Many segmentation algorithms in medical image analysis are based on Bayesian modeling, in which generative image models are constructed and subsequently “inverted” to obtain automated segmentations. Such methods have a prior
that makes predictions about where anatomical structures typically occur throughout the image, such as Markov random field models or probabilistic atlases [1
]. They also include a likelihood
term that models the relationship between segmentation labels and image intensities, often incorporating explicit models of imaging artifacts [3
]. Once the prior and likelihood have been specified, segmentation of a particular image proceeds by inferring the posterior distribution over all possible segmentations using Bayes’ rule, and searching for the segmentation that maximizes this posterior, or estimating the volumes of specific structures.
Although these methods are clearly “Bayesian”, an issue that is usually overlooked is that they only apply Bayesian analysis in an approximate
sense. In particular, these models typically have many free parameters for which suitable values are unknown a priori. In a true Bayesian approach, such parameters need to be integrated over when inferring the segmentation posterior. But, in practice, their optimal
values are first estimated and only the resulting point estimates are used to compute the segmentation posterior instead. In recent years generative models have started to include deformable registration methods that warp probabilistic atlases into the domain of the image being analyzed, often adding thousands of free parameters to the model [4
]. Since many plausible atlas warps beside the truly optimal one may exist, computing segmentations based on a single warp may lead to biased results. Furthermore, the numerical optimizers computing such high-dimensional atlas warps may not necessarily find the global optimum, further contributing to segmentation errors.
In this paper, we investigate the effect of using a more accurate approximation of the segmentation posterior in Bayesian segmentation models than the point estimates of the free model parameters. In particular, we will approximate the integral over atlas deformations in a recently proposed method for hippocampal subfield segmentation [7
] using Markov chain Monte Carlo (MCMC) sampling, and compare the results to those obtained using the most probable warp only. We show that MCMC sampling yields hippocampal subfield volume estimates that better discriminate controls from subjects with Alzheimer’s disease, while providing informative “error bars” on those estimates as well.
To the best of our knowledge, the issue of integrating over free parameters in Bayesian segmentation models has not been addressed before in the literature. The closest work related to the techniques used in this paper infers the posterior distribution of deformation fields in the context of computing location-specific smoothing kernels [8
], quantifying registration uncertainties [9
], or constructing Bayesian deformable models [10