Ricci Flow for 3D Shape Analysis

Abstract

Ricci flow is a powerful curvature flow method in geometric analysis. This work is the first application of surface Ricci flow in computer vision. We show that previous methods based on conformal geometries, such as harmonic maps and least-square conformal maps, which can only handle 3D shapes with simple topology are subsumed by our Ricci flow based method which can handle surfaces with arbitrary topology. Because the Ricci flow method is intrinsic and depends on the surface metric only, it is invariant to rigid motion, scaling, and isometric and conformal deformations. The solution to Ricci flow is unique and its computation is robust to noise. Our Ricci flow based method can convert all 3D problems into 2D domains and offers a general framework for 3D surface analysis. Large non-rigid deformations can be registered with feature constraints, hence we introduce a method that constrains Ricci flow computation using feature points and feature curves. Finally, we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration.