  Recovering high-resolution images from limited sensory data typically leads
to a serious ill-posed inverse problem, demanding inversion algorithms that
effectively capture the prior information. Learning a good inverse mapping from
training data faces severe challenges, including: (i) scarcity of training
data; (ii) need for plausible reconstructions that are physically feasible;
(iii) need for fast reconstruction, especially in real-time applications. We
develop a successful system solving all these challenges, using as basic
architecture the recurrent application of proximal gradient algorithm. We learn
a proximal map that works well with real images based on residual networks.
Contraction of the resulting map is analyzed, and incoherence conditions are
investigated that drive the convergence of the iterates. Extensive experiments
are carried out under different settings: (a) reconstructing abdominal MRI of
pediatric patients from highly undersampled Fourier-space data and (b)
superresolving natural face images. Our key findings include: 1. a recurrent
ResNet with a single residual block unrolled from an iterative algorithm yields
an effective proximal which accurately reveals MR image details. 2. Our
architecture significantly outperforms conventional non-recurrent deep ResNets
by 2dB SNR; it is also trained much more rapidly. 3. It outperforms
state-of-the-art compressed-sensing Wavelet-based methods by 4dB SNR, with 100x
speedups in reconstruction time.
