  Modern image and video compression codes employ elaborate structures existing
in such signals to encode them into few number of bits. Compressed sensing
recovery algorithms on the other hand use such signals' structures to recover
them from few linear observations. Despite the steady progress in the field of
compressed sensing, structures that are often used for signal recovery are
still much simpler than those employed by state-of-the-art compression codes.
The main goal of this paper is to bridge this gap through answering the
following question: Can one employ a given compression code to build an
efficient (polynomial time) compressed sensing recovery algorithm? In response
to this question, the compression-based gradient descent (C-GD) algorithm is
proposed. C-GD, which is a low-complexity iterative algorithm, is able to
employ a generic compression code for compressed sensing and therefore elevates
the scope of structures used in compressed sensing to those used by compression
codes. The convergence performance of C-GD and its required number of
measurements in terms of the rate-distortion performance of the compression
code are theoretically analyzed. It is also shown that C-GD is robust to
additive white Gaussian noise. Finally, the presented simulation results show
that combining C-GD with commercial image compression codes such as JPEG2000
yields state-of-the-art performance in imaging applications.
