Introduction

Inverse problems in imaging arise in various contexts in biological and astronomical imaging such as super-resolution microscopy, MRI reconstruction, CAT reconstruction, and deconvolution. These problems are generally solved by two approaches. In the first approach, the inverse problem is solved by using off-the-shelf regularization based methods, e.g., Wavelet regularization, Total Variation regularization, Tikhonov regularization, and many others. These regularizations correspond to certain ad-hoc beliefs on the image characteristics which often lead to artefacts in the reconstructions. For example, first-order Total Variation promotes piece-wise constant estimates. In the second approach, training data containing typical measured images and the corresponding ground-truth images are collected and desired mapping from the measured image to the required image is learned. The map is generally modelled as Deep Neutral Network (DNN) and the parameters of the network are learned by minimizing an error metric corresponding to the error between the recovered estimate and the ground truth. Among these two classes of methods, the deep learning methods are believed to give better results in terms of various comparison metrics in various inverse problems; however, their practical use by the biological and medical community in sensitive applications has been limited. This is mainly because of Black-box like behaviour of the DNN. Also, due to the large number of parameters in the network, a substantial amount of training data is required for training the DNN which is also not readily publicly available. In this project, we adopt a middle path between the above two approaches. The project aims to learn a prior for the regularization terms which does not require as many parameters as the deep learning-based methods but is sufficiently flexible to avoid artefacts in the reconstruction. Towards this goal, we first explore the Total Generalized Variation (TGV) regularization as a framework, which generalizes the Total Variation regularization.