Planar Curve Registration using Bayesian Inversion

• URL: http://arxiv.org/abs/2307.04909v1
• Date: Mon, 10 Jul 2023 21:26:43 GMT
• Title: Planar Curve Registration using Bayesian Inversion
• Authors: Andreas Bock and Colin J. Cotter and Robert C. Kirby
• Abstract summary: We study parameterisation-independent closed 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. We adopt ensemble Kalman inversion using a negative Sobolev mismatch penalty to measure the discrepancy between the target and the ensemble mean shape.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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