VAEs with structured image covariance applied to compressed sensing MRI.

Objective.This paper investigates how generative models, trained on ground-truth images, can be used as priors for inverse problems, penalizing reconstructions far from images the generator can produce. The aim is that learned regularization will provide complex data-driven priors to inverse problems while still retaining the control and insight of a variational regularization method. Moreover, unsupervised learning, without paired training data, allows the learned regularizer to remain flexible to changes in the forward problem such as noise level, sampling pattern or coil sensitivities in MRI.Approach.We utilize variational autoencoders that generate not only an image but also a covariance uncertainty matrix for each image. The covariance can model changing uncertainty dependencies caused by structure in the image, such as edges or objects, and provides a new distance metric from the manifold of learned images.Main results.We evaluate these novel generative regularizers on retrospectively sub-sampled real-valued MRI measurements from the fastMRI dataset. We compare our proposed learned regularization against other unlearned regularization approaches and unsupervised and supervised deep learning methods.Significance.Our results show that the proposed method is competitive with other state-of-the-art methods and behaves consistently with changing sampling patterns and noise levels.

Probabilistic non-linear registration with spatially adaptive regularisation.

This paper introduces a novel method for inferring spatially varying regularisation in non-linear registration. This is achieved through full Bayesian inference on a probabilistic registration model, where the prior on the transformation parameters is parameterised as a weighted mixture of spatially localised components. Such an approach has the advantage of allowing the registration to be more flexibly driven by the data than a traditional globally defined regularisation penalty, such as bending energy. The proposed method adaptively determines the influence of the prior in a local region. The strength of the prior may be reduced in areas where the data better support deformations, or can enforce a stronger constraint in less informative areas. Consequently, the use of such a spatially adaptive prior may reduce unwanted impacts of regularisation on the inferred transformation. This is especially important for applications where the deformation field itself is of interest, such as tensor based morphometry. The proposed approach is demonstrated using synthetic images, and with application to tensor based morphometry analysis of subjects with Alzheimer's disease and healthy controls. The results indicate that using the proposed spatially adaptive prior leads to sparser deformations, which provide better localisation of regional volume change. Additionally, the proposed regularisation model leads to more data driven and localised maps of registration uncertainty. This paper also demonstrates for the first time the use of Bayesian model comparison for selecting different types of regularisation.