Non-uniform Geometric Matchings

Abstract

In this paper we introduce a generalization of the well studied class of geometric matching problems. The input to a geometric matching problem is usually two geometric objects P, Q drawn from a class of geometric objects $\mathcal{G}$, a transformation class $\mathcal{T}$ applicable to $\mathcal{G}$ and a distance measure $\operatorname{dist}_\mathcal{G}:\mathcal{G}\times\mathcal{G}\rightarrow\mathbb{R}^+$. The task is to compute the transformations $t\in\mathcal{T}$ minimizing $\operatorname{dist}_\mathcal{G}(t(P),Q)$.

Here, we extend this concept to non-uniform geometric matching problems. In this setting, a partition of P into k pieces P ₁,…,P _k is given and the task is to compute a sequence of transformations t ₁,…,t _k such that $\operatorname{dist}_\mathcal{G}(\bigcup_i t_i(P_i), Q)$ is minimized. But instead of solving k usual geometric matching problems independently and taking the maximum of the computed distances, the objective function of a non-uniform geometric matching problem also requires the computed transformations to be similar with respect to a suitable similarity measure on $\mathcal{T}$.

Computing a set of similar transformations to match an object P to Q allows to lower the influence of measurement errors and to model local deformations and has various applications, for example in medical navigation systems.

We present constant factor approximations and approximation schemes for point sequences under translations and constant factor approximations for point sets under translations.