# Probabilistic Diffeomorphic Registration: Representing Uncertainty

## TL;DR

This paper introduces a Bayesian variational framework for representing uncertainty in large deformation diffeomorphic image registration, enabling estimation of the full posterior distribution of deformations.

## Contribution

It proposes a novel SDE-based Gaussian process prior for deformations, allowing direct approximation of the full posterior distribution instead of MAP or Monte Carlo methods.

## Key findings

- Effective uncertainty quantification demonstrated on landmark-based registration.
- Applicable to simulated and real 3D medical images.
- Provides a probabilistic interpretation of deformation fields.

## Abstract

This paper presents a novel mathematical framework for representing uncertainty in large deformation diffeomorphic image registration. The Bayesian posterior distribution over the deformations aligning a moving and a fixed image is approximated via a variational formulation. A stochastic differential equation (SDE) modeling the deformations as the evolution of a time-varying velocity field leads to a prior density over deformations in the form of a Gaussian process. This permits estimating the full posterior distribution in order to represent uncertainty, in contrast to methods in which the posterior is approximated via Monte Carlo sampling or maximized in maximum a-posteriori (MAP) estimation. The frame-work is demonstrated in the case of landmark-based image registration, including simulated data and annotated pre and intra-operative 3D images.