Curve Space: Classifying Curves On Surfaces

Abstract

We design signatures for curves defined on genus zero surfaces. The signature classifies curves according to the conformal geometry of the given curves and their embedded surface. Based on Teichmuller theory, our signature describes not only the curve shape but also the intrinsic relationship between the curve and its embedded surface. Furthermore, the signature metric is stable, it is close to identity between surfaces sharing similar Riemannian geometry metrics. Based on this, we propose a surface matching framework: first, with curve signatures, we match the partitioning of two surfaces defined by simple closed curves on them; second, the segmented subregions are pairwisely matched and then compared on canonical planar domains. 1. Introduction. Shape analysis and shape comparison are fundamental prob- lems in computer vision, graphics and modeling fields with many important appli- cations. Lots of 2D and 3D shape analysis techniques have been developed in the past couple of decades, most of which are based on comparing curvature or spacial positions of the points on the curve. A complete different way is to consider all the closed curves on the surface. The curve space on surface conveys rich geometric information of the surface itself and is easy to process. The philosophy of analyzing shapes by their associated curve spaces has deep root in algebraic topology (8), infinite dimensional Morse theory (18) and Teichmuller space theory in complex geometry (31). Suppose M is a surface (a 2-manifold), a closed curve on M is a map