Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Abstract

This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and cur- vatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing for- mulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in ac- curacy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhance- ment, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.