Matching shapes with a reference point
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Approximate decision algorithms for point set congruence
Computational Geometry: Theory and Applications
Geometric pattern matching under Euclidean motion
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
RAPID: randomized pharmacophore identification for drug design
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Geometric matching under noise: combinatorial bounds and algorithms
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Pattern Matching for Spatial Point Sets
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
2D-pattern matching image and video compression: theory, algorithms, and experiments
IEEE Transactions on Image Processing
MICCAI'06 Proceedings of the 9th international conference on Medical Image Computing and Computer-Assisted Intervention - Volume Part II
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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.