Langevin equations for landmark image registration with uncertainty

TL;DR

This paper introduces a Bayesian approach using Langevin equations to model uncertainty in landmark-based image registration, providing a probabilistic framework for more robust shape analysis.

Contribution

It formulates Langevin equations for landmark registration, introduces computational approximations, and develops a Bayesian method to quantify uncertainty and average multiple landmark sets.

Findings

Provides a probabilistic model for landmark registration uncertainty

Develops computationally efficient approximations for Langevin equations

Enables Bayesian inference for shape averaging and uncertainty quantification

Abstract

Registration of images parameterised by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows one landmark set to the other. This is sometimes known as the geodesic interpolating spline and can be solved via a Hamiltonian boundary-value problem to give a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks.