  In this paper, we propose a novel deep convolutional neural network
(CNN)-based algorithm for solving ill-posed inverse problems. Regularized
iterative algorithms have emerged as the standard approach to ill-posed inverse
problems in the past few decades. These methods produce excellent results, but
can be challenging to deploy in practice due to factors including the high
computational cost of the forward and adjoint operators and the difficulty of
hyper parameter selection. The starting point of our work is the observation
that unrolled iterative methods have the form of a CNN (filtering followed by
point-wise non-linearity) when the normal operator (H*H, the adjoint of H times
H) of the forward model is a convolution. Based on this observation, we propose
using direct inversion followed by a CNN to solve normal-convolutional inverse
problems. The direct inversion encapsulates the physical model of the system,
but leads to artifacts when the problem is ill-posed; the CNN combines
multiresolution decomposition and residual learning in order to learn to remove
these artifacts while preserving image structure. We demonstrate the
performance of the proposed network in sparse-view reconstruction (down to 50
views) on parallel beam X-ray computed tomography in synthetic phantoms as well
as in real experimental sinograms. The proposed network outperforms total
variation-regularized iterative reconstruction for the more realistic phantoms
and requires less than a second to reconstruct a 512 x 512 image on GPU.
