Abstract
Hamilton’s Ricci flow deforms the Riemannian metric proportional to the curvature, such that the curvature evolves according to a non-linear diffusion and reaction equation, eventually the curvature becomes constant everywhere. Ricci flow has been applied in the proof of Poincar´e’s conjecture. It is a powerful tool to design Riemannian metrics by prescribing curvatures, and valuable for many fundamental tasks related to geometric processing. Surface Ricci flow gives the solution to a highly non-linear geometric PDE: Yamabe’s equation. The discrete surface Ricci flow formulates the flow as the gradient flow of a convex energy, the existence and the uniqueness of the solution has been proved recently based on Teichm¨uller theory, which leads to a novel proof of classical surface uniformization theorem. The computational algorithm is based on the convex optimization, and the theoretic framework has been applied in graphics, vision, geometric modeling, networking and medical imaging fields.References
Demos and Data Samples
Surface Ricci flow source code with data samples can be downloaded from here.Gallery
The gallery for Surface Ricci flow can be found here.Abstract
Optimal mass transportation map transforms one probability measure to the other in the most economic way. If the domain and range are Euclidean regions, for L2 transportation cost, optimal transportation map is the gradient map of a convex energy. The equation governing the energy is the Monge-Ampere equation. In classical convex geometry, Monge-Ampere equation is closely related to Minkowski problem and Alexandrov problem. From this point of view, optimal mass transportation and Alexdrov problem are equivalent. This tutorial introduces a variational framework to solve the problem. A special convex energy is found, whose unique minimizer gives the optimal transportation map. The algorithm is closely related to upper envelope, power Voronoi diagram, power Delaunay triangulation in classical computational geometry. The discrete optimal mass transportation map offers a constructive proof for the Alexandrov theory in convex geometry. More importantly, discrete theories lead to computational algorithms, which can be applied to solve real problems, such as measure-preserving parameterization in graphics, surface/volume registration in vision, geometric clustering in pattern recognition and medical imaging.References
Demos and Data Samples
Surface and volume optimal transportation map code and data samples can be found here.