3D shape matching and Teichm\"uller spaces of pointed Riemann surfaces

TL;DR

This paper explores 3D shape matching through conformal geometry, introducing Teichmüller spaces of pointed Riemann surfaces as a unified framework, bridging theoretical foundations and practical applications.

Contribution

It presents a novel application of Teichmüller space geometry to 3D shape matching, integrating local and global conformal methods into multimedia analysis.

Findings

Teichmüller space provides a natural framework for 3D shape matching.

Local and global conformal approaches complement each other.

The geometric framework enhances shape comparison accuracy.

Abstract

Shape matching represents a challenging problem in both information engineering and computer science, exhibiting not only a wide spectrum of multimedia applications, but also a deep relation with conformal geometry. After reviewing the theoretical foundations and the practical issues involved in this fashinating subject, we focus on two state-of-the-art approaches relying respectively on local features (landmark points) and on global properties (conformal parameterizations). Finally, we introduce the Teichm\"uller space of n-pointed Riemann surfaces of genus g into the realm of multimedia, showing that its beautiful geometry provides a natural unified framework for three-dimensional shape matching.