# A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images

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## Abstract

**:**

## 1. Introduction

## 2. FOTV Regularization Recovery Model

#### 2.1. TV Regularization Recovery Model

#### 2.2. TV-Regularization Recovery Model

#### 2.3. FOTV-Regularization Recovery Model

#### 2.4. Solution of the FOTV-Regularization Recovery Model

Algorithm 1: Split Bregman Algorithm for Solving Recovery Models for Anisotropic FOTV Regularization |

Initialization: ${u}^{0}=f,d=0$ While $||{u}^{k}-{u}^{k-1}|{|}_{2}\ge tol$ $\begin{array}{l}{u}^{k+1}={G}^{k}\\ {d}^{k+1}=shrink({D}^{v}{u}^{k+1}+{b}^{k},\frac{1}{\lambda})\\ {b}^{k+1}={b}^{k}+({D}^{v}{u}^{k+1}-{d}^{k+1})\end{array}$ End while |

## 3. GM-APD Depth Image FOTV Restoration Algorithm

#### 3.1. Depth-Image Extraction from Low SBR and Few-Frame Data Using a Spatial-Domain Differential Peak-Picking Method

#### 3.2. FOTV-Regularization Recovery Algorithm

## 4. Simulation and Experimental Verification

#### 4.1. Evaluation Metrics

#### 4.1.1. K

#### 4.1.2. PSNR

#### 4.1.3. SSIM

#### 4.2. Simulation Analysis

#### 4.2.1. Depth Image Extraction

#### 4.2.2. Depth-Image Recovery Using the FOTV Method

- Selection of optimal fractional order for fractional calculus

- 2.
- FOTV-recovery algorithm

#### 4.3. Experimental Verification

#### 4.3.1. Experimental Platform

#### 4.3.2. Outdoor Experiment

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Imaging results for different SBRs and statistical frame numbers. (

**a**) SBR = 0.2; (

**b**) SBR = 0.11; (

**c**) SBR = 0.1.

**Figure 9.**Structural similarity index measure (SSIM) curve of the peak method and the devised method.

**Figure 11.**Target reduction degree (K) curve for different fractional order under different SBRs conditions.

**Figure 12.**Structural similarity index measure (SSIM) curve for different fractional order under different SBRs conditions.

**Figure 13.**Peak signal–to–noise ratio (PSNR) curve for different fractional order under different SBRs conditions.

**Figure 14.**Target reduction degree (K) curve for different fractional order under different Frames conditions.

**Figure 15.**Structural similarity index measure (SSIM) curve for different fractional order under different Frames conditions.

**Figure 16.**Peak signal–to–noise ratio (PSNR) curve for different fractional order under different Frames conditions.

SBR | 0.1 | 0.11 | 0.2 | ||||||
---|---|---|---|---|---|---|---|---|---|

Evaluation metrics | K | SSIM | PSNR | K | SSIM | PSNR | K | SSIM | PSNR |

Optimal order | 0.5 | 1.3 | 1.7 | 0.5 | 1.3 | 1.7 | 0.1 | 1.7 | 1.7 |

Statistical Frame Numbers | Evaluation Metrics | Optimal Order |
---|---|---|

40 | K | 0.1 |

SSIM | 0.1 | |

PSNR | 1.7 | |

70 | K | 1.3 |

SSIM | 1.3 | |

PSNR | 1.7 |

Number of Frames | 30 | 50 | 70 | ||||||
---|---|---|---|---|---|---|---|---|---|

Algorithm | Original image | TV | FOTV | Original image | TV | FOTV | Original image | TV | FOTV |

K | 0.5000 | 0.5237 | 0.5768 | 0.6885 | 0.7277 | 0.8655 | 0.7612 | 0.7905 | 0.9232 |

PSNR | 19.6235 | 20.2623 | 23.1700 | 22.4426 | 23.3513 | 28.4516 | 24.1864 | 25.0615 | 29.8380 |

SSIM | 0.9423 | 0.9463 | 0.9746 | 0.9744 | 0.9774 | 0.9942 | 0.9839 | 0.9861 | 0.9960 |

**Table 4.**Comparison between [8] and the algorithm in this paper.

Conditions | [8] | Ours |
---|---|---|

SBR | 0.12 | 0.1 |

Frames | 200 | 100 |

K | 0.95 | 0.9777 |

PSNR | 20.83 | 33.3639 |

SSIM | 0.940 | 0.9982 |

**Table 5.**Evaluation metrics for the peak-picking method and spatial-domain differential peak-picking method.

Evaluation Metrics | Peak Picking Method | Spatial-Domain Differential Peak Picking Method |
---|---|---|

K | 0.1058 | 0.3051 |

PSNR | 14.0479 | 17.3686 |

SSIM | 0.4065 | 0.7637 |

Evaluation Metric | TV Recovering | FOTV Recovering |
---|---|---|

K | 0.2327 | 0.4109 |

PSNR | 17.3441 | 17.9471 |

SSIM | 0.7659 | 0.8186 |

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## Share and Cite

**MDPI and ACS Style**

Xie, D.; Wang, X.; Wang, C.; Yuan, K.; Wei, X.; Liu, X.; Huang, T.
A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images. *Fractal Fract.* **2023**, *7*, 445.
https://doi.org/10.3390/fractalfract7060445

**AMA Style**

Xie D, Wang X, Wang C, Yuan K, Wei X, Liu X, Huang T.
A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images. *Fractal and Fractional*. 2023; 7(6):445.
https://doi.org/10.3390/fractalfract7060445

**Chicago/Turabian Style**

Xie, Da, Xinjian Wang, Chunyang Wang, Kai Yuan, Xuyang Wei, Xuelian Liu, and Tingsheng Huang.
2023. "A Fractional-Order Total Variation Regularization-Based Method for Recovering Geiger-Mode Avalanche Photodiode Light Detection and Ranging Depth Images" *Fractal and Fractional* 7, no. 6: 445.
https://doi.org/10.3390/fractalfract7060445