Planar Curve Registration using Bayesian Inversion

TL;DR

This paper presents a Bayesian inversion approach for parameterisation-independent planar curve matching, utilizing a large deformation diffeomorphic metric mapping framework and ensemble Kalman inversion to accurately align curves.

Contribution

It introduces a novel Bayesian inversion method for curve registration that leverages the Wu-Xu finite element, improving efficiency and mesh-independence in the process.

Findings

Successfully matches planar curves using Bayesian inversion.

Demonstrates the effectiveness of the Wu-Xu element in curve matching.

Validates the approach with multiple numerical examples.

Abstract

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.