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Discrete & Computational Geometry - ACM Symposium on Computational Geometry, Waterloo
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STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
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SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
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IEEE Transactions on Pattern Analysis and Machine Intelligence
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IEEE Computational Science & Engineering
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WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
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ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
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FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
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STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
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CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
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Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as LCP (largest common point set problem): Let P and Q be two point sets in ${\mathbb R}^{3}$, and let ε ≥ 0 be a tolerance parameter, the problem is to find a rigid motion μ that maximizes the cardinality of subset I of Q, such that the Hausdorff distance dist(P,μ(I))≤ε. We denote the size of the optimal solution to the above problem by LCP(P,Q). The problem is called exact-LCP for ε = 0, and tolerant-LCP when ε0 and the minimum interpoint distance is greater than ε. A β-distance-approximation algorithm for tolerant-LCP finds a subset I⊆Q such that |I| ≥LCP(P,Q) and dist(P,μ(I))≤βε for some β ≥ 1. This paper has three main contributions. (1) We introduce a new algorithm, called T-hashing, which gives the fastest known deterministic 4-distance-approximation algorithm for tolerant-LCP. (2) For the exact-LCP, when the matched set is required to be large, we give a simple sampling strategy that improves the running times of all known deterministic algorithms, yielding the fastest known deterministic algorithm for this problem. (3) We use expander graphs to speed-up the T-hashing algorithm for tolerant-LCP when the size of the matched set is required to be large, at the expense of approximation in the matched set size. Our algorithms also work when the transformation μ is allowed to be scaling transformation.