# Segmentation of Concrete Cracks by Using Fractal Dimension and UHK-Net

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## Abstract

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## 1. Introduction

## 2. Proposed Method

#### 2.1. Fractal Dimension of Concrete Crack Image

#### 2.2. U-Net Network Model

#### 2.3. UHK-Net Network Model

## 3. Performance Evaluation Results

#### 3.1. Model Preparation

#### 3.2. Evaluation of Computational Complexity of Different Methods

#### 3.3. Comparison Analysis Based on the Visualization

#### 3.4. Quantitative Evaluation of Different Methods

## 4. Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The grid size with $\mathsf{\epsilon}=\frac{1}{4},\frac{1}{8},\cdots $ to cover the concrete crack image. (

**a**) $\mathsf{\epsilon}=\frac{1}{4}$, (

**b**) $\mathsf{\epsilon}=\frac{1}{8}$.

**Figure 5.**Four basic Haar-like feature structures. (

**A**) Edge feature of left and right, (

**B**) Edge feature of up and down, (

**C**) Line feature, (

**D**) Center-surround feature.

**Figure 9.**Training and validation results by different methods. (

**a**) U-Net, (

**b**) FCN, (

**c**) YOLO v5, (

**d**) UHK-Net.

**Figure 10.**Segmentation results of the crack by four networks, (

**a**) is the original crack images, (

**b**–

**e**) are the FCN, U-Net, YOLO v5, and proposed method, respectively.

**Figure 12.**Segmentation process of the proposed method. (

**a**–

**d**) are the original images, (

**e**–

**h**) are the detection results of cracks, and (

**i**–

**l**) are the segmentation results of concrete cracks.

**Figure 13.**Segmentation results of simple concrete mesh cracks. (

**a**–

**d**) are the original images, (

**e**–

**h**) are the segmentation results of cracks.

**Figure 14.**Segmentation results of complex concrete mesh cracks. (

**a**–

**d**) are the original images, (

**e**–

**h**) are the segmentation results of cracks.

$\mathbf{\epsilon}$ | $\frac{1}{2}$ | $\frac{1}{4}$ | $\frac{1}{8}$ | $\frac{1}{16}$ | $\frac{1}{32}$ | $\frac{1}{64}$ | $\frac{1}{128}$ | $\frac{1}{256}$ |
---|---|---|---|---|---|---|---|---|

$\mathrm{N}\left(\mathsf{\epsilon}\right)$ | 3 | 5 | 11 | 25 | 58 | 114 | 267 | 602 |

Layer | Output Size | Operation Type | Operation Size | Depth |
---|---|---|---|---|

Input | 512 × 512 × 3 | non | non | non |

Convl | 512 × 512 × 36 | conv | 3 × 3 | 3 |

UHK-Netl | 512 × 512 × 36 | conv | 3 × 3 | 36 |

TD1 | 256 × 256 × 36 | conv | 3 × 3 | 36 |

256 × 256 × 36 | pool | 4 × 4 | non | |

UHK-Net2 | 256 × 256 × 36 | conv | 3 × 3 | 36 |

TD2 | 128 × 128 × 36 | conv | 3 × 3 | 36 |

128 × 128 × 36 | pool | 4 × 4 | non | |

UHK-Net3 | 128 × 128 × 36 | conv | 5 × 5 | 36 |

TUI | 256 × 256 × 36 | deconv | 3 × 3 | 36 |

UHK-Net4 | 256 × 256 × 36 | conv | 3 × 3 | 36 |

TU2 | 512 × 512 × 36 | deconv | 3 × 3 | 36 |

UHK-Net5 | 512 × 512 × 36 | conv | 3 × 3 | 36 |

Conv2 | 512 × 512 × 36 | conv | 3 × 3 | 36 |

Output | 512 × 512 × 3 | non | non | non |

Different Methods | U-Net | FCN | YOLO v5 | UHK-Net |
---|---|---|---|---|

Training time (h) | 12.7 | 10.5 | 7.1 | 7 |

Segmentation time (s) | 1.4 | 1.1 | 0.8 | 0.9 |

Different Methods | PA | MPA | MIoU |
---|---|---|---|

FCN | 0.9366 | 0.8971 | 0.8406 |

U-Net | 0.9542 | 0.9037 | 0.8594 |

YOLO v5 | 0.9608 | 0.9143 | 0.8831 |

Proposed method | 0.9723 | 0.9298 | 0.9012 |

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**MDPI and ACS Style**

An, Q.; Chen, X.; Wang, H.; Yang, H.; Yang, Y.; Huang, W.; Wang, L.
Segmentation of Concrete Cracks by Using Fractal Dimension and UHK-Net. *Fractal Fract.* **2022**, *6*, 95.
https://doi.org/10.3390/fractalfract6020095

**AMA Style**

An Q, Chen X, Wang H, Yang H, Yang Y, Huang W, Wang L.
Segmentation of Concrete Cracks by Using Fractal Dimension and UHK-Net. *Fractal and Fractional*. 2022; 6(2):95.
https://doi.org/10.3390/fractalfract6020095

**Chicago/Turabian Style**

An, Qing, Xijiang Chen, Haojun Wang, Huamei Yang, Yuanjun Yang, Wei Huang, and Lei Wang.
2022. "Segmentation of Concrete Cracks by Using Fractal Dimension and UHK-Net" *Fractal and Fractional* 6, no. 2: 95.
https://doi.org/10.3390/fractalfract6020095