Approximate one-to-one point pattern matching

Given a set A = {a1,..., an} of n image points and a set B = {b1,..., bn} of n model points, the problem is to find a transformation matching (a one-to-one mapping) each point a ∈ A to some point b ∈ B such that the length of the longest edge in the matching is minimized (so-called bottleneck distance). The geometric transformations we allow are translation, rotation, reflexion and scaling. In this paper, we give (1 + ε)-approximation algorithms for the case when the points log n log diam(B)) time, respectively, dopt where diam(B) is the diameter of B and dopt is the bottleneck distance in an optimal matching. are given in R2, two of which run in O ( n3.5 ε4 log n) and O ( n2.5 ε