The variational particle-mesh method for matching curves

TL;DR

This paper introduces a novel particle-mesh discretisation method for diffeomorphic shape matching, ensuring conservation properties and improving the numerical representation of shape evolution in non-rigid registration tasks.

Contribution

It presents a new particle-mesh algorithm for shape matching that preserves key conservation properties and introduces a discretisation for geometric current matching.

Findings

The method conserves momenta related to particle-relabelling symmetry.

Numerical examples demonstrate the effectiveness of the discretisation.

The approach improves shape evolution accuracy in non-rigid registration.

Abstract

Diffeomorphic matching (only one of several names for this technique) is a technique for non-rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time-series of vector fields. One seeks the flow between two topologically-equivalent curves or surfaces which minimises some metric defined on the vector fields, \emph{i.e.} the flow closest to the identity in some sense. In this paper, we describe a new particle-mesh discretisation for the evolution of the geodesic flow and the embedded shape. Particle-mesh algorithms are very natural for this problem because Lagrangian particles (particles moving with the flow) can represent the movement of the shape whereas the vector field is Eulerian and hence best represented on a static mesh. We explain the derivation of the method, and prove conservation properties: the discrete method has a set of conserved momenta corresponding to the particle-relabelling symmetry which converge to conserved quantities in the continuous problem. We also introduce a new discretisation for the geometric current matching condition of (Vaillant and Glaunes, 2005). We illustrate the method and the derived properties with numerical examples.