Discrete Differential-Geometry Operators in nD

This paper provides a consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures, on arbitrary 2D and 3D meshes embedded in n dimensions. We present a consistent derivation of these first and second order differential properties using Voronoi cells and the mixed FiniteElement/Finite-Volume method. The new operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect the intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We finally present applications of these operators, such as mesh smoothing and enhancement, quality checking, and denoising of arbitrary vectors fields, such as flow fields or tensor images.