Geometry-aware topological decompositions of meshes

Topology captures a surface’s global features invariant to local deformation, and many geometry processing applications can benefit from topological information. However, traditional topological data analysis methods, e.g., persistent homology, when applied to surfaces, suffer from their massive computation cost and their lack of exact correspondence with surface geometry. In this dissertation, we use edge cycles as a compact representation of the surface’s topology and apply it in two topological decompositions of meshes. We propose an iterative method to localize tunnel and handle cycles, which respectively capture the surface's exterior and interior spaces. We then present the tori decomposition that segments the surface into genus-1 components. We formulate the tori decomposition as a min-cut problem in the dual graph and design geometry-aware edge weights to make the decomposition fit to the geometry. We also propose a framework to decompose the surface into contractible solids. Unlike previous methods which rely on volumetric representation, we solve the problem on the surface. We find a redundant set of cycles, which form an oversegmentation of the surface, and then we apply a dynamic programming method to merge the cells to form a contractible decomposition. All of our algorithms are based on efficient surface-embedded graph algorithms, and we demonstrate their robustness on numerous models