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.

本研究采用 LDDMM functional 描述曲线的运动，在此基础上通过 Hamilton's equations 采用 Wu-Xu 元素求解曲线匹配问题，再利用贝叶斯反演解决动量问题，并采用集合 kalman 反演确认该方法可行。