Triangle Mesh Duality: Reconstruction and Smoothing

Current scan technologies provide huge data sets which have to be processed considering several application constraints. The different steps required to achieve this purpose use a structured approach where fundamental tasks, e.g. surface reconstruction, multi-resolution simplification, smoothing and editing, interact using both the input mesh geometry and topology. This paper is twofold; firstly, we focus our attention on duality considering basic relationships between a 2-manifold triangle mesh M and its dual representation AT. The achieved combinatorial properties represent the starting point for the reconstruction algorithm which maps M' into its primal representation M, thus defining their geometric and topological identification. This correspondence is further analyzed in order to study the influence of the information in M and M for the reconstruction process. The second goal of the paper is the definition of the "dual Laplacian smoothing", which combines the application to the dual mesh M of well-known smoothing algorithms with an inverse transformation for reconstructing the regularized triangle mesh. The use of M instead of M exploits a topological mask different from the 1-neighborhood one, related to Laplacian-based algorithms, guaranteeing good results and optimizing storage and computational requirements.