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.

翻译：针对线性逆问题的求解，特别是断层成像和压缩感知中出现的类型，本文提出了两种新方法。第一种方法是迭代算法，通过最小化基于复合高斯先验分布的正则化最小二乘目标函数来实现。复合高斯先验涵盖了图像重建中许多常用先验，包括基于稀疏性的方法。所提出的迭代算法引出了本文的第二种方法，即对应于该迭代算法"展开"或"解折叠"的深度神经网络。展开的深度神经网络具有可解释的层结构，且性能优于标准深度学习方法。本文包含详细的计算理论，为两种算法的构建和性能提供了深刻见解。结论表明，这两种算法在断层图像形成和压缩感知任务中均优于其他最先进方法，尤其在训练样本稀少的困难情形下表现突出。