  Inverse problems are essential to imaging applications. In this paper, we
propose a model-based deep learning network, named FISTA-Net, by combining the
merits of interpretability and generality of the model-based Fast Iterative
Shrinkage/Thresholding Algorithm (FISTA) and strong regularization and
tuning-free advantages of the data-driven neural network. By unfolding the
FISTA into a deep network, the architecture of FISTA-Net consists of multiple
gradient descent, proximal mapping, and momentum modules in cascade. Different
from FISTA, the gradient matrix in FISTA-Net can be updated during iteration
and a proximal operator network is developed for nonlinear thresholding which
can be learned through end-to-end training. Key parameters of FISTA-Net
including the gradient step size, thresholding value and momentum scalar are
tuning-free and learned from training data rather than hand-crafted. We further
impose positive and monotonous constraints on these parameters to ensure they
converge properly. The experimental results, evaluated both visually and
quantitatively, show that the FISTA-Net can optimize parameters for different
imaging tasks, i.e. Electromagnetic Tomography (EMT) and X-ray Computational
Tomography (X-ray CT). It outperforms the state-of-the-art model-based and deep
learning methods and exhibits good generalization ability over other
competitive learning-based approaches under different noise levels.
