Geometric Matching Algorithms for Two Realistic Terrains

Abstract

We consider a geometric matching of two realistic terrains, each of which is modeled as a piecewise-linear bivariate function. For two realistic terrains f and g where the domain of g is relatively larger than that of f, we seek to find a translated copy $$f'$$ of f such that the domain of $$f'$$ is a sub-domain of g and the $$L_\infty $$ or the $$L_1$$ distance of $$f'$$ and g restricted to the domain of $$f'$$ is minimized. In this paper, we show a tight bound on the number of different combinatorial structures that f and g can have under translation in their projections on the xy-plane. We give a deterministic algorithm and a randomized algorithm that compute an optimal translation of f with respect to g under $$L_\infty $$ metric. We also give a deterministic algorithm that computes an optimal translation of f with respect to g under $$L_1$$ metric.