Gaussian mixtures are classically used to model multi-modal data. In most state-of-the-art image analysis methods, they are the cornerstone of the segmentation component. Typically, in medical images, voxels are assumed to belong to a finite number of classes, which all have a particular intensity distribution modeled by a Gaussian. Since both class labelling and Gaussian parameters (mean and covariance) are unknown, methods usually rely on an Expectation-Maximisation scheme to iterate between computing the expected labels and computing the Gaussian parameters that maximise the model likelihood. The problem can be tackled in a more Bayesian way by placing a Normal-Wishart prior over the prameters of each Gaussian, in which case variational inference is used. This allows to regularise the parameter estimation by penalising unlikely values. Additionnaly, if a set of several images is segmented at once, it may be of interest to also optimise the Normal-Wishart prior parameters by, e.g., maximising the model evidence.
In two cases, it might be interesting to place an Inverse-Wishart prior on the Wishart prior parameters:
If all classes are expected to possess roughly similar variances, the Inverse-Wishart prior could be shared by all (prior) covariance matrices. This would penalise covariance matrices that deviate from the mean.
If images were acquired in different centers, or with different sequences, it makes sense to model this by associating a Normal-Wishart prior with each center/sequence and placing a shared Normal-Inverse-Wishart prior over all their parameters.
Created by Yaël Balbastre on 11 April 2018. Last edited on 13 April 2018.