To definite and compute differential invariants, like curvatures, for triangular meshes (or polyhedral surfaces) is a key problem in CAGD and the computer vision. The Gaussian curvature and the mean curvature are determined by the differential of the Gauss map of the underlying surface. The Gauss map assigns to each point in the surface the unit normal vector of the tangent plane to the surface at this point. We follow the ideas developed in Chen and Wu Chen2 (2004) and Wu, Chen and Chi Wu (2005) to describe a new and simple approach to estimate the differential of the Gauss map and curvatures from the viewpoint of the gradient and the centroid weights. This will give us a much better estimation of curvatures than Taubin's algorithm Taubin (1995).
 An empirical study of the accuracy of five different curvature estimation techniques, using synthetic range images and images obtained from three range sensors, is presented. The results obtained highlight the problems inherent in accurate estimation of curvatures, which are second-order quantities, and thus highly sensitive to noise contamination. The numerical curvature estimation methods are found to perform about as accurately as the analytic techniques, although ensemble estimates of overall surface curvature such as averages are unreliable unless trimmed estimates are used. The median proved to be the best estimator of location. As an exception, it is shown theoretically that zero curvature can be fairly reliably detected, with appropriate selection of threshold values. > This paper takes a systematic look at calculating the curvature of surfaces represented by triangular meshes. We have developed a suite of test cases for assessing the sensitivity of curvature calculations, to noise, mesh resolution, and mesh regularity. These tests are applied to existing discrete curvature approximation techniques and three common surface fitting methods (polynomials, radial basis functions and conics). We also introduce a modification to the standard parameterization technique. Finally, we examine the behaviour of the curvature calculation techniques in the context of segmentation. Approximations to the surface normal and to the Gaussian curvature of a smooth surface are often required when the surface is defined by a set of discrete points. The accuracy of an approximation can be measured using asymptotic analysis. The errors of several approximations to the surface normal and to the Gaussian curvature are compared. © 2000 Elsevier Science B.V. All rights reserved. This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures 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 formulations. 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 accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.