Compressed Sensing MRI Reconstruction Regularized by VAEs with Structured Image Covariance

Learned regularization for MRI reconstruction can 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. One such approach uses generative models, trained on ground-truth images, as priors for inverse problems, penalizing reconstructions far from images the generator can produce. In this work, we utilize variational autoencoders (VAEs) 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. We demonstrate these novel generative regularizers on radially sub-sampled MRI knee measurements from the fastMRI dataset and compare them to other unlearned, unsupervised and supervised methods. 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.