#
A Reweighted Symmetric Smoothed Function Approximating L_{0}-Norm Regularized Sparse Reconstruction Method

^{*}

## Abstract

**:**

## 1. Introduction

- Greedy algorithms with sparsity as a prior condition;
- Relaxation method.

- (1)
- For ${L}_{p}$-RLS, the value of p cannot be too small because, the smaller p is, the less smooth ${\left|\right|\mathit{x}\left|\right|}_{p,\u03f5}^{p}$ is, which makes the optimization effect worse [25], so ${\left|\right|\mathit{x}\left|\right|}_{p,\u03f5}^{p}$ cannot closely approach the ${L}_{0}$ norm, and reconstruction accuracy cannot be further improved;
- (2)
- For ${L}_{2}$-SL0, although the algorithm can more closely approach the ${L}_{0}$-norm, the convergence of the adopted optimization method is not good, resulting in limited reconstruction accuracy.

## 2. RRCTSL0 Algorithm

#### 2.1. New Smoothed ${L}_{0}$-Norm Function Model

**Property**

**1.**

**Proof.**

#### 2.2. New Reweighted Function Design

- It has a proper range that can give each signal component a proper reweighted value, and, when the signal component is close to zero, the reweighted value is not too large.
- It does not need the adjustment of parameters like $\zeta $, and the denominator does not equal to zero.

#### 2.3. New Proposed RRCTSL0 Algorithm and Its Steps

#### 2.4. Selection of Parameters $\lambda $ and $\sigma $

## 3. Numerical Simulation and Analysis

#### 3.1. Convergence-Performance Comparison of the Algorithms

#### 3.2. Accuracy Performance Comparison of the Algorithms

#### 3.3. Applications of the Proposed RRCTSL0 Algorithm

#### 3.3.1. Real Sparse Signal Recovery

#### 3.3.2. Real-Image Recovery

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Different delta-approximating (DA) functions are plotted in this figure for $\sigma =0.1$, $a=2$, displaying the ${L}_{0}$-norm and ${L}_{0.5}$-norm of 2D.

**Figure 3.**Normalized mean square error (NMSE) of recovery signal changes with iterations. (

**a**) Comparison between SL0, ${L}_{2}$-SL0, ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm; (

**b**) effect of the RRCTSL0 algorithm under sparsity $k=[5,15,25,35]$.

**Figure 4.**Signal-to-noise ratio (SNR) analysis. (

**a**) Signal SNR for the SL0, ${L}_{2}$-SL0, ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm with noise intensity $\xi $ from 0 to 0.5 (the interval is 0.05) during 100 runs; (

**b**) SNR for proposed RRCTSL0 algorithm with $a=[0.5,1,1.5,2,2.5,3,3.5,4,4.5,5]$ while $\xi =0.05$.

**Figure 5.**Signal NMSE analysis for the SL0, ${L}_{2}$-SL0, ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm with sparsity k from 1 to 71 at an interval of 5, during 100 runs in the noise case $\xi =0.1$.

**Figure 6.**Signal CPU running time (CRT) analysis for the SL0, ${L}_{2}$-SL0, ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm with the length of signal n equaling [170, 220, 270, 320, 370, 420, 470, 520], during 100 runs in noise case $\xi =0.1$.

**Figure 8.**Signal-recovery effect by proposed RRCTSL0 algorithm when noise is incrementing according to sequence $\xi $ = [0, 0.05, 0.1, 0.2, 0.5]. (

**a**) $\xi $ = 0; (

**b**) $\xi $ = 0.05; (

**c**) $\xi $ = 0.1; (

**d**) $\xi $ = 0.2; (

**e**) $\xi $ = 0.5.

**Figure 9.**Reconstruction time-frequency characteristics of the proposed RRCTSL0 algorithm when noise is incrementing according to sequence $\xi $ = [0, 0.05, 0.1, 0.2, 0.5]. (

**a**) Time-frequency characteristic of the original signal; (

**b**) $\xi $ = 0; (

**c**) $\xi $ = 0.05; (

**d**) $\xi $ = 0.1; (

**e**) $\xi $ = 0.2; (

**f**) $\xi $ = 0.5.

**Figure 10.**Signal recovery effect by different algorithms when $\xi =0.1$. (

**a**) Signal recovery by the SL0 algorithm; (

**b**) signal recovery by the ${L}_{2}$-SL0 algorithm; (

**c**) signal recovery by the ${L}_{p}$-RLS algorithm; (

**d**) signal recovery by the RRCTSL0 algorithm.

**Figure 11.**Reconstruction time-frequency characteristics of different algorithms when $\xi =0.1$. (

**a**) SL0; (

**b**) ${L}_{2}$-SL0; (

**c**) ${L}_{p}$-RLS; (

**d**) RRCTSL0.

**Figure 12.**Real images used in the recovery experiments: (

**a**) Lena ($256\times 256$); (

**b**) Peppers ($256\times 256$).

**Figure 14.**Lena image-recovery effect by the proposed RRCTSL0 algorithm when noise was incrementing according to sequence $\xi $ = [0, 0.05, 0.1, 0.2, 0.5]. (

**a**) $\xi $ = 0; (

**b**) $\xi $ = 0.05; (

**c**) $\xi $ = 0.1; (

**d**) $\xi $ = 0.2; (

**e**) $\xi $ = 0.5.

**Figure 15.**Peppers image-recovery effect by the proposed RRCTSL0 algorithm when noise was incrementing according to sequence $\xi $ = [0, 0.05, 0.1, 0.2, 0.5]. (

**a**) $\xi $ = 0; (

**b**) $\xi $ = 0.05; (

**c**) $\xi $ = 0.1; (

**d**) $\xi $ = 0.2; (

**e**) $\xi $ = 0.5.

**Table 1.**Regularized Reweighted Composite Trigonometric Smoothed ${L}_{0}$-Norm Minimization (RRCTSL0) Algorithm Using the CG Method.

Initialization: $\mathbf{\Phi},\phantom{\rule{4pt}{0ex}}\mathit{x},\phantom{\rule{4pt}{0ex}}\mathit{y},\phantom{\rule{4pt}{0ex}}\mathit{\nu},\phantom{\rule{4pt}{0ex}}\mathit{\varsigma},\phantom{\rule{4pt}{0ex}}\mathit{\tau},\phantom{\rule{4pt}{0ex}}{\mathit{\sigma}}_{\mathit{T}},\phantom{\rule{4pt}{0ex}}\mathit{T},\phantom{\rule{4pt}{0ex}}\mathit{\lambda},\phantom{\rule{4pt}{0ex}}\mathit{a}$, and ${\mathit{x}}^{\ast}=\mathit{x}$ |
---|

Step 1: Set $t=0,\phantom{\rule{4pt}{0ex}}{\sigma}_{1}$; |

Step 2: Compute $\mathit{W}$ using (7), ${\sigma}_{t}$ for $t=2,3,\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}T-1$ using Equation (15); |

Step 3: For $t=1,2,\dots ,T$ |

(1) Set $\sigma ={\sigma}_{t},\phantom{\rule{4pt}{0ex}}\Gamma =0,\phantom{\rule{4pt}{0ex}}{\mathit{x}}_{\left(\Gamma \right)}={\mathit{x}}^{\ast}$ |

(2) Compute Residual $Res=\left|\right|{\varrho}_{\left(\Gamma \right)}{\mathit{d}}_{\left(\Gamma \right)}{\left|\right|}_{2}^{2}$, and iterative termination threshold $err$ |

(3) While $Res>err$ |

(a) Compute ${\mathit{x}}_{(\Gamma +1)}$ using Equations (16)–(23), $\mathit{W}$ using Equation (7) |

(b) Set $\Gamma =\Gamma +1$ |

(c) Compute $Res=\left|\right|{\varrho}_{\left(\Gamma \right)}{\mathit{d}}_{\left(\Gamma \right)}{\left|\right|}_{2}^{2}$ |

(4) Set ${\mathit{x}}^{\ast}={\mathit{x}}_{\left(\Gamma \right)}$ |

Step 4: Output $\mathit{x}={\mathit{x}}^{\ast}$ |

**Table 2.**Peak SNR (PSNR) and Structural Similarity Index (SSIM) of the recovered Lena image by the SL0, ${L}_{2}$-SL0, and ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm with $\xi =0.01$.

CR | PSNR (dB) | SSIM (%) | ||||||
---|---|---|---|---|---|---|---|---|

SL0 | ${\mathit{L}}_{2}$-SL0 | ${\mathit{L}}_{\mathit{p}}$-RLS | RRCTSL0 | SL0 | ${\mathit{L}}_{2}$-SL0 | ${\mathit{L}}_{\mathit{p}}$-RLS | RRCTSL0 | |

0.4 | 29.075 | 29.255 | 32.369 | 34.825 | 98.24 | 98.28 | 99.16 | 99.53 |

0.5 | 30.379 | 30.688 | 34.664 | 36.669 | 98.70 | 98.77 | 99.51 | 99.69 |

0.6 | 33.140 | 30.232 | 36.699 | 36.789 | 99.31 | 98.64 | 99.69 | 99.70 |

**Table 3.**PSNR and SSIM of the recovered Peppers image by the SL0, ${L}_{2}$-SL0, and ${L}_{p}$-RLS algorithms and the proposed RRCTSL0 algorithm with $\xi =0.01$.

CR | PSNR (dB) | SSIM (%) | ||||||
---|---|---|---|---|---|---|---|---|

SL0 | L2-SL0 | ${\mathit{L}}_{\mathit{q}}$-RLS | RRCTSL0 | SL0 | L2-SL0 | ${\mathit{L}}_{\mathit{q}}$-RLS | RRCTSL0 | |

0.4 | 21.811 | 26.343 | 33.887 | 34.673 | 93.13 | 97.33 | 99.53 | 99.61 |

0.5 | 28.405 | 29.769 | 34.588 | 35.046 | 98.35 | 98.80 | 99.60 | 99.64 |

0.6 | 32.276 | 33.188 | 34.872 | 35.160 | 99.33 | 99.45 | 99.63 | 99.65 |

**Table 4.**PSNR and SSIM of the recovery image by the proposed RRCTSL0 algorithm when noise was incrementing according to sequence $\xi =[0,0.05,0.1,0.2,0.5]$.

$\mathit{\xi}$ | Photo | PSNR (dB) | SSIM (%) |
---|---|---|---|

0 | Lena | 38.492 | 99.80 |

Peppers | 39.367 | 99.87 | |

0.05 | Lena | 29.203 | 98.28 |

Peppers | 28.826 | 97.53 | |

0.1 | Lena | 24.272 | 94.78 |

Peppers | 24.305 | 95.64 | |

0.2 | Lena | 18.727 | 83.43 |

Peppers | 19.782 | 86.07 | |

0.5 | Lena | 12.489 | 49.50 |

Peppers | 13.416 | 55.34 |

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**MDPI and ACS Style**

Xiang, J.; Yue, H.; Yin, X.; Ruan, G.
A Reweighted Symmetric Smoothed Function Approximating *L*_{0}-Norm Regularized Sparse Reconstruction Method. *Symmetry* **2018**, *10*, 583.
https://doi.org/10.3390/sym10110583

**AMA Style**

Xiang J, Yue H, Yin X, Ruan G.
A Reweighted Symmetric Smoothed Function Approximating *L*_{0}-Norm Regularized Sparse Reconstruction Method. *Symmetry*. 2018; 10(11):583.
https://doi.org/10.3390/sym10110583

**Chicago/Turabian Style**

Xiang, Jianhong, Huihui Yue, Xiangjun Yin, and Guoqing Ruan.
2018. "A Reweighted Symmetric Smoothed Function Approximating *L*_{0}-Norm Regularized Sparse Reconstruction Method" *Symmetry* 10, no. 11: 583.
https://doi.org/10.3390/sym10110583