Planar Curve Registration using Bayesian Inversion
Andreas Bock Colin J. Cotter Robert C. Kirby
We study parameterisation-independent closed planar curve matching as a
Bayesian inverse problem. The motion of the curve is modelled via a curve on
the diffeomorphism group acting on the ambient space, leading to a large
deformation diffeomorphic metric mapping (LDDMM) functional penalising the
kinetic energy of the deformation. We solve Hamilton's equations for the curve
matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite
element spaces for $2m^\text{th}$ order partial differential equations on
$\mathbb{R}^n$ simplicial grids when $m=n+1$, Mathematics of Computation 88
(316) (2019) 531-551] which provides mesh-independent Lipschitz constants for
the forward motion of the curve, and solve the inverse problem for the momentum
using Bayesian inversion. Since this element is not affine-equivalent we
provide a pullback theory which expedites the implementation and efficiency of
the forward map. We adopt ensemble Kalman inversion using a negative Sobolev
norm mismatch penalty to measure the discrepancy between the target and the
ensemble mean shape. We provide several numerical examples to validate the
approach.