# Denoising-based Turbo Compressed Sensing

## TL;DR

This paper introduces Denoising-based Turbo Compressed Sensing (D-Turbo-CS), a flexible algorithm that improves sparse signal recovery by employing generic denoisers, extending Turbo-CS to more complex signal structures without prior distribution knowledge.

## Contribution

The paper proposes a novel D-Turbo-CS algorithm that uses generic denoisers, enabling efficient recovery of complex signals like images and low-rank matrices without prior distribution information.

## Key findings

- D-Turbo-CS outperforms existing algorithms in reconstruction quality.
- D-Turbo-CS has faster running times compared to traditional methods.
- The algorithm's dynamics are accurately described by a simple MSE evolution recursion.

## Abstract

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.