Ricci Flow In Computer Graphics

Surface conformal parameterization and surface retrieval are hot and fundamental problems in computer graphics field. In this paper, we introduces discrete Ricci flow as an efficient and powerful mathematics tools, by which we can computer conformal metric and conformal structure of the surface. By the conformal metric we can conformally parameter the surface to the plane and by the conformal structure we can compare different surfaces directly. At the same time, we compute the Teichmtiller coordinates of surfaces for tackling shape classification, comparison and retrieval problems.In the parameterization we introduce a novel metric based class. We think that conformal surface parameterization is equivalent to finding a proper Riemannian metric on the surface, such that the metric is conformal to the original metric and induces zero Gaussian curvature for all interior points. Discrete Ricci flow is a theoretic tool to compute such a conformal flat metric. Our algorithm can parameterize surfaces with different topological structures in an unified way. In addition, we can configure different target curvatures to get versatile parameterization results.In surface shape identification we introduce the uniform flat metric and the hyperbolic uniformization metric. We use discrete Ricci flow to find these two special metric from special prescribed target curvature. We show that these two metrics are intrinsic to the geometry of a surface. They can be applied to compute the fingerprints of surfaces for the purposes of shape identification and comparison.We proposes a novel approach with conformal classification of surfaces based on Teichmüller shape space theory. The coordinates of shape space are algebraically deduced from geodesic lengths of a set of special curves under a special metric, which is obtained by using curvature flow method. The well defined coordinates in the shape space can be applied as a shape descriptor, which is general for surfaces with arbitrary topologies, intrinsic to the geometric structures, and invariant under translation, rotation, scaling and isometric deformations.