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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. In addition, we give exercises in large-scale reconstruction of real CT data using the Python package Core Imaging Library (CIL). The goal is that participants will get a basic understanding of the formulation, implementation, and use of basic and advanced CT reconstruction algorithms, and thus be able to use them to perform data analysis for their own CT problems. As part of the course, participants will acquire their own X-ray CT data at the DTU 3D Imaging Facility and reconstruct it using the methods from the course.

Learning objectives:
A student who has met the objectives of the course will be able to:
  • Describe 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.
  • Use the software package Core Imaging Library (CIL) 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. The use of GPU computing. The software package Core Imaging Library (CIL) 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.

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Course organizer
Jakob Sauer , Per Christian
Place/Venue
Anker Engelunds Vej 1
City
2800 Kgs. Lyngby
Country
Denmark
Workload
5
Link
http://kurser.dtu.dk/course/02946