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An Efficient Approximation Algorithm for Point Pattern Matching Under Noise
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 LCP (largest common point
set problem): Let \PP and \QQ be two point sets in , and let
be a tolerance parameter, the problem is to find a rigid
motion that maximizes the cardinality of subset \II of , such that
the Hausdorff distance \distance(\PP,\mu(\II)) \leq \epsilon. We denote the
size of the optimal solution to the above problem by \LCP(P,Q). The problem
is called exact-LCP for , and \tolerant-LCP when and
the minimum interpoint distance is greater than . A
-distance-approximation algorithm for tolerant-LCP finds a subset I
\subseteq \QQ such that |I|\geq \LCP(P,Q) and \distance(\PP,\mu(\II)) \leq
\beta \epsilon for some .
This paper has three main contributions. (1) We introduce a new algorithm,
called {\DA}, which gives the fastest known deterministic
4-distance-approximation algorithm for \tolerant-LCP. (2) For the exact-LCP,
when the matched set is required to be large, we give a simple sampling
strategy that improves the running times of all known deterministic algorithms,
yielding the fastest known deterministic algorithm for this problem. (3) We use
expander graphs to speed-up the \DA algorithm for \tolerant-LCP when the size
of the matched set is required to be large, at the expense of approximation in
the matched set size. Our algorithms also work when the transformation is
allowed to be scaling transformation.Comment: 18 page