Neumann Networks for Inverse Problems in Imaging

TL;DR

This paper introduces Neumann networks, a novel data-driven approach inspired by Neumann series, for solving ill-posed linear inverse problems in imaging, outperforming traditional and existing deep learning methods.

Contribution

The paper proposes a new Neumann network architecture that directly solves inverse problems using a truncated Neumann series and provides theoretical analysis of its approximation capabilities.

Findings

Neumann networks outperform traditional inverse solvers and deep learning methods.

Theoretical proof shows Neumann networks can approximate the optimal estimator.

Empirical results confirm the model's effectiveness on standard datasets.

Abstract

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory.