Sparse Model-Based Reconstruction in Dynamic Positron Emission Tomography

This thesis deals with the inverse problem of dynamic positron emission tomography, which we attempt to solve with variational methods. Based on kinetic modeling we express the unknown image as a linear combination of a large dictionary, which contains the given basis functions. Since only a few of them are necessary to express the data for each pixel, we motivate the usage of `1,∞-regularization to promote sparsity of the coefficients. In a more general approach we analyze, which conditions are necessary for exact recovery of the coefficients. The main difficulty is that the basis functions are extremely similar. Thus, most of the known conditions for exact recovery are not applicable in this case. We provide three different splitting algorithms, which demonstrate that the use of `1,∞-regularization is a reasonable approach. In order to analyze these algorithms, we finally test them on artificial data.