A TGV-Based Framework for Variational Image Decompression, Zooming, and Reconstruction. Part I: Analytics

A variational model for image reconstruction is introduced and analyzed in function space. Specific to the model is the data fidelity, which is realized via a basis transformation with respect to a Riesz basis followed by interval constraints. This setting in particular covers the task of reconstructing images constrained to data obtained from JPEG or JPEG 2000 compressed files. As image prior, the total generalized variation (TGV) functional of arbitrary order is employed. The present paper, the first of two works that deal with both analytical and numerical aspects of the model, provides a comprehensive analysis in function space and defines concrete instances for particular applications. A new, noncoercive existence result and optimality conditions, including a characterization of the subdifferential of the TGV functional, are obtained in the general setting.