Triangle Mesh Duality: Reconstruction and Smoothing

Abstract

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 ${\mathcal M}$ and its dual representation ${\mathcal M}'$. The achieved combinatorial properties represent the starting point for the reconstruction algorithm which maps ${\mathcal M}'$ into its primal representation ${\mathcal M}$, thus defining their geometric and topological identification. This correspondence is further analyzed in order to study the influence of the information in ${\mathcal M}$ and ${\mathcal M}'$ 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 ${\mathcal M}'$ of well-known smoothing algorithms with an inverse transformation for reconstructing the regularized triangle mesh. The use of ${\mathcal M}'$ instead of ${\mathcal M}$ exploits a topological mask different from the 1-neighborhood one, related to Laplacian-based algorithms, guaranteeing good results and optimizing storage and computational requirements.