Approximate center points in dense point sets

Verbarg, Knut

doi:10.1016/s0020-0190(97)00017-3

Cited by 12 publications

(7 citation statements)

References 5 publications

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“…Valtr [47]- [49] proved several other combinatorial bounds for dense planar sets that improve the corresponding worst-case bounds. Verbarg [50] describes an efficient algorithm to find approximate center points in dense point sets. Cardoze and Schulman [17], Indyk et al [42], and Gavrilov et al [36] describe algorithms for approximate geometric pattern matching whose running times depend favorably on the spread of the input set.…”

Section: Sublinear Spreadmentioning

confidence: 99%

Nice Point Sets Can Have Nasty Delaunay Triangulations

Erickson¹

2003

Discrete and Computational Geometry

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Section: Sublinear Spreadmentioning

confidence: 99%

Nice Point Sets Can Have Nasty Delaunay Triangulations

Erickson¹

2003

Discrete and Computational Geometry

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“…Several approaches using random sampling are known [10,14,2]. Verbarg showed that for dense points, the mean is a good approximate centerpoint [16]; it is a β-center where β depends on the density.…”

Section: Related Workmentioning

confidence: 99%

Approximate center points with proofs

Sheehy

2009

Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry

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We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an approximate centerpoint of a set S ∈ R d with running time sub-exponential in d. The algorithm is a derandomization of the IteratedRadon algorithm of Clarkson et al and is guaranteed to terminate with an O(1/d2 )-center. Moreover, it returns a polynomial-time checkable proof of the approximation guarantee, despite the coNP-Completenes of testing centerpoints in general. We also explore the use of higher order Tverberg partitions to improve the runtime of the deterministic algorithm and improve the approximation guarantee for the randomized algorithm. In particular, we show how to im-

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“…Valtr and others [14], [39], [40], [41] investigated bounds on the number of approximate incidences between families of wellseparated lines and points; assuming that the point sets are dense, i.e., their diameter is O( √ n), he showed that several key results developed for the exact incidence model have counterparts in the approximate setting.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Combinatorial and Experimental Methods for Approximate Point Pattern Matching

et al. 2003

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Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modeling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high-degree polynomials, and require robust calculations of intersection points of high-degree surfaces.We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate.We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average-case analysis and a detailed empirical study of our methods. Introduction.Given point sets P and Q, the problem of pattern matching is to determine whether P is similar to some portion of Q. Similarity is typically defined in terms of a distance measure, and P and Q may undergo transformations like rotations, translations, and scaling before the distance between them is computed. This general formulation captures a wide variety of problems, arising in image registration [33], model-based object recognition [4], [21], computer vision [35], molecular modeling [27], and many other areas.In the domain of computational geometry, these problems were mostly studied in an exact formulation: point coordinates are given with infinite precision, and two points match if their coordinates are identical. The disadvantage with this formulation is twofold: Firstly, this is an inaccurate representation of reality, where the input points are typically noisy, and any match requires some notion of a tolerance parameter. Secondly, the algorithms designed to compute similarity were complex, with running times in the order of large polynomials, and required sophisticated algebraic manipulation of high-degree manifolds in a robust fashion.This provides the motivation for approximate pattern matching. In an approximate setting, one might replace points by balls of radius ε (where ε is a tolerance parameter)

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Approximate center points in dense point sets

Cited by 12 publications

References 5 publications

Nice Point Sets Can Have Nasty Delaunay Triangulations

Nice Point Sets Can Have Nasty Delaunay Triangulations

Approximate center points with proofs

Combinatorial and Experimental Methods for Approximate Point Pattern Matching

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