Inverse Problems in Imaging - M1 - 6EC

Prerequisites
The course is aimed at Master and starting PhD students in Mathematics (and related studies like Physics and Technical Medicine) at the comprehensive as well as the technical universities. We expect a good working knowledge of 1. Calculus and Linear algebra 2. Functional analysis or numerical methods 
When in doubt, please contact the lecturers to discuss if you background is suitable for the course.


Aim of the course
This course is about inverse problems in imaging. The mathematical reconstruction and processing of images is of fundamental importance in state of the art applications in health and geosciences, e.g. in medical tomography, in high-resolution microscopy or in geophysical inversion. In many cases, underlying inverse problems can be formulated and solved using variational methods and partial differential equations. This course offers a theoretical as well as an applied insight into inverse problems and variational methods for mathematical imaging. It addresses reconstruction problems of different imaging modalities (e.g. CT or PET) in biomedicine and geophysics. The course covers the full chain of solving inverse problems in imaging, namely 

Problem identification → Modeling and discretization → Analysis → Numerical optimization.

Where variational principles, regularization theory and numerical optimization (scientific computing) form the underlying joint core. The course connects and extends upon the main concepts of basic courses on differential equations and numerics. The main learning goal for the students is to model, analyze and use state-of-the-art variational methods, PDEs and optimization techniques to solve challenging inverse problems in imaging. Upon completing this course, students achieved the following learning goals: 

  • Problem identification: Identification of imaging problems as mathematical inverse operator problems (e.g. integral equations, dynamical systems); 
  • Modeling and discretization: Problem formulation arising in applications using the language of nonlinear variational methods and partial differential equations; use Bayesian modeling to take data and model uncertainty into account; continuous versus discrete modeling;
  • Analysis: Understanding the main concepts of nonlinear regularization theory in an analogous way in PDEs as well as in variational methods, and how it influences existence (duality, weak topologies, Theorem of Banach-Alaoglu) and uniqueness results;
  • Numerical optimization: Formulating optimality conditions (variations) for constrained convex variational methods (saddle point problems) and to solve them via primal-dual methods or discretized higher-order methods. At the end of the course participants will be able to tackle inverse problems for imaging in biomedicine or geophysics with a new repertoire of state-of-the-art mathematical tools.

Rules about Homework & Exam

Homework (10 %): A total of 4 homework exercises will be assigned throughout the course. Homework must be handed in individually and may be discussed orally. 

Projects (30 %): Towards the end of the course students will work in project groups on specific inverse problems arising in imaging. In this way, the participants learn to work on specific imaging problem in a group applying all new problem solving skills. The participants present their results to each other as 10-minute poster pitches. A written report must be handed in as well. 

Exam (60 %): In the exam we will ask about all topics covered in class and their connection to the final project of the participant. The exam is thus a mix of theoretical questions and ‘essay’-style questions to allow the student to elaborate on the connection with their project. The exam may be discussed orally. At least a 5.0 in the exam is necessary to pass this course. 

In order to pass the course a student must:

  • score an average of at least 5.0 in the homework assignments;
  • score at least 5.0 in the final project; 
  • obtain a final mark of at least 5.5.

A student failing any of the conditions above will be given the opportunity to submit a retake homework assignment and/or a retake project. In particular:

An additional oral exam is scheduled, and homework assignments and/or a final project are submitted prior to this exam. The oral will concern material of the homework and/or final projects. For each piece of resubmitted or late-submitted assessed work, students can attain a maxmimum mark of 6.0.

The details of the final examination are as follows: Your final project is an individual project, for which you write a report, and present to your fellow students. Based on your report and the presentation we will ask a few questions, which according to the Mastermath rules is the formal oral examination. A prerequisite to give a final oral presentation is to have handed in the report for the final project in advance (see schedule).