A Reparameterisation Based Approach to Geodesic Constrained Solvers for Curve Matching

摘要

We present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane, targeted at robust registration between curves that require large deformations. This condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the Hamiltonian equations of motion; this means that each iteration of the nonlinear optimisation algorithm produces a geodesic (up to numerical discretisation). We ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition (conjugate momentum is normal to the curve). Explicitly introducing and solving for a reparameterisation variable allows the use of a point-to-point matching condition. The equations are discretised using the variational particle-mesh method. We provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations, to show that it is a viable, efficient and accurate method for obtaining geodesics between curves.