Scientific Computing for XRay Computed TomographyTechnical University of DenmarkGeneral course objectives: Xray Computed Tomography (CT) is used routinely in medicine, materials science and many other applications to reconstruct an object's interior using mathematical methods and numerical algorithms. This course focuses on the formulation, implementation, and use of standard reconstruction methods for CT such as Filtered Back Projection, Algebraic Iterative Reconstruction methods, and regularization methods. We give a rigorous mathematical description of the CT reconstruction problem, the associated mathematical formulations, and the underlying computational algorithms  supplemented with handson MATLAB computer exercises that illustrate these methods. The goal is that the participants will get a basic understanding of the formulation, implementation, and use of basic CT reconstruction algorithms, and thus be able to use them to perform data analysis for their own CT problems. Learning objectives: A student who has met the objectives of the course will be able to:
Contents: Introduction to CT and some of its applications. The CTscanner. The Radon transform and its inverse, Filtered Back Projection. Discretization of the CT problem. The Singular Value Decomposition (SVD) and its use for studying the CT problem. Stability and the need for filtering; truncated SVD. Algebraic iterative reconstruction algorithms  foundations and convergence properties. Their behavior for noisy data; semiconvergence and stopping rules. Block algebraic methods for largescale CT problems; the use of GPU computing. The software package ASTRA and its algebraic reconstruction algorithms. Noise models, priors and regularization. Variational formulations and Bayesian modeling. Cases: Total Variation and Tikhonov regularization. Introduction to convex optimization and numerical optimization algorithms. Artifacts in reconstructions and model calibration. 
