Differential quantities, including normals, curvatures, principal directions, and associated matrices, play a fundamental role in geometric processing and physics-based modeling. Computing these differential quantities consistently on surface meshes is important and challenging, and some existing methods often produce inconsistent results and require ad hoc fixes. In this paper, we show that the computation of the gradient and Hessian of a height function provides the foundation for consistently computing the differential quantities. We derive simple, explicit formulas for the transformations between the first- and second-order differential quantities (i.e., normal vector and curvature matrix) of a smooth surface and the first- and second-order derivatives (i.e., gradient and Hessian) of its corresponding height function. We then investigate a general, flexible numerical framework to estimate the derivatives of the height function based on local polynomial fittings formulated as weighted least squares approximations. We also propose an iterative fitting scheme to improve accuracy. This framework generalizes polynomial fitting and addresses some of its accuracy and stability issues, as demonstrated by our theoretical analysis as well as experimental results.
Many methods have been proposed for computing or estimating the first- and second-order differential quantities of a surface. In recent years, there have been significant interests in the convergence and consistency of these methods. We do not attempt to give a comprehensive review of these methods but consider only a few of them that are more relevant to our proposed approach; readers are referred to @cite_2 and @cite_1 for comprehensive surveys. Among the existing methods, many of them estimate the different quantities separately of each other. For the estimation of the normals, a common practice is to estimate vertex normals using a weighted average of face normals, such as area weighting or angle weighting. These methods in general are only first-order accurate, although they are the most efficient.
@cite_1 @cite_2
2155553113 2048436819
This paper takes a systematic look at methods for estimating the curvature of surfaces represented by triangular meshes. We have developed a suite of test cases for assessing both the detailed behavior of these methods, and the error statistics that occur for samples from a general mesh. Detailed behavior is represented by the sensitivity of curvature calculation methods to noise, mesh resolution, and mesh regularity factors. Statistical analysis breaks out the effects of valence, triangle shape, and curvature sign. These tests are applied to existing discrete curvature approximation techniques and common surface fitting methods. We provide a summary of existing curvature estimation methods, and also look at alternatives to the standard parameterization techniques. The results illustrate the impact of noise and mesh related issues on the accuracy of these methods and provide guidance in choosing an appropriate method for applications requiring curvature estimates. In a variety of practical situations such as reverse engineering of boundary representation from depth maps of scanned objects, range data analysis, model-based recognition and algebraic surface design, there is a need to recover the shape of visible surfaces of a dense 3D point set. In particular, it is desirable to identify and fit simple surfaces of known type wherever these are in reasonable agreement with the data. We are interested in the class of quadric surfaces, that is, algebraic surfaces of degree 2, instances of which are the sphere, the cylinder and the cone. A comprehensive survey of the recent work in each subtask pertaining to the extraction of quadric surfaces from triangulations is presented.