NL-CS Net: Deep Learning with Non-Local Prior for Image Compressive
  Sensing
Shuai Bian Shouliang Qi Chen Li Yudong Yao Yueyang Teng
Deep learning has been applied to compressive sensing (CS) of images
successfully in recent years. However, existing network-based methods are often
trained as the black box, in which the lack of prior knowledge is often the
bottleneck for further performance improvement. To overcome this drawback, this
paper proposes a novel CS method using non-local prior which combines the
interpretability of the traditional optimization methods with the speed of
network-based methods, called NL-CS Net. We unroll each phase from iteration of
the augmented Lagrangian method solving non-local and sparse regularized
optimization problem by a network. NL-CS Net is composed of the up-sampling
module and the recovery module. In the up-sampling module, we use learnable
up-sampling matrix instead of a predefined one. In the recovery module,
patch-wise non-local network is employed to capture long-range feature
correspondences. Important parameters involved (e.g. sampling matrix, nonlinear
transforms, shrinkage thresholds, step size, $etc.$) are learned end-to-end,
rather than hand-crafted. Furthermore, to facilitate practical implementation,
orthogonal and binary constraints on the sampling matrix are simultaneously
adopted. Extensive experiments on natural images and magnetic resonance imaging
(MRI) demonstrate that the proposed method outperforms the state-of-the-art
methods while maintaining great interpretability and speed.