# Scientific Computing for X-Ray Computed Tomography

## Technical University of Denmark

General course objectives:
X-ray 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 hands-on 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:
• Understand the underlying physics of a CT scanner.
• Formulate the corresponding mathematical model and the Radon transform.
• Formulate the inverse Radon transform and the Filtered Back Projection algorithm.
• Discretize the Radon transform to obtain a system of linear equations.
• Use the singular value decomposition (SVD) to analyze the reconstruction problem.
• Formulate and use algebraic iterative methods that include simple constraints.
• Formulate the convergence behavior of these methods.
• Understand the basic ideas and implementation of block methods for large-scale problems.
• Use the software package ASTRA for large-scale problems.
• Formulate variational problems based on Bayesian noise modeling.
• Formulate and use Tikhonov regularization and Total Variation regularization.
• Use modern numerical methods from convex optimization for CT reconstruction.

Contents:
Introduction to CT and some of its applications. The CT-scanner. 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; semi-convergence and stopping rules. Block algebraic methods for large-scale 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.
 Lecturer Per Christian , Yiqiu , Jakob Sauer Place/Venue Anker Engelunds Vej 1 City Kgs. Lyngby Country Denmark ECTS 5 points Link http://www.kurser.dtu.dk/02946.aspx?menulangu... Language English Block-scheduling No Cost Not available