A Generative Variational Model for Inverse Problems in Imaging

TL;DR

This paper introduces a novel variational model inspired by generative neural networks for solving inverse imaging problems, which learns directly from data without prior training and is supported by rigorous mathematical analysis and numerical experiments.

Contribution

It presents a new generative variational model that learns convolution kernels directly from data, avoiding traditional training, with theoretical analysis and practical algorithms for inverse imaging tasks.

Findings

The model achieves effective inpainting, denoising, and super-resolution.

Mathematical proofs ensure regularity, stability, and convergence.

Numerical results demonstrate competitive performance across tasks.

Abstract

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.