  In this paper we develop a novel computational sensing framework for sensing
and recovering structured signals. When trained on a set of representative
signals, our framework learns to take undersampled measurements and recover
signals from them using a deep convolutional neural network. In other words, it
learns a transformation from the original signals to a near-optimal number of
undersampled measurements and the inverse transformation from measurements to
signals. This is in contrast to traditional compressive sensing (CS) systems
that use random linear measurements and convex optimization or iterative
algorithms for signal recovery. We compare our new framework with
$\ell_1$-minimization from the phase transition point of view and demonstrate
that it outperforms $\ell_1$-minimization in the regions of phase transition
plot where $\ell_1$-minimization cannot recover the exact solution. In
addition, we experimentally demonstrate how learning measurements enhances the
overall recovery performance, speeds up training of recovery framework, and
leads to having fewer parameters to learn.
