Inverse Problems
Summer Semester 2023
The course will treat the classical theory for linear inverse problems as well as variational methods. Inverse problems occur in many applications in physics, engineering, biology and medical imaging. Loosely speaking, solving the forward problem consists of computing the outcome of a known model given the model parameters. The inverse problem consists of computation of the unknown parameter of interest given the physical model and noisy measurements of the outcome. One widely used approach are variational methods resulting in minimization problems. Typical examples are parameter identification problems such as computer tomography, deconvolution problems, denoising of images etc. In particular the following topics are discussed:
- Examples of ill-posed inverse problems
- Reconstruction in Computer tomography
- Ill-posed operator equations
- Regularization of linear inverse problems: filter methods, iterative methods
- Variational regularization methods: theory and algorithms
- Application of regularization methods to medical imaging applications
- Outlook: Variational methods for nonlinear inverse problems and connections to deed learning
Note: One half of the course will take place together with the course about (convex) optimization where the necessary theory and tools from convex analysis/optimization are provided which we will need to understand variational regularization methods. There is only one common Moodle course named "Convex optimization & Inverse problems" for both courses. Please register to this course in Moodle (different passwords for each course are provided in Stine).
Exercises:
- One exercises sheet every week;
- The exercises consits of both theoretical and computer (Matlab) exercises.
- You need to mark at least 60% of the overall exercises and 50% of the computer exercises.
Actual exercise sheet: check the Moodle course
Exams: 24./25.7.23 (first round), August (second round)
Literature:
- Engl, Hanke, Neubauer, Regularization of inverse problems
- Rieder, Keine Probleme mit inversen Problemen
- Mueller, Siltanen, Linear and nonlinear inverse problems with practical applications
- Scherzer et al, Variational methods in imaging
- H. Bauschke and P. Combettes, Convex analysis and Monotone Operator Theory in Hilbert Spaces
- J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples
Other useful material
- Nice introduction to inverse problems by Samuli Siltanen
- and to computer tomography CT (including videos)