Pattern Recognition 43 (2010) 153--159 Contents lists available at ScienceDirect Pattern Recognition journal homepage: www.elsevier.com/locate/pr Approximate input sensitive algorithms for point pattern matching  Dror Aiger a,b, ∗ , Klara Kedem a,c a Department of Computer Science, Ben Gurion University, Be'er Sheva, Israel b Orbotech LTD, Yavne, Israel c Computer Science Department, Cornell University, USA ARTICLE INFO ABSTRACT Article history: Received 31 January 2009 Received in revised form 1 May 2009 Accepted 17 May 2009 Keywords: Geometric pattern matching Hausdorff distance Approximation Randomization We study input sensitive algorithms for point pattern matching under various transformations and the Hausdorff metric as a distance function. Given point sets P and Q in the plane, the problem of point pattern matching is to determine whether P is similar to some portion of Q, where P may undergo trans- formations from a group G of allowed transformations. All algorithms are based on methods for extracting small subsets from Q that can be matched to a small subset of P. The runtime is proportional to the number k of these subsets. Let d be the number of points in P that are needed to define a transformation in G. The key observation is that for some set B ⊂ P of cardinality larger than d, the number of subsets of Q of this cardinality that match B, is practically small, as the problem becomes more constrained. We present methods to extract efficiently all these subsets in Q. We provide algorithms for homothetic, rigid and similarity transformations in the plane and give a general method that works for any dimension and for any group of transformations. The runtime of our algorithms depends roughly linearly on the number of subsets k, in addition to an n log n factor. Thus our approximate matching algorithms run roughly in time O(n log n + km log n), where m and n are the number of points in P and Q, respectively. The constants hidden in the big O vary depending on the group of transformations G. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction A central problem in pattern recognition, computer vision, and robotics, is the question of whether two point sets P and Q resemble each other. One approach to this problem, first used by Huttenlocher and Kedem [17] and further developed by Huttenlocher et al. [18], is based on the minimum Hausdorff distance between point sets in the plane. The Hausdorff distance between two point sets P and Q is defined as H(P, Q ) = max(h(P, Q ), h(Q , P)) where the directional distance from P to Q is h(P, Q ) = max p∈P min q∈Q d(p, q) Here, d(·, ·) represents a more familiar metric on points; for instance, the standard Euclidean metric (the L 2 metric) or the L 1 or L ∞ metrics.  This work was partly supported by the MAGNET program of the Israel Ministry of Industry and Trade (IMG4 consortium). ∗ Corresponding author at: Department of Computer Science, Ben Gurion University, Be'er Sheva, Israel. E-mail address: aiger@cs.bgu.ac.il (D. Aiger). 0031-3203/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.patcog.2009.05.014 The minimum Hausdorff distance problem is to find D(P, Q , G) = min T∈G h(T (P), Q ) where G is the group of allowed transformations. The pattern match- ing problem is, given a parameter  > 0, to find a transformation T ∈ G such that h(T (P), Q )  . Algorithms for solving the minimum Hausdorff distance exactly for transformations other than translation have high computation time even for rigid transformations. The best known algorithm for rigid transformations was given by Chew et al. [10], and requires O(m 3 n 2 log 2 mn) time, where |P|=m and |Q |=n. To overcome the high complexity of exact algorithms, approximation and randomization were considered by several authors for both the minimum Hausdorff distance and the pattern matching problems [3,8,9,14–16,20–22]. Let  ∗ be the minimum directional Hausdorff distance between P and Q. The approximate algorithm finds a transformation that maps P within distance (1 + ) ∗ from Q for some  > 0. The alignment scheme [19,15,14] is a common heuristic for geo- metric matching that works as follows: pick a small subset B of P, called a base. For every subset of Q of size |B| compute the trans- formation T that maps B to the respective subset of Q, and verify the quality of each transformation T by computing h(T (P), Q ), or by counting the number of points, p i in P, such that h(T (p i ), Q )  . Based on this scheme, Goodrich et al. [14] considered the