Robust shape approximation and mapping between surfaces
Approximation robuste de surfaces avec garanties
Résumé
This thesis is divided into two independent parts.In the first part, we introduce a method that, given an input tolerance volume, generates a surface triangle mesh guaranteed to be within the tolerance, intersection free and topologically correct. A pliant meshing algorithm is used to capture the topology and discover the anisotropy in the input tolerance volume in order to generate a concise output. We first refine a 3D Delaunay triangulation over the tolerance volume while maintaining a piecewise-linear function on this triangulation, until an isosurface of this function matches the topology sought after. We then embed the isosurface into the 3D triangulation via mutual tessellation, and simplify it while preserving the topology. Our approach extends toDépôt de thèseDonnées complémentairessurfaces with boundaries and to non-manifold surfaces. We demonstrate the versatility and efficacy of our approach on a variety of data sets and tolerance volumes.In the second part we introduce a new approach for creating a homeomorphic map between two discrete surfaces. While most previous approaches compose maps over intermediate domains which result in suboptimal inter-surface mapping, we directly optimize a map by computing a variance-minimizing mass transport plan between two surfaces. This non-linear problem, which amounts to minimizing the Dirichlet energy of both the map and its inverse, is solved using two alternating convex optimization problems in a coarse-to-fine fashion. Computational efficiency is further improved through the use of Sinkhorn iterations (modified to handle minimal regularization and unbalanced transport plans) and diffusion distances. The resulting inter-surface mapping algorithm applies to arbitrary shapes robustly and efficiently, with little to no user interaction.
Cette thèse comprend deux parties indépendantes.Dans la première partie nous contribuons une nouvelle méthode qui, étant donnée un volume de tolérance, génère un maillage triangulaire surfacique garanti d’être dans le volume de tolérance, sans auto-intersection et topologiquement correct. Un algorithme flexible est conçu pour capturer la topologie et découvrir l’anisotropie dans le volume de tolérance dans le but de générer un maillage de faible complexité.Dans la seconde partie nous contribuons une nouvelle approche pour calculer une fonction de correspondance entre deux surfaces. Tandis que la plupart des approches précédentes procède par composition de correspondance avec un domaine simple planaire, nous calculons une fonction de correspondance en optimisant directement une fonction de sorte à minimiser la variance d’un plan de transport entre les surfaces
Origine : Version validée par le jury (STAR)