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Sparse signal reconstruction, as the main link of compressive sensing (CS) theory, has attracted extensive attention in recent years. The essence of sparse signal reconstruction is how to recover the original signal accurately and effectively from an underdetermined linear system equation (ULSE). For this problem, we propose a new algorithm called regularization reweighted smoothed

CS [

For solving the ULSE in (

This rather wonderful transformation is actually supported by outstanding theory [

Greedy search

Relaxation method for the

Greedy search requires known sparsity as a constraint and the main methods are the approximate algorithms based on greedy matching pursuit (GMP) algorithms, such as OMP [

Based on this, the characteristics of GMP algorithms can be summarized as follows [

Sparsity is used as a prior information

Least square error is employed as an iterative criterion

The main advantage of GMP algorithms is that it is simple to calculate, but its application range is limited due to its low reconstruction accuracy in the case of noise.

At present, relaxation method for the

The above sparse signal recovery methods focus more on signal recovery in the absence of noise; they work not well in noise case. However, sparse signal recovery under noise is a very realistic and inevitable problem. Fortunately, the regularization mechanism makes it possible to solve this problem. The regularization mechanism relaxes

For regularization, there are many sparsity regularizers for relaxing the

In the case of no noise,

The tanh function proposed in [

In this paper, we propose a CIPF function as new sparsity regularizer and show their effectiveness and advantages over other popular regularizers in promoting sparse solutions with both theoretical analyses and experimental evaluations.

This paper is organized as follows. Section

In this paper, based on the

According to [

It follows immediately from

Based on the

In (

Figure

Different function used in the literature to approximate the

In conclusion, the merits of CIPF model can be summarized as follows:

It closely approximates

It is simpler than Gauss and tanh function model.

These merits make it possible to reduce the computational complexity on the premise of ensuring the sparse signal reconstruction accuracy, which is of practical significance for sparse signal reconstruction.

Candès et al. [

Candès et al.:

Pant et al.:

From the two reweighted functions, we can find a phenomenon: a large signal entry

Combined with the above idea, we propose a new reweighted function, which is given by

As for Candès et al., when signal entry is zero or close to zero, the value of

As for

As explained above, the proposed objective function can be eventually described as

Solving problem of ULSE is to solve the optimization problem in (

The problem firstly can be solved by using a sequential

Given

According to CG algorithm, the solution

From above equations, we can see that

The above is the optimization process analysis of MCG method. For MCG method, the direction of each iteration of this method is the combination of the last iteration direction and the negative gradient direction, which can overcome the sawtooth phenomenon caused by the steepest descent method.

According to the explanation above, we can conclude the steps of proposed RRSL0 algorithm, which is given in Algorithm

Initialization:

Step

Step

Step

(

(

(

(a) Compute

(b) Compute

(

Step

The selection of parameters

The choice of parameter

According to (

For initial

From the equation, we can see constant

As for final value

The numerical simulation platform is MATLAB 2017b, which is installed on the WINDOWS 10, 64-bit operating system. The CPU of simulation computer is Intel (R) Core (TM) i5-3230M, and the frequency is 2.6 GHz. In this section, the performance of RRSL0 algorithm is verified by signal and image recovery in the noiseless case and noise case.

Here some state-of-the-art algorithms are selected for comparison. The parameters are selected to obtain best performance for each algorithm: for BPDN algorithm [

For signal recovery under no noise conditions, we evaluate performance of algorithms by

In this section, the SL0, BPDN, L2-SL0, and Lp-RLS are used for comparison. Here we fix

Signal CRT analysis for BPDN, SL0, L2-SL0,

Signal length (n) | CPU running time (seconds) | ||||
---|---|---|---|---|---|

BPDN | SL0 | L2-SL0 | Lp-RLS | RRSL0 | |

170 | 0.1958 | 0.0572 | 0.0909 | 0.0636 | 0.0628 |

220 | 0.2899 | 0.1385 | 0.2298 | 0.1503 | 0.1425 |

270 | 0.4953 | 0.2285 | 0.4257 | 0.3050 | 0.2908 |

320 | 0.7675 | 0.3196 | 0.6385 | 0.5121 | 0.5088 |

370 | 1.0586 | 0.4559 | 0.9260 | 0.9023 | 0.8924 |

420 | 1.4767 | 0.6133 | 1.1327 | 1.0906 | 1.0170 |

470 | 1.9414 | 0.7956 | 1.4782 | 1.4178 | 1.3438 |

520 | 2.6194 | 1.0383 | 2.0894 | 1.9097 | 1.8818 |

Signal recovery in noiseless case. RSR and NMSE of recovery signal are displayed at intervals of 5 with sparsity from 1 to 61. The comparison algorithms are BPDN, SL0, L2-SL0, Lp-RLS, and RRSL0. The experimental results were obtained by 100 independent repeated experiments.

RSR of recovery signal

NMSE of recovery signal

Figure

The NMSE of all algorithms is shown in Figure

Table

Convergence verification for

In this section, we discuss signal recovery performance in noise case. We add noise

Figure

NMSE analysis for BPDN, SL0, L2-SL0,

Noise intensity ( | NMSE of signal recovery | ||||
---|---|---|---|---|---|

BPDN | L0 | L2-SL0 | Lp-RLS | RRSL0 | |

0 | 2.553e-05 | 1.311e-07 | 4.563e-07 | 1.563e-07 | 2.338e-07 |

0.01 | 4.659e-04 | 3.054e-04 | 2.352e-04 | 7.930e-05 | 6.032e-05 |

0.05 | 1.607e-03 | 4.079e-03 | 9.012e-04 | 2.384e-04 | 1.150e-04 |

0.1 | 7.633e-03 | 8.424e-03 | 3.309e-03 | 7.793e-04 | 5.522e-04 |

0.2 | 1.381e-02 | 2.090e-02 | 6.091e-03 | 1.130e-03 | 8.892e-04 |

0.5 | 3.063e-02 | 6.908e-02 | 1.083e-02 | 5.743e-03 | 3.041e-03 |

Signal recovery effect by different algorithms when noise intensity

Reconstruction time-frequency characteristics of different algorithms when noise intensity

Real images are considered to be approximately sparse under some proper basis, such as the Discrete Cosine Transform (DCT) basis and Discrete Wavelet Transform (DWT) basis. Here we choose DWT basis to recover these images. We compare the recovery performances based on the 2 real images in Figure

Original images.

Original Boat

Original Barbara

For performance of image recovery, we valuate it by

Among this,

Figure

PSNR analysis of recovered Boat image by BPDN, SL0, L2-SL0,

Noise intensity ( | PSNR of Boat image recovery | ||||
---|---|---|---|---|---|

BPDN | SL0 | L2-SL0 | | RRSL0 | |

0 | 28.4928 | 31.5901 | 31.7175 | 32.2332 | 33.3495 |

0.01 | 27.9775 | 29.8589 | 30.2423 | 30.9742 | 31.5447 |

0.05 | 24.0531 | 27.1226 | 27.7101 | 28.2422 | 29.2505 |

0.1 | 20.3769 | 21.6485 | 23.9331 | 25.8639 | 26.4745 |

0.2 | 16.3044 | 17.9303 | 20.6744 | 23.0823 | 23.7415 |

0.5 | 9.2283 | 10.7863 | 13.4863 | 14.4974 | 16.3646 |

SSIM analysis of recovered Boat image by BPDN, SL0, L2-SL0,

Noise intensity ( | SSIM of Boat image recovery | ||||
---|---|---|---|---|---|

BPDN | SL0 | L2-SL0 | | RRSL0 | |

0 | 0.9865 | 0.9901 | 0.9902 | 0.9914 | 0.9934 |

0.01 | 0.9764 | 0.9852 | 0.9865 | 0.9887 | 0.9898 |

0.05 | 0.9418 | 0.9714 | 0.9756 | 0.9789 | 0.9827 |

0.1 | 0.8827 | 0.9083 | 0.9403 | 0.9631 | 0.9669 |

0.2 | 0.7452 | 0.8077 | 0.8868 | 0.9284 | 0.9396 |

0.5 | 0.3571 | 0.4427 | 0.5966 | 0.6492 | 0.7351 |

PSNR analysis of recovered Barbara image by BPDN, SL0, L2-SL0,

Noise intensity ( | PSNR of Barbara image recovery | ||||
---|---|---|---|---|---|

BPDN | SL0 | L2-SL0 | | RRSL0 | |

0 | 28.6613 | 29.5761 | 31.7921 | 32.3566 | 33.4074 |

0.01 | 27.7887 | 30.4746 | 30.9875 | 31.7365 | 32.6767 |

0.05 | 26.0316 | 26.1561 | 26.3147 | 27.3533 | 29.8613 |

0.1 | 20.7076 | 21.6914 | 23.8343 | 26.9408 | 27.5167 |

0.2 | 16.6564 | 16.6721 | 19.1297 | 19.2624 | 19.7801 |

0.5 | 9.1316 | 9.3525 | 13.1838 | 15.7972 | 16.9070 |

SSIM analysis of recovered Barbara image by BPDN, SL0, L2-SL0,

Noise intensity ( | SSIM of Barbara image recovery | ||||
---|---|---|---|---|---|

BPDN | SL0 | L2-SL0 | | RRSL0 | |

0 | 0.9834 | 0.9920 | 0.9926 | 0.9936 | 0.9984 |

0.01 | 0.9817 | 0.9904 | 0.9914 | 0.9927 | 0.9942 |

0.05 | 0.9797 | 0.9744 | 0.9753 | 0.9739 | 0.9888 |

0.1 | 0.9161 | 0.9314 | 0.9544 | 0.9728 | 0.9810 |

0.2 | 0.8165 | 0.8170 | 0.9051 | 0.9142 | 0.9278 |

0.5 | 0.3863 | 0.3992 | 0.6309 | 0.7712 | 0.8190 |

Images recovery effect by BPDN, SL0, L2-SL0,

Recovered Boat with noise intensity

Recovered Barbara with noise intensity

Recovered Boat with noise intensity

Recovered Barbara with noise intensity

In this paper, we propose the RRSL0 algorithm to recover sparse signal from given

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

All authors have made great contributions to the work. Jianhong Xiang, Huihui Yue, and Xiangjun Yin conceived and designed the experiments; Jianhong Xiang and Huihui Yue performed the experiments and analyzed the data; Xiangjun Yin and Linyu Wang gave insightful suggestions for the work; Jianhong Xiang, Huihui Yue, and Xiangjun Yin wrote the paper.

This paper is supported by the National Key Laboratory of Communication Anti-jamming Technology.