  We present a new method for image reconstruction which replaces the projector
in a projected gradient descent (PGD) with a convolutional neural network
(CNN). CNNs trained as high-dimensional (image-to-image) regressors have
recently been used to efficiently solve inverse problems in imaging. However,
these approaches lack a feedback mechanism to enforce that the reconstructed
image is consistent with the measurements. This is crucial for inverse
problems, and more so in biomedical imaging, where the reconstructions are used
for diagnosis. In our scheme, the gradient descent enforces measurement
consistency, while the CNN recursively projects the solution closer to the
space of desired reconstruction images. We provide a formal framework to ensure
that the classical PGD converges to a local minimizer of a non-convex
constrained least-squares problem. When the projector is replaced with a CNN,
we propose a relaxed PGD, which always converges. Finally, we propose a simple
scheme to train a CNN to act like a projector. Our experiments on sparse view
Computed Tomography (CT) reconstruction for both noiseless and noisy
measurements show an improvement over the total-variation (TV) method and a
recent CNN-based technique.
