Incorporating prior information into inverse problems, e.g. via maximum-a-posteriori estimation, is an important technique for facilitating robust inverse problem solutions. In this paper, we devise two novel approaches for linear inverse problems that permit problem-specific statistical prior selections within the compound Gaussian (CG) class of distributions. The CG class subsumes many commonly used priors in signal and image reconstruction methods including those of sparsity-based approaches. The first method developed is an iterative algorithm, called generalized compound Gaussian least squares (G-CG-LS), that minimizes a regularized least squares objective function where the regularization enforces a CG prior. G-CG-LS is then unrolled, or unfolded, to furnish our second method, which is a novel deep regularized (DR) neural network, called DR-CG-Net, that learns the prior information. A detailed computational theory on convergence properties of G-CG-LS and thorough numerical experiments for DR-CG-Net are provided. Due to the comprehensive nature of the CG prior, these experiments show that DR-CG-Net outperforms competitive prior art methods in tomographic imaging and compressive sensing, especially in challenging low-training scenarios.

翻译：将先验信息纳入逆问题（例如通过最大后验估计）是促进鲁棒逆问题求解的重要技术。本文针对线性逆问题设计了两种新方法，允许在复合高斯（CG）分布类中进行问题特定的统计先验选择。CG类涵盖了信号与图像重构方法中许多常用先验，包括基于稀疏性的方法。第一种方法是一种迭代算法，称为广义复合高斯最小二乘法（G-CG-LS），它最小化一个正则化最小二乘目标函数，其中正则化项强制执行CG先验。随后将G-CG-LS展开或解折叠，以提供第二种方法——一种新颖的深度正则化（DR）神经网络，称为DR-CG-Net，用于学习先验信息。本文提供了G-CG-LS收敛特性的详细计算理论，并对DR-CG-Net进行了全面的数值实验。由于CG先验的综合性，这些实验表明DR-CG-Net在断层成像和压缩感知中优于现有的竞争方法，尤其在挑战性的低训练场景下表现突出。