  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.
