Bayesian data assimilation in shape registration

TL;DR

This paper introduces a Bayesian approach to shape registration via geodesic curve matching, enabling the recovery of the posterior distribution of initial conditions and reparameterizations, and demonstrating the use of MCMC for uncertainty quantification.

Contribution

It develops a Bayesian framework for shape registration that explicitly models reparameterization and uses MCMC to characterize the full posterior distribution, including multimodal cases.

Findings

Posterior distributions can be multimodal and irregular.

MLEs identify high-density regions but do not capture full uncertainty.

Full posterior sampling improves understanding of shape registration uncertainty.

Abstract

In this paper we apply a Bayesian framework to the problem of geodesic curve matching. Given a template curve, the geodesic equations provide a mapping from initial conditions for the conjugate momentum onto topologically equivalent shapes. Here, we aim to recover the well-defined posterior distribution on the initial momentum which gives rise to observed points on the target curve; this is achieved by explicitly including a reparameterisation in the formulation. Appropriate priors are chosen for the functions which together determine this field and the positions of the observation points, the initial momentum $p_0$ and the reparameterisation vector field $\nu$, informed by regularity results about the forward model. Having done this, we illustrate how Maximum Likelihood Estimators (MLEs) can be used to find regions of high posterior density, but also how we can apply recently developed \SLC{Markov chain Monte Carlo (MCMC)} methods on function spaces to characterise the whole of the posterior density. These illustrative examples also include scenarios where the posterior distribution is multimodal and irregular, leading us to the conclusion that knowledge of a state of global maximal posterior density does not always give us the whole picture, and full posterior sampling can give better quantification of likely states and the overall uncertainty inherent in the problem.