Geometry and Topology-based Segmentation of 2-Manifold Triangular Meshes in R3

Abstract

This manuscript reports a geometrical and a topological methods to segment a closed triangular 2-manifold mesh M ⊂ R3. The mesh M does not self-intersect) and has no border (i.e. watertight. Geometrical and topological segmentation methods require a Boundary Representation (BRep) from M. Building the BRep for M uniforms the triangle orientations, and makes explicit triangle and edge - counter edge adjacency. In the context of Reverse Engineering, the sub-meshes produced by the segmentation are subsequently used to fit parametric surfaces, which are in turn trimmed by the sub-mesh boundaries (forming FACEs). A Full Parametric Boundary Representation requires a seamless set of FACEs, to build watertight SHELLs. The fitting of parametric surfaces to the triangular sub-meshes (i.e. sub-mesh parameterization) requires quasi-developable sub-meshes.As a result, our geometric segmentation places 2 neighboring triangles in the same sub-mesh if their dihedral angle is π ± η for a small η (angle between their triangle normal vectors is a small η angle). On the other hand, our topological segmentation heuristic classifies triangles in a common sub-mesh if the value of the First eigenfunction of the triangulation graph Laplacian in these triangles falls in the same bin of a histogram formed with the eigenfunction values. The segmentation will obviously depend on the histogram bin distribution. The data sets processed indicate that geometrical segmentation is more convenient for mechanical parts with analytical surfaces. Conversely, topological segmentation works better for organic or artistic shapes. Future work is needed on the tuning of both the dihedral threshold η (for geometrical segmentation) and on the bin distributions of the eigenfunction (for topological segmentation).