Discrete geodesic calculus in the space of viscous fluidic objects

TL;DR

This paper introduces a computationally efficient discrete geodesic calculus for shape space, enabling shape morphing, extrapolation, and feature transfer for viscous fluidic objects using a novel dissimilarity measure.

Contribution

It develops a new discrete geodesic calculus based on a local approximation of the Riemannian metric, specifically designed for shape spaces of viscous objects, with applications to shape analysis.

Findings

Demonstrates topology-preserving shape morphing.

Shows efficient shape extrapolation via discrete geodesic flow.

Enables transfer of geometric features between shapes.

Abstract

Based on a local approximation of the Riemannian distance on a manifold by a computationally cheap dissimilarity measure, a time discrete geodesic calculus is developed, and applications to shape space are explored. The dissimilarity measure is derived from a deformation energy whose Hessian reproduces the underlying Riemannian metric, and it is used to define length and energy of discrete paths in shape space. The notion of discrete geodesics defined as energy minimizing paths gives rise to a discrete logarithmic map, a variational definition of a discrete exponential map, and a time discrete parallel transport. This new concept is applied to a shape space in which shapes are considered as boundary contours of physical objects consisting of viscous material. The flexibility and computational efficiency of the approach is demonstrated for topology preserving shape morphing, the representation of paths in shape space via local shape variations as path generators, shape extrapolation via discrete geodesic flow, and the transfer of geometric features.