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.

翻译：本文研究参数化无关的闭合平面曲线匹配问题，将其建模为贝叶斯逆问题。曲线运动通过作用于环境空间的微分同胚群上的曲线来建模，由此引入惩罚形变动能的大型形变微分同胚度量映射（LDDMM）泛函。我们采用Wu-Xu有限元[吴思宇, 许进超, 用于$\mathbb{R}^n$单纯形网格上$2m^\text{th}$阶偏微分方程的非协调有限元空间——当$m=n+1$时，数学计算88(316)(2019)531-551]求解曲线匹配问题的哈密顿方程，该单元可为曲线正向运动提供网格无关的Lipschitz常数，并通过贝叶斯逆推求解动量逆问题。由于该单元非仿射等价，我们提出拉回理论以加速正向映射的实现与计算效率。采用负Sobolev范数失配惩罚的集合卡尔曼逆推方法，度量目标形状与集合平均形状之间的差异。通过多个数值算例验证了该方法的有效性。