Measure-Valued Variational Models with Applications in Image Processing

This thesis introduces the notion of measure-valued image processing models both as a mathematical framework for the study of images that take values in a space of probability measures, such as data from diffusion-weighted MRI, and as a global solution strategy for a broad class of image processing problems, such as image enhancement, inpainting and registration, that are often hard to solve globally optimally due to non-convexities. A novel total variation regularization energy is proposed for edge-preserving denoising and restoration of Q-ball data from diffusion-weighted MRI that naturally comes in the form of probability measures describing the direction of diffusivity of water in fibrous organic tissue. Furthermore, the global solution strategy that is known as functional lifting is adapted to a large class of manifold-valued imaging problems with an efficient discretization and implementation strategy based on finite elements. A generalization of functional lifting to energies that depend on second-order derivatives is proposed and its application to image registration problems is discussed. Finally, it is demonstrated that a large class of state-of-the-art functional lifting approaches including all of the previous models can be described through measure-valued image processing models whose mathematical description and analysis involves tools from functional analysis and the theory of optimal transport, especially dynamical models of optimal transportation.