

Section: Updating the Dynamic Partition And The Structure Of T T And mentioning

confidence: 99%

“…Using just one phase of a static maximum cardinality matching algorithm per update leads to a dynamic algorithm with O(n + m) worst-case update time (see, e.g., [2]). This is still the best known algorithm.…”

Section: Maximum Cardinality Matchingmentioning

confidence: 99%

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Average-Case Analysis of Dynamic Graph Algorithms

Alberts

Henzinger

1998

Algorithmica

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Abstract. We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., not random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2-edge connectivity, kedge connectivity, k-vertex connectivity, and bipartiteness. Given a random graph G with m 0 edges and n vertices and a sequence of l update operations such that the graph contains m i edges after operation i, the expected time for performing the updates for any l is O(l log n + l i=1 n/ √ m i ) in the case of minimum spanning forests, connectivity, 2-edge connectivity, and bipartiteness. The expected time per update operation is O(n) in the case of maximum matching. We also give improved bounds for k-edge and k-vertex connectivity. Additionally we give an insertions-only algorithm for maximum cardinality matching with worst-case O(n) amortized time per insertion.

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“…In pattern recognition and computer vision literature, the problem of symmetry detection was studied mainly in images [15], two-dimensional [31,2,1] and three-dimensional shapes [27,10,17]. A wide spectrum of methods employed for this purpose includes approaches based on dual spaces [7], genetic algorithms [9], moments [5], pair matching [13,6], and local shape descriptors [33].…”

Section: Introductionmentioning

confidence: 99%

Shape Palindromes: Analysis of Intrinsic Symmetries in 2D Articulated Shapes

Hooda

Bronstein

et al. 2012

Lecture Notes in Computer Science

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Abstract. Analysis of intrinsic symmetries of non-rigid and articulated shapes is an important problem in pattern recognition with numerous applications ranging from medicine to computational aesthetics. Considering articulated planar shapes as closed curves, we show how to represent their extrinsic and intrinsic symmetries as self-similarities of local descriptor sequences, which in turn have simple interpretation in the frequency domain. The problem of symmetry detection and analysis thus boils down to analysis of descriptor sequence patterns. For that purpose, we show two efficient computational methods: one based on Fourier analysis, and another on dynamic programming. Metaphorically, the later can be compared to finding palindromes in text sequences.

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Congruence, similarity, and symmetries of geometric objects

Cited by 185 publications

References 10 publications

Efficient algorithms for robust feature matching

Efficient algorithms for robust feature matching

Average-Case Analysis of Dynamic Graph Algorithms

Shape Palindromes: Analysis of Intrinsic Symmetries in 2D Articulated Shapes

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