Congruence, similarity and symmetries of geometric objects
Discrete & Computational Geometry - ACM Symposium on Computational Geometry, Waterloo
Approximate matching of polygonal shapes (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
SCG '92 Proceedings of the eighth 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
Average-case analysis of algorithms for matchings and related problems
Journal of the ACM (JACM)
Efficient 2-dimensional approximate matching of half-rectangular figures
Information and Computation
Randomized algorithms
Algorithmic number theory
Improvements on bottleneck matching and related problems using geometry
Proceedings of the twelfth annual symposium on Computational geometry
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
Geometric Manipulation of Flexible Ligands
FCRC '96/WACG '96 Selected papers from the Workshop on Applied Computational Geormetry, Towards Geometric Engineering
Pattern Matching for Spatial Point Sets
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Efficient colored point set matching under noise
ICCSA'07 Proceedings of the 2007 international conference on Computational science and its applications - Volume Part I
Noisy colored point set matching
Discrete Applied Mathematics
A near-linear time ε-approximation algorithm for geometric bipartite matching
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Coherence Pattern-Guided Compressive Sensing with Unresolved Grids
SIAM Journal on Imaging Sciences
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The problem of geometric point set matching has been studied extensively in the domain of computational geometry, and has many applications in areas such as computer vision, computational chemistry, and pattern recognition. One of the commonly used metrics is the bottleneck distance, which for two point sets P and Q is the minimum over all one-to-one mappings f : P → Q of maxp∈Pd(p,f(p)), where d is the Euclidean distance. Much effort has gone into developing efficient algorithms for minimising the bottleneck distance between two point sets under groups of transformations. However, the algorithms that have thus far been developed suffer from running times that are large polynomials in the size of the input, even for approximate formulations of the problem.In this paper we define a point set similarity measure that includes both the bottleneck distance and the Hausdorff distance as special cases. This measure relaxes the condition that the mapping must be one-to-one, but guarantees that only a few points are mapped to any point. Using a novel application of Hall's Theorem to reduce the geometric matching problem to a combinatorial matching problem, we present near-linear time approximation schemes for minimising this distance between two point sets in the plane under isometries; we note here that the best known algorithms for congruence under the bottleneck measure run in time Õ(n2.5).We also obtain a combinatorial bound on the metric entropy of certain families of geometric objects. This result yields improved algorithms for approximate congruence, and may be of independent interest.