Discrete & Computational Geometry
The upper envelope of Voronoi surfaces and its applications
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Convex Optimization
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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 G, a transformation class T applicable to G and a distance measure distG : G × G → R+. The task is to compute the transformations t ∈ T minimizing distG(t(P),Q). Here, we extend this concept to non-uniform geometric matching problems. In this setting, a partition of P into k pieces P1, . . . , Pk is given and the task is to compute a sequence of transformations t1, . . . , tk such that distG(∪i ti(Pi),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 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.