Richard Schmähl

Zusammenfassung

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 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.

Zusammenfassung

Building on the well-known total variation, this paper develops a general regularization technique based on nonlinear isotropic diffusion (NID) for inverse problems with piecewise smooth solutions. The novelty of our approach is to be adaptive (we speak of A-NID), i.e., the regularization varies during the iterates in order to incorporate prior information on the edges, deal with the evolution of the reconstruction and circumvent the limitations due to the nonconvexity of the proposed functionals. After a detailed analysis of the convergence and well-posedness of the method, the latter is validated by simulations performed on synthetic and real data on computerized tomography.