Exact and approximate geometric pattern matching for point sets in the plane under similarity transformations

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some δ > 0, find a similarity transformation (translation, rotation and scale) such that h(T(P),Q) < δ, where h(.,.) is the directional Hausdorff distance with L _∞ as the underlying metric. Similarity transformations have not been dealt with in the context of the directional Hausdorff distance and we fill the gap here. We present efficient, exact and approximate algorithms for this problem imposing a reasonable separation restriction on the set Q. For the exact case if the minimum L_∞ distance between every pair of points in Q is 8δ then the problem can be solved in O (n² m log n) time where m and n are the number of points in P and Q respectively. If the points in Q are just cδ apart from each other for any 0 < c < 1 we get a randomized approximate solution with expected runtime O(n ²c^-ε^-8 log⁴ mn), where ε > 0 controls the approximation and the answer is correct with high probability.