  This paper is concerned with the development, analysis and numerical
realization of a novel variational 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 multi-layer convolutions and non-linear
penalties, and penalizes an associated cost. In contrast to conventional
neural-network-based approaches, however, the convolution 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
inpainting, denoising, deblurring under noise, super-resolution and JPEG
decompression with multiple test images.

Optimization and Control