(Nonlocal) Total Variation in Medical Imaging

This thesis deals with the reconstruction of images where the measured data are corrupted by Poisson noise and a specific signal-dependent speckle noise, which occurs e.g. in medical ultrasound images. Since both noise types fundamentally differ from the frequently studied additive Gaussian noise in mathematical image processing, adapted variational models are required to handle these types of noise accurately. The first part of this thesis introduces variational regularization frameworks for inverse problems with data corrupted by Poisson and ultrasound speckle noise. Due to the strong nonlinearity of both data fidelity terms, a forward-backward splitting approach is used to provide efficient numerical schemes allowing the use of arbitrary convex regularization energies, in particular singular ones. Moreover, analytical results such as the well-posedness of the variational problems as well as the positivity preservation and convergence of the proposed iteration methods are proved. Finally, an iterative extension of both frameworks is proposed in order to refine the systematic errors of variational regularization techniques, using inverse scale space methods and Bregman distance iterations. The second part of this thesis considers the use of the (nonlocal) total variation functional as regularization energy in both previously developed frameworks. In particular, a modified version of the projected gradient descent algorithm of Chambolle and an augmented Lagrangian method are presented to solve the weighted (nonlocal) ROF model arising in both previously developed frameworks. In the case of the total variation regularization strategy, analytical results obtained previous in the general context of a convex regularization functional are carried over to the TV seminorm. In the case of the nonlocal regularization approach, a continuous framework of nonlocal derivative operators on directed graphs is introduced. This framework generalizes the nonlocal operators on undirected graphs in continuous and discrete setting and is consistent to the discrete local derivative operators. Finally, the performance of the proposed algorithms is illustrated by 2D and 3D synthetic and real data reconstructions. To validate the method proposed for inverse problems with data corupted by Poisson noise, simulated PET data (2D) and real cardiac H2 O (2D) and F-FDG (3D) PET measurements with low count rates are used. Additionally, a denoising and reconstruction comparison between TV and nonlocal TV regularization is presented using 2D synthetic Poisson data. In the case of denoising problems in medical US imaging, results on real patient data (2D) are illustrated.