Interpolation of digital elevation models using AMLE and related methods

Interpolation of digital elevation models becomes necessary in many situations, for instance, when constructing them from contour lines (available e.g., from nondigital cartography), or from disparity maps based on pairs of stereoscopic views, which often leaves large areas where point correspondences cannot be found reliably. The absolutely minimizing Lipschitz extension (AMLE) model is singled out as the simplest interpolation method satisfying a set of natural requirements. In particular, a maximum principle is proven, which guarantees not to introduce unnatural oscillations which is a major problem with many classical methods. We then discuss the links between the AMLE and other existing methods. In particular, we show its relation with geodesic distance transformation. We also relate the AMLE to the thin-plate method, that can be obtained by a prolongation of the axiomatic arguments leading to the AMLE, and addresses the major disadvantage of the AMLE model, namely its inability to interpolate slopes as it does for values. Nevertheless, in order to interpolate slopes, we have to give up the maximum principle and authorize the appearance of oscillations. We also discuss the possible link between the AMLE and Kriging methods that are the most widely used in the geoscience literature. We end by numerical comparison between the different methods.