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 (arbitrary) δ > 0, find a similarity transformation T (translation, rotation and scale) such that h (T (P), Q) < δ, where h (ṡ, ṡ) is the directional Hausdorff distance with L_∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L_∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O (m n² log n) time, where m and n are the numbers of points in P and Q respectively. If the L_∞ distance between every pair of points in Q is at least cδ, for some c, 0 < c < 1, we present a randomized approximate solution with expected runtime O (n² c^{- 4} ε^{- 8} log⁴ m n), where ε > 0 controls the approximation. Our approximation is on the size of the subset, B ⊆ P, such that h (T (B), Q) < δ and | B | > (1 - ε) | P | with high probability.