A Generative Variational Model for Inverse Problems in Imaging

This paper is concerned with the development, analysis, and numerical realization of a novel varia-tional model for the regularization of inverse problems in imaging. The proposed model is inspired by the architecture of generative convolutional neural networks; it aims to generate the unknown from variables in a latent space via multilayer convolutions and nonlinear penalties, and penalizes an associated cost. In contrast to conventional neural-network-based approaches, however, the con-volution kernels are learned directly from the measured data such that no training is required. The present work provides a mathematical analysis of the proposed model in a function space setting, including proofs for regularity and existence/stability of solutions, and convergence for vanishing noise. Moreover, in a discretized setting, a numerical algorithm for solving various types of inverse problems with the proposed model is derived. Numerical results are provided for applications in in -painting, denoising, deblurring under noise, superresolution, and JPEG decompression with multiple test images.