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 tosome point b ∈ B such that the length of the longest edge in the matching is minimized (so-calledbottleneck distance). The geometric transformations we allow are translation, rotation, reflexionand scaling. In this paper, we give (1 + ε)-approximation algorithms for the case when the pointsare given in R, two of which run in O( 3.5ε4 logn) and O( nε4 log n log diam(B)dopt ) time, respectively,where diam(B) is the diameter of B and dopt is the bottleneck distance in an optimal matching.