Image reconstruction from downsampled and corrupted measurements, such as fast MRI and incomplete data for CT, is mathematically ill-posed inverse problem. In this talk, I will discuss two data driven approaches taking different directions of solving image reconstruction problem: Bayesian Inversion and Deep learning.

Bayesian inference methods have been popular for inverse problems due to the abilities of characterizing the uncertainty of solutions. We proposed a hybrid prior distribution which combines the nonlocal total variation regularization and the Gaussian measure, which results in a well-behaved posterior distribution in infinite dimension. The proposed priori also provides the flexibility to incorporate structure information from a reference image. We applied the prior to solve limited tomography reconstruction problem with difficulties of severe data missing. Both MAP and CM estimates are computed through two efficient methods and the uncertainty level are quantified based on the posterior distribution. The numerical experiments validate the advantages and feasibility of the proposed NLTG prior.

The second method is based on Deep learning and operator splitting. We propose to train a network to refine intermediate images from classical reconstruction procedure to the ground truth, i.e. the intermediate images that satisfy the data consistence will be fed into some chosen denoising networks or generative networks for denoising and removing artifact in each iterative stage. The proposed approach involves only techniques of conventional image reconstruction and usual image representation/denoising deep network learning. Extensive experiments on MRI reconstruction applied with both stack auto-encoder networks and generative adversarial nets demonstrate the efficiency and accuracy of the proposed method compared with other image reconstruction algorithms.