A Metric for genus-zero surfaces

TL;DR

This paper introduces a new metric based on symmetric distortion energy for comparing genus-zero surfaces, establishing a conformal diffeomorphism that minimizes this energy and providing a mathematically rigorous way to measure surface similarity.

Contribution

The paper presents a novel metric for genus-zero surfaces based on symmetric distortion energy and proves the existence of optimal conformal diffeomorphisms minimizing this energy.

Findings

The symmetric distortion energy defines a valid metric on genus-zero surfaces.

Existence of a conformal diffeomorphism minimizing the energy is established.

Properties of the metric and diffeomorphisms are suitable for practical applications.

Abstract

We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing diffeomorphisms give a metric on the space of genus-zero Riemannian surfaces. This metric and the corresponding optimal diffeomorphisms are shown to have properties that are highly desirable for applications.