1. Even faster point set pattern matching in 3-d Niagara University and SUNY - Buffalo Laurence Boxer boxer@niagara.edu Research partially supported by a grant from the Niagara University Research Council

2. The problem: Given a pattern set P and a sample set S in with, identify all subsets of S that are congruent to P.

3. Growth of relevant functions n: size of sample set Regard these functions as measures of time Time units not given. They rely on factors like speed of computer - important, but irrelevant to analysis of algorithms.

4. History - 1 P.J. de Rezende & D.T. Lee, Point set pattern matching in d dimensions, Algorithmica 13 (1995): running time

5. History - 2 L. Boxer, Point set pattern matching in 3-D, Pattern Recognition Letters 17 (1996): Running time Key to improved running time: derivation of smaller upper bound on output, based on upper bound for # of segments of same length in S, due to Clarkson, et al., Combinatorial complexity bounds for arrangements of curves and surfaces, Discrete & Computational Geometry 5 (1990). Output bound:

6. Current paper (Proc. SPIE Vision Geometry 1999) Running time: Improved running time due to derivation of yet smaller upper bound on output, from upper bound on triangles in S, in T. Akutsu et al., Distributions of distances and triangles in a point set and algorithms for computing the largest common point sets, Discrete & Computational Geometry 20 (1998). Output bound:

7. Efficient parallel solutions on CGM Running time: Ideal: Speedup linear in p (= # of processors) In practice: Often must introduce sorts for global data communications

8. Work to be done: Narrow the gap between the output bound and the running time Obtain comparable results for the harder but more-useful approximate matching version of the problem - currently working on interesting cases in which this can be done; different type of algorithm is required