Advanced Image Reconstruction and Denoising Bregmanized ( Higher Order ) Total Variation and Application in PET

In this thesis (higher order) total variation regularization methods are examined in the context of image reconstruction and denoising. Therefore, variational schemes consisting of a data fidelity term and a regularization term, additionally weighted with a constant, are considered. With the staircasing effect and the loss of contrast, two well known drawbacks of the total variation (TV) as regularizer are addressed and thus higher order extensions to prevent the former one, namely infimal convolution total variation (ICTV) and generalized total variation (GTV), are discussed. We provide a detailed comparison of the functionals and carry out an investigation of exact recovery based on the concept of eigenfunctions. In particular, we present a function which is an eigenfunction of GTV but not an eigenfunction of ICTV. In the case of total variation the concept of Bregman iterations is capable to compensate for the loss of contrast. We thus extend this approach to GTV, where it is additionally able to compensate for an distinctive effect of GTV: the slight tilting of former constant parts in certain cases. A quasi Newton method based on penalty respectively barrier approximation is introduced to solve the resulting variational problems. In addition to the proof of convergence, it is shown that the arising Lagrange multipliers yield decompositions of the image and the gradient of the image, respectively, which lets them serve as excellent edge detectors. The presented methods are finally applied to image reconstruction in positron emission tomography (PET). In particular in cases where standard reconstruction algorithms (as e.g. EM) only produce noisy images, extensions with TV and GTV regularization, further improved by Bregman iterations, produce superior results.