Planar point location using persistent search trees
Communications of the ACM
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Ray shooting and parametric search
SIAM Journal on Computing
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Applications of parametric searching in geometric optimization
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Geometric pattern matching under Euclidean motion
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
Journal of the ACM (JACM)
Geometric matching under noise: combinatorial bounds and algorithms
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Applying Parallel Computation Algorithms in the Design of Serial Algorithms
Journal of the ACM (JACM)
Approximate subset matching with Don't Cares
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
The Medial axis of a union of balls
Computational Geometry: Theory and Applications
Linear-size approximate voronoi diagrams
SODA '02 Proceedings of the thirteenth 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
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating extent measures of points
Journal of the ACM (JACM)
Nearest-neighbor searching under uncertainty
PODS '12 Proceedings of the 31st symposium on Principles of Database Systems
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We study the shape matching problem under the Hausdorff distance and its variants. In the first part of the article, we consider two sets A,B of balls in Rd, d=2,3, and wish to find a translation t that minimizes the Hausdorff distance between A+t, the set of all balls in A shifted by t, and B. We consider several variants of this problem. First, we extend the notion of Hausdorff distance from sets of points to sets of balls, so that each ball has to be matched with the nearest ball in the other set. We also consider the problem in the standard setting, by computing the Hausdorff distance between the unions of the two sets (as point sets). Second, we consider either all possible translations t (as is the standard approach), or consider only translations that keep the balls of A+t disjoint from those of B. We propose several exact and approximation algorithms for these problems. In the second part of the article, we note that the Hausdorff distance is sensitive to outliers, and thus consider two variants that are more robust: the root-mean-square (rms) and the summed Hausdorff distance. We propose efficient approximation algorithms for computing the minimum rms and the minimum summed Hausdorff distances under translation, between two point sets in Rd. In order to obtain a fast algorithm for the summed Hausdorff distance, we propose a deterministic efficient dynamic data structure for maintaining an &epsis;-approximation of the 1-median of a set of points in Rd, under insertions and deletions.