The method of the approximate inverse for limited-angle CT
Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel (LARK) is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy, named constrained limited-angle reconstruction kernel (CLARK), by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.