Approximate decision algorithms for point set congruence

Heffernan, Paul J.; Schirra, Stefan

doi:10.1016/0925-7721(94)90004-3

Cited by 56 publications

(41 citation statements)

References 5 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

135

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

135

103

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“…An approximation scheme An (ε, β)-approximation (defined by Heffernan and Schirra [21]) for d F (P, Q) under translations can be obtained from the following observation: Lemma 4.9 Given polygonal chains P, Q, let t be the translation that maps the first point of Q to the first point of P . Then d F (P, Q + t) ≤ 2d * , where d * = min translations t d F (P, Q + t).…”

Section: Theorem 47mentioning

confidence: 99%

Pattern Matching for Sets of Segments

2004

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In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration.We propose a new measure of segment similarity called a coverage measure, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n 3 polylog n) in the general case, and run in time O(n 2 polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n 3/2 polylog n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time O(n 4 log 2 n).In the second part of this paper we address the problem of matching polygonal chains. We study the well known Fréchet distance , and present the first algorithm for computing the Fréchet distance under general translations. Our methods also yield algorithms for computing a generalization of the Fréchet distance, and we also present a simple approximation algorithm for the Fréchet distance that runs in time O(n 2 polylog n).

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“…Heffernan and Schirra [HS94] proposed an approximate decision algorithm that tests for congruence within a user-supplied absolute error . Their algorithm might decline to give a response if the actual distance is too close to .…”

Section: Introductionmentioning

confidence: 99%

Improved Approximation Bounds for Planar Point Pattern Matching

Cho

Mount

2007

Algorithmica

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Abstract. We consider the well known problem of matching two planar point sets under rigid transformations so as to minimize the directed Hausdorff distance. This is a well studied problem in computational geometry. Goodrich, Mitchell, and Orletsky [GMO94] presented a very simple approximation algorithm for this problem, which computes transformations based on aligning pairs of points. They showed that their algorithm achieves an approximation ratio of 4. We consider a minor modification to their algorithm, which is based on aligning midpoints rather than endpoints. We show that this algorithm achieves an approximation ratio not greater than 3.13. Our analysis is sensitive to the ratio between the diameter of the pattern set and the Hausdorff distance, and we show that the approximation ratio approaches 3 in the limit. Finally, we provide lower bounds that show that our approximation bounds are nearly tight.

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Approximate decision algorithms for point set congruence

Cited by 56 publications

References 5 publications

Efficient algorithms for robust feature matching

Efficient algorithms for robust feature matching

Pattern Matching for Sets of Segments

Improved Approximation Bounds for Planar Point Pattern Matching

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