A Metric on Shape Space with Explicit Geodesics

TL;DR

This paper introduces a new metric on plane curves that aligns with classical manifolds, allowing explicit geodesic computation and curvature analysis, with applications to shape comparison and analysis.

Contribution

It defines a novel metric on shape space that isometric to classical manifolds, enabling explicit geodesic and curvature calculations for shape analysis.

Findings

Explicit geodesics are derived in parametric and quotient spaces.

The sectional curvature of the shape space of closed curves is positive.

Experimental algorithms for computing minimal geodesics between curves are presented.

Abstract

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided