Theory: Fundamentals of functional analysis, calculus of variations and convex analysis; continuous and discrete image models.
Algorithms: Proximal-point and splitting methods
Applications: Convex models in image processing (denoising, deblurring, decompression, MRI, tomography etc.)
The focus of this lecture lies in modern methods for addressing typical problems in image processing such as denoising, deblurring, image reconstruction from incomplete measurements etc. We concentrate on the so-called variational methods. The lecture is divided into four main chapters on the basics of calculus of variations and of convex analysis, on numerical algorithms, on image models and on image reconstruction models. During the exercises, the understanding of the presented material should be deepened with a focus on the numerical implementation.
The content of the lecture is targeted at students of mathematics and scientific computing with a long-term interest in mathematical imaging, to prepare them for more advanced topics closer to research. The lecture notes are available and are self-contained and basic mathematical tools from functional and convex analysis will be provided. In an effort to help students draw relationships between the theoretical concepts and practical applications, the course is accompanied by an optional programming project.