This paper proposes a generalized version of SO(3)-equivariant architectures including Spherical CNN. By utilizing the algebraic properties of Fourier transform and the tools in non-commutative harmonic analysis, the authors are able to construct (and prove) a most generalized version of SO(3)-equivariant architecture. Specifically, it only requires that, when an input image is rotated, each fragment (i.e., the output, Fourier coefficient vectors) of each layer will be multiplied by a Wigner-D matrix. To include non-linearities without performing inverse Fourier transform, the authors propose to use Clebsch-Gordon transformation. The experiments show that the proposed CG-Net can outperform Spherical CNN in several tasks.

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.