Gauge Invariant Framework for Shape Analysis of Surfaces

TL;DR

This paper introduces a gauge-invariant framework for shape analysis of spherical surfaces by defining a Riemannian metric directly on shape space, enabling computation of geodesics that are invariant to surface parameterizations.

Contribution

It proposes a novel shape space Riemannian metric that is directly defined on the quotient space, simplifying geodesic computation and linking elastic metrics with mathematical theory.

Findings

Framework is invariant to surface parameterizations

Provides a geometric interpretation of elastic metrics

Demonstrates effectiveness on real and simulated 3D objects

Abstract

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energy that measures only the normal components of velocities along the path. In other words, this paper defines and solves for geodesics directly on the shape space and avoids complications resulting from the quotient operation. This comprehensive framework is invariant to arbitrary parameterizations of surfaces along paths, a phenomenon termed as gauge invariance. Additionally, this paper makes a link between different elastic metrics used in the computer science literature on one hand, and the mathematical literature on the other hand, and provides a geometrical interpretation of the terms involved. Examples using real and simulated 3D objects are provided to help illustrate the main ideas.