  Turbo compressed sensing (Turbo-CS) is an efficient iterative algorithm for
sparse signal recovery with partial orthogonal sensing matrices. In this paper,
we extend the Turbo-CS algorithm to solve compressed sensing problems involving
more general signal structure, including compressive image recovery and
low-rank matrix recovery. A main difficulty for such an extension is that the
original Turbo-CS algorithm requires prior knowledge of the signal distribution
that is usually unavailable in practice. To overcome this difficulty, we
propose to redesign the Turbo-CS algorithm by employing a generic denoiser that
does not depend on the prior distribution and hence the name denoising-based
Turbo-CS (D-Turbo-CS). We then derive the extrinsic information for a generic
denoiser by following the Turbo-CS principle. Based on that, we optimize the
parametric extrinsic denoisers to minimize the output mean-square error (MSE).
Explicit expressions are derived for the extrinsic SURE-LET denoiser used in
compressive image denoising and also for the singular value thresholding (SVT)
denoiser used in low-rank matrix denoising. We find that the dynamics of
D-Turbo-CS can be well described by a scaler recursion called MSE evolution,
similar to the case for Turbo-CS. Numerical results demonstrate that D-Turbo-CS
considerably outperforms the counterpart algorithms in both reconstruction
quality and running time.
