Improved Approximation Bounds for Planar Point Pattern Matching |
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Authors: | Minkyoung Cho David M Mount |
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Affiliation: | (1) Department of Computer Science, University of Maryland, College Park, MD 20742, USA;(2) Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742, USA |
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Abstract: | We analyze the performance of simple algorithms for matching two planar point sets under rigid transformations so as to minimize
the directed Hausdorff distance between the sets. This is a well studied problem in computational geometry. Goodrich, Mitchell,
and Orletsky 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 introduce a modification to their
algorithm, which is based on aligning midpoints rather than endpoints. This modification has the same simplicity and running
time as theirs, and we show that it achieves a better approximation ratio of roughly 3.14. We also analyze the approximation
ratio in terms of a instance-specific parameter that is based on the ratio between diameter of the pattern set to the optimum
Hausdorff distance. We show that as this ratio increases (as is common in practical applications) the approximation ratio
approaches 3 in the limit. We also investigate the performance of the algorithm by Goodrich et al. as a function of this ratio,
and present nearly matching lower bounds on the approximation ratios of both algorithms.
This work was supported by the National Science Foundation under grants CCR-0098151 and CCF-0635099. |
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Keywords: | Point pattern matching Approximation algorithms Hausdorff distance |
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