# A New Regularized Reconstruction Algorithm Based on Compressed Sensing for the Sparse Underdetermined Problem and Applications of One-Dimensional and Two-Dimensional Signal Recovery

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}norm. Compared with the previous convex optimization method, in this algorithm, the discrete l

_{0}norm is replaced by the continuous Gaussian function with parameters to construct a new objective function, then the steepest descent method is used to minimize the approximate continuous function, and finally, the minimal solution is projected into the solution space to satisfy the constraints. From the implementation process of the algorithm, the algorithm combines the rapidity of the greedy algorithm and the accuracy of convex optimization and is a better reconstruction algorithm. There are many improved algorithms based on the SL0 algorithm. Recently, in [5], the author proposed a regularized smooth ${l}_{0}$ reconstruction algorithm called ReSL0 [5], which can reconstruct the signal in the presence of noise, and the reconstruction effect is impressive. However, the ReSL0 algorithm uses the Gauss approximation function and the steepest descent optimization method of the SL0 algorithm. On the one hand, through our research and investigation, we find that many approximation functions are better than the Gauss function, such as the approximation function in [19]; on the other hand, although the steepest descent optimization method adopted by ReSL0 does not require the accurate initial value, the optimization algorithm itself has drawbacks. In the early stage of the algorithm, the steepest descent method does have the best approach; however, in the later stage of optimization, there will be a jagged optimization path, and the convergence becomes very slow. The well-known Newton method has second-order convergence, but the Newton method requires a more accurate initial value, and it is not easy to get a perfect initial value. For this reason, we add the combined optimization method of the steepest descent method and Newton method to the ReSL0 algorithm and select the approximation function as used in [19]. Thus, the combined-optimization ReSL0 algorithm is proposed in this paper, which is called the CReSL0 algorithm.

## 2. Preliminaries

_{0}-minimization problem, as below:

## 3. ReSL0 Algorithm

Algorithm 1. The pseudo-code of the ReSL0 algorithm. |

Initialization: |

(1) Set $L\text{},\text{}\mu 0$ and ${\widehat{x}}^{0}={A}^{H}{(A{A}^{H})}^{-1}y$. |

(2) Set $0<\rho <1,{\sigma}_{1},{\sigma}_{J}$ and ${\sigma}_{j}=\rho {\sigma}_{j-1}$ where $j>1$. |

While ${\sigma}_{j}>{\sigma}_{J}$ |

(1) Let $\sigma ={\sigma}_{j}$. |

(2) Initialization: $x={\widehat{x}}^{j-1}$. |

-for $k=1,\dots ,L$ |

(a) $x\leftarrow x+\mu x\mathrm{exp}(-{x}^{2}/(2{\sigma}^{2}))$. |

(b) $x=x-{A}^{H}{(A{A}^{H}{I}_{n}+{\lambda}^{-1}{I}_{m})}^{-1}(Ax-y)$. |

(3) Set ${\widehat{x}}^{j}=x$. |

The estimated value is $\widehat{x}={\widehat{x}}^{j}$. |

## 4. CReSL0 Algorithm

#### 4.1. Selection of Approximation Function

#### 4.2. Selection of Optimization Method

Algorithm 2. The pseudocode of CReSL0 algorithm. |

Initialization: |

(1) Set $L,\text{}\mu 0$ and ${\widehat{x}}^{0}={A}^{H}{(A{A}^{H})}^{-1}y$. |

(2) Set ${\sigma}_{1},,,,,{\sigma}_{J}$. |

While ${\sigma}_{j}>{\sigma}_{J}$ |

(1) Let $\sigma ={\sigma}_{j}$. |

(2) Initialization: $x={\widehat{x}}^{j-1}$. |

-for $k=1,\dots ,\beta ,\dots ,L$ |

If $k\le \beta $ (a) $x\leftarrow x+\mu \nabla {F}_{\sigma}(x)$. |

(b) $x\leftarrow x-{A}^{H}{(A{A}^{H}{I}_{n}+{\lambda}^{-1}{I}_{m})}^{-1}(Ax-y)$. |

Else (c) $x\leftarrow x-{G}^{-1}\Delta {Y}_{\sigma}(x)$. |

(d) $x\leftarrow x-{A}^{H}{(A{A}^{H}{I}_{n}+{\lambda}^{-1}{I}_{m})}^{-1}(Ax-y)$. |

(3) Set ${\widehat{x}}^{j}=x$. |

The estimated value is $\widehat{x}={\widehat{x}}^{j}$. |

#### 4.3. Selection of Parameters

## 5. Simulation and Results

#### 5.1. Reconstruction of One-Dimensional Gauss Signal

#### 5.2. Reconstruction of Two-Dimensional Image Signal

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Candés, E. Compressive sampling. In Proceedings of the international congress of mathematicians, Madrid, Spain, 22–30 August 2006; pp. 1433–1452. [Google Scholar]
- Baraniuk, R. Compressive sensing. IEEE Signal Process. Mag.
**2007**, 24, 118–121. [Google Scholar] [CrossRef] - Candès, E.; Romberg, J.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math.
**2006**, 59, 1207–1223. [Google Scholar] [CrossRef][Green Version] - Wang, H.; Guo, Q.; Zhang, G.X.; Li, G.X.; Xiang, W. Thresholded smoothed ℓ° norm for accelerated Sparse recovery. IEEE Commun, Lett.
**2015**, 19, 953–956. [Google Scholar] [CrossRef] - Bu, H.X.; Tao, R.; Bai, X.; Zhao, J. Regularized smoothed ℓ° norm algorithm and its application to CS-based radar imaging. Signal Process.
**2016**, 122, 115–122. [Google Scholar] [CrossRef] - Goyal, P.; Singh, B. Subspace pursuit for sparse signal reconstruction in wireless sensor networks. Procedia. Comput. Sci.
**2018**, 125, 228–233. [Google Scholar] [CrossRef] - Wei-Hong, F.U.; Ai-Li, L.I.; Li-Fen, M.A.; Huang, K.; Yan, X. Underdetermined blind separation based on potential function with estimated parameter’s decreasing sequence. Syst. Eng. Electron.
**2014**, 36, 619–623. [Google Scholar] - Mallat, S.; Zhang, Z. Matching pursuit in time–frequency dictionary. IEEE Trans. Signal Process.
**1993**, 41, 3397–3415. [Google Scholar] [CrossRef] - Pati, Y.C.; Rezaiifar, R.; Krishnaprasad, P.S. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Proceedings of the 27th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 1–3 November 1993; pp. 40–44. [Google Scholar]
- Dai, W.; Milenkovic, O. Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans. Inf. Theory
**2009**, 5, 2230–2249. [Google Scholar] [CrossRef] - Needell, D.; Tropp, J.A. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Commun. ACM
**2010**, 12, 93–100. [Google Scholar] [CrossRef] - Do, T.T.; Lu, G.; Nguyen, N.; Tran, T.D. Sparsity adaptive matching pursuit algorithm for practical compressed sensing. In Proceedings of the 2008 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA, 26–29 October 2008; pp. 581–587. [Google Scholar]
- Wang, J.; Li, P. Recovery of Sparse Signals Using Multiple Orthogonal Least Squares. IEEE Trans. Signal Process.
**2017**, 65, 2049–2061. [Google Scholar] [CrossRef] - Ekanadham, C.; Tranchina, D.; Simoncelli, E.P. Recovery of Sparse Translation-Invariant Signals with Continuous Basis Pursuit. IEEE Trans. Signal Process.
**2011**, 10, 4735–4744. [Google Scholar] [CrossRef] - Pant, J.K.; Lu, W.S.; Antoniou, A. New Improved Algorithms for Compressive Sensing Based on lp Norm. IEEE Trans. Circuits Syst. II Express Briefs
**2014**, 61, 198–202. [Google Scholar] [CrossRef] - Figueiredo, M.A.T.; Nowak, R.D.; Wright, S.J. Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems. IEEE J. Sel. Top. Signal Process
**2008**, 1, 586–597. [Google Scholar] [CrossRef] - Kim, D.; Fessler, J.A. Another look at the fast iterative shrinkage/thresholding algorithm (FISTA). Siam J. Optim
**2018**, 28, 223–250. [Google Scholar] [CrossRef] - Mohimani, G.H.; Babaie-Zadeh, M.; Jutten, C. Fast sparse representation based on smoothed ℓ0 norm, in: Independent Component Analysis and Signal Separation. In Proceedings of the International Conference on Latent Variable Analysis and Signal Separation, London, UK, 9–12 September 2007; pp. 389–396. [Google Scholar]
- Wang, L.; Yin, X.; Yue, H.; Xiang, J. A regularized weighted smoothed L0 norm minimization method for underdetermined blind source separation. Sensors
**2018**, 18, 4260. [Google Scholar] [CrossRef] [PubMed]

**Figure 2.**(

**a**) Peak reconstruction signal-to-noise ratio (PSNR) values of each algorithm for different sparsities; (

**b**) running time of each algorithm for different sparsities.

**Figure 3.**(

**a**) PSNR of each algorithm for different signal lengths; (

**b**) running time of each algorithm for different signal lengths.

**Table 1.**The PSNR, time and PSNR/time of each algorithm for five images. ReSL0: regularized smooth L0; CReSL0: combined regularized smooth L0; WReSL0: weighted regularized smooth L0.

Figure 4 | PSNR (dB) | Time (s) | PSNR/Time | ||||||
---|---|---|---|---|---|---|---|---|---|

CReSL0 | ReSL0 | WReSL0 | CReSL0 | ReSL0 | WReSL0 | CReSL0 | ReSL0 | WReSL0 | |

a | 35.64 | 34.36 | 34.94 | 6.56 | 5.65 | 7.16 | 5.43 | 6.08 | 4.88 |

b | 37.01 | 35.37 | 36.01 | 6.58 | 5.58 | 7.18 | 5.62 | 6.34 | 5.02 |

c | 36.82 | 35.78 | 36.31 | 6.39 | 5.60 | 7.37 | 5.76 | 6.39 | 4.93 |

d | 33.39 | 31.88 | 32.49 | 6.48 | 5.77 | 7.25 | 5.15 | 5.53 | 4.48 |

e | 33.03 | 32.01 | 32.68 | 6.54 | 5.74 | 7.18 | 5.05 | 5.58 | 4.55 |

CR | PSNR (dB) | Time (s) | PSNR/Time | ||||||
---|---|---|---|---|---|---|---|---|---|

CReSL0 | ReSL0 | WReSL0 | CReSL0 | ReSL0 | WReSL0 | CReSL0 | ReSL0 | WReSL0 | |

0.4 | 27.55 | 26.85 | 27.23 | 3.23 | 2.71 | 4.45 | 8.53 | 9.91 | 6.12 |

0.6 | 32.04 | 30.82 | 31.42 | 4.75 | 4.11 | 5.85 | 6.75 | 7.50 | 5.37 |

0.8 | 37.68 | 36.30 | 36.94 | 7.50 | 6.54 | 8.18 | 5.02 | 5.55 | 4.52 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, B.; Wang, L.; Yu, H.; Xin, F.
A New Regularized Reconstruction Algorithm Based on Compressed Sensing for the Sparse Underdetermined Problem and Applications of One-Dimensional and Two-Dimensional Signal Recovery. *Algorithms* **2019**, *12*, 126.
https://doi.org/10.3390/a12070126

**AMA Style**

Wang B, Wang L, Yu H, Xin F.
A New Regularized Reconstruction Algorithm Based on Compressed Sensing for the Sparse Underdetermined Problem and Applications of One-Dimensional and Two-Dimensional Signal Recovery. *Algorithms*. 2019; 12(7):126.
https://doi.org/10.3390/a12070126

**Chicago/Turabian Style**

Wang, Bin, Li Wang, Hao Yu, and Fengming Xin.
2019. "A New Regularized Reconstruction Algorithm Based on Compressed Sensing for the Sparse Underdetermined Problem and Applications of One-Dimensional and Two-Dimensional Signal Recovery" *Algorithms* 12, no. 7: 126.
https://doi.org/10.3390/a12070126