Computing Teichmüller Shape Space

Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmüller theory. Two surfaces are conformal equivalent, if there exists a bijective angle-preserving map between them. The Teichmüller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to Identity. A curve in the Teichmüller space represents a deformation process from one class to the other. In this work, we apply Teichmüller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic, invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmüller coordinates is practical, stable and efficient. The algorithms for the Teichmüller coordinates of surfaces with positive or zero Euler numbers have been studied before. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with −1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmüller space