Constructing reparametrization invariant metrics on spaces of plane curves

TL;DR

This paper introduces a family of reparametrization invariant Sobolev-type metrics on the space of plane curves, providing explicit formulas, curvature analysis, and an efficient numerical algorithm for computing geodesics between shapes.

Contribution

It develops a new class of metrics on shape space that are invariant under reparametrization, with explicit geodesic formulas and a practical algorithm for shape comparison.

Findings

Explicit geodesic distance formula for open curves

Non-negative sectional curvatures on unparametrized curves

Efficient numerical algorithm using RATTLE for shape matching

Abstract

Metrics on shape space are used to describe deformations that take one shape to another, and to determine a distance between them. We study a family of metrics on the space of curves, that includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space $\text{Imm}(S^1,\mathbb R^2)$ of parametrized plane curves and the quotient space $\text{Imm}(S^1,\mathbb R^2)/\text{Diff}(S^1)$ of unparametrized curves. For the space of open parametrized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parametrized and are non-negative on the space of unparametrized open curves. For the metric, which is induced by the "R-transform", we provide a numerical algorithm that computes geodesics between unparameterised, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests that demonstrate it's efficiency and robustness.