Sampling Possible Reconstructions of Undersampled Acquisitions in MR Imaging With a Deep Learned Prior

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 capture the uncertainty in the inversion process. 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. We evaluate our method on two datasets; with images from the Human Connectome Project and in-house measured multi-coil images. We compare to five alternative methods. Results indicate that the proposed method produces images that match the measured k-space data better than the alternatives, while showing realistic structural variability. Furthermore, in contrast to the compared methods, the proposed method yields higher uncertainty in the undersampled phase encoding direction, as expected.