# A Matching Pursuit Algorithm for Backtracking Regularization Based on Energy Sorting

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## Abstract

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`(`ROMP) algorithm backtracking. The support set is continuously updated and expanded during each iteration. While the signal energy distribution is not uniform, or the energy distribution is in an extreme state, the reconstructive performance of the ROMP algorithm becomes unstable if the maximum energy is still taken as the selection criterion. The proposed method for the regularized orthogonal matching pursuit algorithm can be adopted to improve those drawbacks in signal reconstruction due to its high reconstruction efficiency. The experimental results show that the algorithm has a proper reconstruction.

## 1. Introduction

## 2. Compressed Sensing Theory

## 3. Reconstruction Processes

- (1)
- Initialization: Set the residual ${r}_{0}=y$, $\Lambda =\varphi $.
- (2)
- Calculate the inner product between the residuals ${r}_{i-1}$ and the atoms of the observation matrix.
- (3)
- Set the threshold value, select the value larger than the threshold value $Th$ from $u$, and make up the set $J$ of the sequence number $j$ corresponding to these values.
- (4)
- Energy sorting and finding subsets ${J}_{0}\in J$.
- (5)
- Update the index set ${\Lambda}_{i}={\Lambda}_{i-1}\cup {J}_{0}$ and update the support set ${\Gamma}_{i}={\Gamma}_{i-1}\cup {J}_{0}$.
- (6)
- Solve the least squares problem $\widehat{\theta}=\mathrm{arg}\mathrm{min}\Vert y-{A}_{t}{\theta}_{t}\Vert $.
- (7)
- Backtracking update support set: Based on the backtracking idea, a new support set is made up of the larger $aL$ elements ($0<a<1$, A is the number of B)
- (8)
- Update the residual ${\widehat{r}}_{t}=y-{A}_{t}\widehat{\theta}$.
- (9)
- Judge whether ${\Vert {\widehat{r}}_{t}\Vert}_{2}\le {\Vert {\widehat{r}}_{t-1}\Vert}_{2}$ is established. If it is established, stop iterating; if it is not established, determine whether the number of initial stages s can be reached. If it is reached, the iteration is stopped; if it is not reached, return to the second step and continue to iterate.

## 4. Experimental Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ESBRMP | backtracking regularization matching pursuit algorithm based on energy sorting |

MR | magnetic resonance |

CS | compressed sensing |

CS-MRI | compressed sensing magnetic resonance imaging |

OMP | orthogonal matching pursuit algorithm |

ROMP | regularized orthogonal matching pursuit algorithm |

CoSaMP | compressive sampling matching pursuit algorithm |

SP | subspace tracking |

LP | linear programming |

RIP | restricted isometry property |

BRAMP | Backtracking Regularized Adaptive Matching Pursuit |

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**Figure 2.**The flow chart of the matching pursuit algorithm for regular backtracking based on the energy ranking (ESBRMP).

**Figure 5.**The ESBRMP algorithm’s relationship between the signal reconfiguration rate and the number of measurements.

**Figure 6.**The ESBRMP algorithm’s relationship between the signal reconfiguration rate and the sparsity.

**Table 1.**Qualities of images reconstructed and running time by different algorithms. PSNR: Peak signal to noise ratio.

Algorithms | M/N = 0.3 | M/N = 0.4 | M/N = 0.5 | |||
---|---|---|---|---|---|---|

PSNR (dB) | T (s) | PSNR (dB) | T (s) | PSNR (dB) | T (s) | |

OMP | 23.7848 | 16.3249 | 26.3581 | 35.4372 | 29.6349 | 62.9146 |

ROMP | 19.3325 | 2.8873 | 22.6383 | 3.0273 | 26.8823 | 3.5338 |

CoSaMP | 22.1336 | 4.1558 | 24.0518 | 6.3511 | 25.8473 | 9.2915 |

ESBRMP | 26.2538 | 3.2903 | 28.6912 | 5.1083 | 31.2703 | 8.9474 |

Algorithms | Lena | Fruits | Cameraman | Pepers |
---|---|---|---|---|

OMP | 29.6349 | 30.9803 | 28.0214 | 29.1471 |

ROMP | 26.8921 | 28.8023 | 24.0257 | 25.7125 |

CoSaMP | 25.8473 | 27.2755 | 24.1903 | 25.0361 |

ESBRMP | 31.2703 | 33.4108 | 29.1827 | 30.5297 |

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**MDPI and ACS Style**

Zhang, H.; Xiao, S.; Zhou, P.
A Matching Pursuit Algorithm for Backtracking Regularization Based on Energy Sorting. *Symmetry* **2020**, *12*, 231.
https://doi.org/10.3390/sym12020231

**AMA Style**

Zhang H, Xiao S, Zhou P.
A Matching Pursuit Algorithm for Backtracking Regularization Based on Energy Sorting. *Symmetry*. 2020; 12(2):231.
https://doi.org/10.3390/sym12020231

**Chicago/Turabian Style**

Zhang, Hanfei, Shungen Xiao, and Ping Zhou.
2020. "A Matching Pursuit Algorithm for Backtracking Regularization Based on Energy Sorting" *Symmetry* 12, no. 2: 231.
https://doi.org/10.3390/sym12020231