Gauge Invariant Framework for Shape Analysis of Surfaces

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.

15 pages, 11 Figures, to appear in IEEE Transactions on Pattern Analysis and Machine Intelligence in a better resolution

References in corpus (3)

• Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
• Constructing reparametrization invariant metrics on spaces of plane curves
• Precise Matching of PL Curves in $R^N$ in the Square Root Velocity Framework

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• Data-driven quasiconformal morphodynamic flows