Browse Theses

Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces coherently and allow general planar algorithmic constructions to be generalized onto the surfaces directly. For example, all oriented surfaces have conformal structure. We can generalize planar texture mapping, texture synthesis, remeshing and mapping algorithms to surfaces based on their conformal structure without angle distortion. Also polar form splines with planar domains can be generalized to manifold splines on the surfaces which admit affine structure and are equipped with affine geometry.

This work presents theoretically rigorous and practically efficient methods for computing general surface geometric structures, including conformal structure, affine structure, hyperbolic structure, real projective structure, and spherical structure. The powerful tool we used is discrete surface Ricci flow. We generalized surface Ricci flow from continuous to discrete setting, and designed a series of algorithms to compute discrete surfaces Ricci flow, which includes discrete Euclidean Ricci flow, discrete hyperbolic Ricci flow, and discrete spherical Ricci flow.

We applied surface geometric structures computed from discrete surface Ricci flow to computer graphics, medical imaging, geometric modeling, and computer vision. We compute globally conformal parametrization for surfaces of general topologies, with less area distortion and control of both the number and location of singularity points; we conformally flatten colon surfaces onto plane, which enhances the navigation of virtual colonoscopy system; we design N-RoSy field on general surfaces based on flat metric induced from surfaces’ conformal structure; we construct manifold spline with single singularity using surface affine structure, which achieves the theoretical minimum of the singularity number; we combine manifold spline and T-spline to polycube T-spline by building polycube map of surface which naturally induces surface affine structure; we compute shape space for general surfaces, where Surfaces are indexed and classified by their conformal structure.