Tangential Approximation of Surfaces

Abstract

In the Computer Vision community it is a common belief that higher order smoothness, such as curvature, should be modeled using higher order interactions. For example, 2nd order derivatives for deformable (active) contours are represented by triple cliques. Similarly, the 2nd order regularization methods in stereo predominantly use MRF models with scalar (1D) disparity labels and triple clique interactions. In this paper we give an overview of an energy minimization framework for tangential approximation of surfaces developed in [21, 22]. The framework uses higher dimensional labels to encode second order smoothness with pairwise interactions. Hence, many generic optimization algorithms (e.g. message passing, graph cut, etc.) can be used to optimize the proposed regularization functionals. The accuracy of our approach for representing curvature is demonstrated by theoretical and empirical results on real data sets from multi-view reconstruction and stereo.