Transformation Hashing: An Efficient Point Pattern Matching Scheme

Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as the largest common point set problem (LCP) under approximate congruence: given two point sets M and Q in R, and a tolerance parameter ǫ ≥ 0, the problem is to find a rigid motion μ such that the cardinality of subset I ⊆ Q, for which each point of μ(I) is within ǫ distance of a point of M , is maximized. The known algorithms (exact or approximate) for this problem are inefficient and unpractical. In practice, the problem is solved heuristically by voting schemes such as generalized Hough transform or geometric hashing which can be rigorously analyzed only for their exact matching (i.e. ǫ = 0) version. With a combinatorial observation, we improve these voting schemes by an extra simple step. We also propose a new improved voting scheme, called transformation hashing. One advantage of this scheme is that the transformations to be clustered have only one degree of freedom. This allows us to analyze the approximate version of the problem rigorously–it guarantees the result with an approximation factor while keeping the practicality as a voting scheme. We also propose an expander-based approach to further speed up the algorithm at the expense of another approximation factor. Our algorithms are deterministic.