Abstract

With growing computational resources on the one hand, and data acquisition strategies approaching physical limits on the other hand, mathematical methods are nowadays indispensable for achieving state of the art results for concrete applications in image processing. There, image reconstruction typically amounts to solve ill-posed operator equations, and variational methods and regularisation are crucial to obtain stable and hence numerically feasible solution schemes.

When dealing with image data, regularisation, besides allowing for stability, also aims to incorporate expected structures of the image-representing functions one aims to recover. Prominent examples of such structures are jump discontinuities, corresponding to sharp edges, and smooth regions. While the former can be modelled well with the popular total variation functional, which penalises the Radon norm of the first order distributional derivative, the latter requires to incorporate higher order differentiation. In that respect, a main difficulty is to do this in a way such that jump discontinuities can still be recovered.

Addressing this challenge, this course will cover analytical aspects and concrete applications of higher order regularisation approaches in imaging. Starting from the total variation functional, we will consider different extensions from the perspective of modelling image data and achieving a regularisation effect in ill-posed problems. After establishing the fundamental theory, we will also deal with a numerical realisation and different applications in medical imaging and other disciplines.