Discrete curvatures of triangle meshes: From approximation of smooth surfaces to digital terrain data

Abstract

The emergence of high resolution LiDAR measurements raises the challenge of extracting accurate geometric information from 3D point cloud data. The standard approach is to generate digital elevation models in the form of regular grids and to compute numerical descriptors such as slope and curvatures. An alternative is the use of triangular representations, which preserve many details that might be lost when passing to regular grids. It is therefore a natural question whether triangulated terrains benefit of appropriate counterparts for the Gaussian and mean curvatures. A possible answer might be provided by an interdisciplinary perspective: triangle meshes are popular in computer graphics and, in this case, several approaches were taken into account for defining suitable concepts of curvature. The aim of this study is to compare methods for computing curvatures of triangle meshes, with an emphasis on the case of terrain data. We first consider several particular surfaces and a synthetic terrain generated by a smooth function and we assess whether the methods provide values that are correlated to the true smooth curvatures. Then we address two study cases relying on point clouds acquired through in situ measurements and we compare the methods with the regular grid approach. The outcomes of the numerical experiments are consistent to each other and lead to the same conclusions. Thus, for the Gaussian curvature the best approximation is provided by two methods derived from the Gauss–Bonnet theorem and by the jet fitting method. In the case of mean curvatures, one of the Gauss–Bonnet schemes, the method based on Euler’s theorem and the tensor approach yield the best matches with the reference methods. We also illustrate how curvatures for triangle meshes can highlight specific terrain features.