Learning the Invisible: Limited Angle Tomography, Shearlets and Deep Learning
Limited angle geometry is still a rather challenging modality in computed tomography (CT), in which entire boundary sections are not captured in the measurements making the reconstruction a severly ill-posed inverse problem. Compared to the standard filtered back-projection, iterative regularization-based methods help in removing artifacts but still cannot deliver satisfactory reconstructions. Based on the result that limited tomographic data sets reveal parts of the wavefront (WF) set in a stable way and artifacts from limited angle CT have directional properties, we present a hybrid reconstruction framework that combines model-based sparse regularization with data-driven deep learning. The core idea is to solve the compressed sensing formulation associated to the limited angle CT problem to recover the so called “visible” part of WF and learning via a convolutional neural network architecture the “invisble” ones, which provably cannot be handled by model-based methods. Such a decomposition into visible and invisible parts is achieved using the shearlet transform that allows to resolve WF sets in the phase space. Our numerical experiments show that our approach surpasses both pure model- and more data-based reconstruction methods, while offering an (heuristic) understanding of why the method works, providing a more reliable approach especially for medical applications.
2020-05-19 at 3:00 pm