Sampling possible reconstructions of undersampled acquisitions in MR imaging

Undersampling the k-space during MR acquisitions saves time, however results in an ill-posed inversion problem, leading to an infinite set of images as possible solutions. Traditionally, this is tackled as a reconstruction problem by searching for a single "best" image out of this solution set according to some chosen regularization or prior. This approach, however, misses the possibility of other solutions and hence ignores the uncertainty in the inversion process. In this paper, we propose a method that instead returns multiple images which are possible under the acquisition model and the chosen prior. To this end, we introduce a low dimensional latent space and model the posterior distribution of the latent vectors given the acquisition data in k-space, from which we can sample in the latent space and obtain the corresponding images. We use a variational autoencoder for the latent model and the Metropolis adjusted Langevin algorithm for the sampling. This approach allows us to obtain multiple possible images and capture the uncertainty in the inversion process under the used prior. We evaluate our method on images from the Human Connectome Project dataset as well as in-house measured multi-coil images and compare to two different methods. The results indicate that the proposed method is capable of producing images that match the ground truth in regions where acquired k-space data is informative and construct different possible reconstructions, which show realistic structural variations, in regions where acquired k-space data is not informative. Keywords: Magnetic Resonance image reconstruction, uncertainty estimation, inverse problems, sampling, MCMC, deep learning, unsupervised learning.