A Compound Gaussian Least Squares Algorithm and Unrolled Network for Linear Inverse Problems

TL;DR

This paper introduces a novel iterative algorithm based on a compound Gaussian prior for linear inverse problems and develops an unrolled neural network from it, both outperforming existing methods in image reconstruction tasks.

Contribution

The paper presents a new compound Gaussian least squares algorithm and its unrolled neural network version, offering interpretable layers and improved performance in inverse problems.

Findings

Both algorithms outperform state-of-the-art methods.

The unrolled network achieves better results with less training data.

Theoretical insights explain the algorithms' effectiveness.

Abstract

For solving linear inverse problems, particularly of the type that appears in tomographic imaging and compressive sensing, this paper develops two new approaches. The first approach is an iterative algorithm that minimizes a regularized least squares objective function where the regularization is based on a compound Gaussian prior distribution. The compound Gaussian prior subsumes many of the commonly used priors in image reconstruction, including those of sparsity-based approaches. The developed iterative algorithm gives rise to the paper's second new approach, which is a deep neural network that corresponds to an "unrolling" or "unfolding" of the iterative algorithm. Unrolled deep neural networks have interpretable layers and outperform standard deep learning methods. This paper includes a detailed computational theory that provides insight into the construction and performance of both algorithms. The conclusion is that both algorithms outperform other state-of-the-art approaches to tomographic image formation and compressive sensing, especially in the difficult regime of low training.