Geometric matching algorithms for two realistic terrains

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∞ or the L1 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 one that compute an optimal translation of f with respect to g under L∞ metric. We also give a deterministic algorithm that computes an optimal translation of f with respect to g under L1 metric.