Maximum Overlap of Convex Polytopes under Translation

Abstract

We study the problem of maximizing the overlap of two convex polytopes under translation in R d for some constant d ≥ 3. Let n be the number of bounding hyperplanes of the polytopes. We present an algorithm that, for any ε > 0, finds an overlap at least the optimum minus ε and reports the translation realizing it. The running time is O(n d/2 +1 log d n) with probability at least 1 − n −O(1) , which can be improved to O(n log 3.5 n) in R 3. The time complexity analysis depends on a bounded incidence condition that we enforce with probability one by randomly perturbing the input polytopes. The perturbation causes an additive error ε, which can be made arbitrarily small by decreasing the perturbation magnitude. Our algorithm in fact computes the maximum overlap of the perturbed polytopes. The running time bounds, the probability bound, and the big-O constants in these bounds are independent of ε.

Shape matching under rigid motion

“…The running times of our algorithms are O(n 2 ε −3 log 1.5 n log(n/ε)) for the translation case, and O(n 3 ε −4 log 5/3 n log 5/3 (n/ε)) for the rigid motion case. The error is additive and it is at most ε • area(P ) with probability 1 − n −O (1) .…”

“…The linearity of determinant gives equation (1). Now, suppose that 1 and 2 are not parallel and s 3 is picked uniformly at random independent of s 1 and s 2 from a subset of R 2 with positive area.…”

“…By representing the overlap function as a sum of algebraic functions, Vigneron [16] devised an algorithm to compute a (1 − ε)-approximate maximum overlap in O(n 6 ε −3 ) time. 1 If the maximum overlap under rigid motion is not Ω(max{area(P ), area(Q)}), then P and Q are hardly similar and knowing this is often sufficient for shape matching. This motivates the approximation of the maximum overlap to within an additive error of ε•max{area(P ), area(Q)} or ε • min{area(P ), area(Q)}.…”

“…This motivates the approximation of the maximum overlap to within an additive error of ε•max{area(P ), area(Q)} or ε • min{area(P ), area(Q)}. Cheong et al [6] proposed algorithms to approximate the maximum overlap such that the additive error is ε•min{area(P ), area(Q)} with probability 1−n −O (1) . The running times are O(n 2 ε −4 log 2 n) time for the translation case and O(n 3 ε −8 log 5 n) time for the rigid motion case 2 , assuming that n ≥ ε −1 .…”

“…In R d for d ≥ 3, Ahn et al [1] showed that the maximum overlap of two convex polytopes under translation can be computed in O(n d/2 +1 log d n) time, where n is the total number of bounding hyperplanes, and the running time can be improved to O(n log 3.5 n) in R 3 . Vigneron's method [16] computes a (1 − ε)-approximate maximum overlap of two possibly non-convex polytopes under rigid motion in O n 2 ε log n ε d 2 /2+d/2+1 time, assuming that the two polytopes are represented by a union of n interior-disjoint d-simplices.…”