Planar Curve Registration using Bayesian Inversion
This addresses curve matching in computer vision or medical imaging, but it appears incremental as it applies existing Bayesian and LDDMM methods with a specific finite element implementation.
The paper tackles the problem of parameterisation-independent closed planar curve matching by formulating it as a Bayesian inverse problem using large deformation diffeomorphic metric mapping (LDDMM) and solving it with ensemble Kalman inversion. It validates the approach with numerical examples, though no concrete performance numbers are provided.
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.