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.

Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces

“…2 If one cannot invert the representation, it is not clear how to transfer geodesics and statistical analyses conducted in Q back to F, thereby removing one of the main motivations for introducing the SRNF. One can always pull the L 2 metric back to F under Q and perform computations there, as in [21], [22], but this is computationally expensive, and rather defeats the purpose of having an L 2 metric in the first place.…”

Gauge Invariance of Degenerate Riemannian Metrics

“…Barbara.Tumpach@math.univ-lille1.fr. 1 This note is based on joint work [1] What is a parameterization? What we mean mathematically by a parameterization of a spherical surface is a diffeomorphism from the unit sphere to this surface.…”

“…To accomplish such so-called gauge invariance, we defined a metric on the space of parameterized surfaces that is degenerate in the direction of reparameterization. 1 What are the surfaces under consideration? The surfaces we will consider in this note are surfaces which are diffeomorphic to the unit sphere.…”

Persistent homology to analyse 3D faces and assess body weight gain

“…their dependence on the adopted shape sampling scheme. Effective methods for applying ESA methodology for surface representation have been proposed in [31,32,48]. In the literature, computer vision tools have been applied to body analysis to estimate height, weight and other parameters enclosed in the body appearance, and most of the methods proposed are based on computing body measurements.…”