Limited Angle Computed Tomography using shearlets and neural networks

Within the medical field, mathematical sciences and computer skills are increasingly leading towards an intersection between each other, especially concerning the improvement of diagnostics. Computed tomography (CT) represents a very important tool, since it allows the visualization of internal areas of our body through the reconstruction of digital images. In this thesis, the ill-posed inverse problem that models 2D limited angle computed tomography is addressed with deep learning and shearlets techniques. The main idea of ​​our work is to start from the practical filtered backprojection (FBP) reconstruction and to post-process this solution using a U-net like convolutional neural network. The network is therefore implemented with the aim of reducing the characteristic artifacts of the FBP solution, caused by the limited angle setting. This approach is compared with a regularized solution that uses the shearlet transform in the l1 norm. The problem is then solved with the Alternating Direction Method of Multipliers (ADMM). The tests are performed in Python on two different datasets that we created ourselves. The results show that the deep learning approach drastically reduces computational times. Moreover, it allows to recover some boundary information of the images that with the iterative reconstruction would inevitably have gone lost.