This thesis deals with the development, analysis and application of a mathematical model for image reconstruction. The model is realized by minimizing the sum of two convex functionals, one ensuring data fidelity and the other being a regularization term. The novelty of the considered approach lies both in the general definition of the data term as well as the application of the non-standard Total Generalized Variation (TGV) functional for regularization in this context. The main part of the work is the extension of theory for the TGV functional, the definition and analysis of the general reconstruction model and its application to imaging tasks. In particular, existence of a solution and optimality conditions for the resulting minimization problem are obtained for a setting that covers diverse applications: Suitable problem formulations for JPEG and JPEG 2000 decompression as well as wavelet based zooming are defined in function space setting and numerical solution schemes for the discretized problems are developed.