Bayesian characterization of uncertainty in intra-subject non-rigid registration

Abstract

In settings where high-level inferences are made based on registered image data, the registration uncertainty can contain important information. In this article, we propose a Bayesian non-rigid registration framework where conventional dissimilarity and regularization energies can be included in the likelihood and the prior distribution on deformations respectively through the use of Boltzmann’s distribution. The posterior distribution is characterized using Markov Chain Monte Carlo (MCMC) methods with the effect of the Boltzmann temperature hyper-parameters marginalized under broad uninformative hyper-prior distributions. The MCMC chain permits estimation of the most likely deformation as well as the associated uncertainty. On synthetic examples, we demonstrate the ability of the method to identify the maximum a posteriori estimate and the associated posterior uncertainty, and demonstrate that the posterior distribution can be non-Gaussian. Additionally, results from registering clinical data acquired during neurosurgery for resection of brain tumor are provided; we compare the method to single transformation results from a deterministic optimizer and introduce methods that summarize the high-dimensional uncertainty. At the site of resection, the registration uncertainty increases and the marginal distribution on deformations is shown to be multi-modal.

Quantifying Registration Uncertainty With Sparse Bayesian Modelling

“…Collapsing temperature parameters λ, β when sampling regressor variables w is highly opportune. In the context of registration, Risholm et al [4] propose a MH scheme where marginalizing over temperature parameters induces the expensive computation of partition functions, for which an intricate procedure based on Laplace approximations is designed. In the proposed model, the computation of partition functions (specifically, marginal likelihoods, a.k.a.…”

“…Exploiting the Gaussian Markov random field structure inherited from a finite-element discretization of the domain, they characterize the posterior distribution of displacements by Gibbs sampling. Risholm et al [4] extend the approach to the case of unknown confidence on the observed data and on model priors respectively, aiming to address the critical issue of finding an objective trade-off between data fit and regularity-inducing priors. The so-called temperature hyperparameters are treated as latent variables and approximately marginalized over, while a Markov chain with full dimensional Metropolis-Hastings transitions traverses the space of transformation parameters.…”

Unified voxel- and tensor-based morphometry (UVTBM) using registration confidence

“…More generally, spatial uncertainty in pairwise registrations can be modeled in a Bayesian framework as the variance in the posterior distribution, because the posterior is defined as the distribution of deformation parameters. Risholm et al (2013) used computationally intensive Markov Chain Monte Carlo methods to sample the posterior distribution and estimate the uncertainty as the variance of parameters in this distribution, and Simpson et al (2012) proposed a more efficient variational Bayes model for the estimation of uncertainty.…”

“…For the sake of extensibility to arbitrary registration methods, we would like to use an approach of measuring uncertainty that is not computationally expensive (Risholm et al, 2013) or require a custom formulation and optimization of the registration (Simpson et al, 2012). Thus, instead of specific estimation of the posterior probability distribution, we represent the registration confidence as the residual spatial variability remaining after group-wise registration.…”

Accuracy Estimation for Medical Image Registration Using Regression Forests

“…Several methods have been suggested to estimate the registration accuracy, such as exploitation of the Bayesian posterior distribution [1] or based on the consistency of multiple registrations [2]. In the stochastic approaches Kybic [3] computed the registration uncertainty by performing multiple registrations with bootstrapping on the cost function samples to generate a set of registration solutions.…”