Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

Application of discrete curvatures to surface mesh segmentation and feature line extraction

Alexandra Bac, Jean-Luc Mari, Marc Daniel

Slides are available by following the link.

We present two applications of discrete curvatures for surface mesh processing. The first one deals with simplifying a mesh while preserving its sharp features. Through the quadratic error metric introduced by Garland et al., such a simplification can be performed by an edge collapse process guided by the metric. Such an approach leads to high quality simplification but remains slow and costly both in terms of space and time. We introduce a two-step method in which we perform an initial adaptive cell segmentation guided by the curvature and direction of each cell (computed by PCA). This pre-segmentation according to local curvatures preserves the quality of simplified meshes while reducing computing time by a factor 3 to 4.

The second application can be considered as a dual problem, as we investigate ways to detect feature lines within a mesh. Robust extraction of the feature lines of a 3D surface model is a challenging problem. Classical approaches generally rely on curvature derivatives, leading to the detection of a salient part as multiple segments despite the fact that it visually appears as a single and fully connected element. We propose a two-step method aiming at extracting feature lines on 3D meshes with connectivity preservation. First, all the mesh vertices are labeled according to their curvature values in order to construct regions of interest on the discrete surface. The second step consists in a skeletonization directly on the mesh that corresponds to a homotopic thinning of the previously binarized areas. Consequently, the resulting lines are highly connected due to the topological properties of the thinning operator.

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