# Image Edge Detection Based on Fractional-Order Ant Colony Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- Section 2 provides an overview of fractional calculus, including its fundamental definition. Additionally, this section introduces the fundamental principles and procedures of the fractional-order ant colony algorithm.
- (b)
- In Section 3, a novel edge detection technique utilizing the fractional-order ant colony algorithm combined with fractional differential mask and coefficient of variation (FACAFCV) is presented. The heuristic function utilized in this approach is constructed by amalgamating the concepts of fractional differential mask and the coefficient of variation (CV).
- (c)
- In Section 4, a reasonable experimental strategy is designed to evaluate the results of edge detection on images with and without noise, using metrics such as recall, precision, and F-measure.
- (d)
- Section 5 performs a comprehensive analysis of our method through three distinct experiments. Firstly, the impact of the fractional differential mask and coefficient of variation on the edge detection performance is examined. Secondly, the performance of both fractional-order ant colony algorithm combined with fractional differential mask and coefficient of variation (FACAFCV) and fractional-order ant colony algorithm combined with coefficient of variation (FACACV) in the presence of multiplicative noise is studied. Finally, a standard benchmark evaluation is carried out on the widely-used dataset to assess the effectiveness of the proposed method.
- (e)
- Section 6 delves into the merits and limitations of our method, as well as future research directions. Furthermore, potential applications of our method are outlined.

## 2. Background

#### 2.1. Fractional Calculus

#### 2.2. Fractional-Order Ant Colony Algorithm (FACA)

## 3. Fractional-Order Ant Colony Algorithm Combined with Fractional Differential Mask and Coefficient of Variation (FACAFCV) for Image Edge Detection

## 4. Experiments Methodology

## 5. Result and Analysis

#### 5.1. Effect of Fractional-Order Coefficients V in FACAFCV

#### 5.2. Comparison of FACAFCV and FACACV on Images with Multiplicative Noise

#### 5.2.1. Synthetic Image

#### 5.2.2. Real Image

#### 5.3. Test on BSDS500 Dataset

## 6. Discussion and Future Directions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Original image and ground truths: (

**a**) Original image; (

**b**) Ground truth obtained by Canny; (

**c**) Ground truth obtained by Roberts; (

**d**) Ground truth obtained by Sobel.

**Figure 3.**The impact of fractional-order coefficients v in FACAFCV: (

**a**) −0.8; (

**b**) −0.6; (

**c**) −0.4; (

**d**) −0.2; (

**e**) 0; (

**f**) 0.2; (

**g**) 0.4; (

**h**) 0.6; (

**i**) 0.8.

**Figure 4.**Average recall curves of edge images obtained by FACAFCV with different fractional differential orders.

**Figure 5.**Average precision curves of edge images obtained by FACAFCV with different fractional differential orders.

**Figure 6.**Average F-measure curves of edge images obtained by FACAFCV with different fractional differential orders.

**Figure 8.**Synthetic image: (

**a**) Low level of noise; (

**b**) Medium level of noise; (

**c**) High level of noise; (

**d**) Low level of noise of FACAFCV; (

**e**) Medium level of noise of FACAFCV; (

**f**) High level of noise of FACAFCV; (

**g**) Low level of noise of FACACV; (

**h**) Medium level of noise of FACACV; (

**i**) High level of noise of FACACV.

**Figure 9.**Average threshold curves of edge images obtained by FACAFCV and FACACV on synthetic images with different noise levels.

**Figure 10.**Average recall curves of edge images obtained by FACAFCV and FACACV on synthetic images with different noise levels.

**Figure 11.**Average precision curves of edge images obtained by FACAFCV and FACACV on synthetic images with different noise levels.

**Figure 12.**Average F-measure curves of edge images obtained by FACAFCV and FACACV on synthetic images with different noise levels.

**Figure 13.**Real image and ground truths: (

**a**) Original image; (

**b**) Ground truth obtained by Canny; (

**c**) Ground truth obtained by Roberts; (

**d**) Ground truth obtained by Sobel.

**Figure 14.**Real image: (

**a**) Low level of noise; (

**b**) Medium level of noise; (

**c**) High level of noise; (

**d**) Low level of noise of FACAFCV; (

**e**) Medium level of noise of FACAFCV; (

**f**) High level of noise of FACAFCV; (

**g**) Low level of noise of FACACV; (

**h**) Medium level of noise of FACACV; (

**i**) High level of noise of FACACV.

**Figure 15.**Average threshold curves of edge images obtained by FACAFCV and FACACV on real images with different noise levels.

**Figure 16.**Average recall curves of edge images obtained by FACAFCV and FACACV on real images with different noise levels.

**Figure 17.**Average precision curves of edge images obtained by FACAFCV and FACACV on real images with different noise levels.

**Figure 18.**Average F-measure curves of edge images obtained by FACAFCV and FACACV on real images with different noise levels.

Parameters | Value |
---|---|

$v$ | 0.75 |

${Q}_{a}$ | $\u230a\sqrt{R\times C}\u230b$ |

${Q}_{p}$ | $\u230a2{Q}_{a}\u230b$ |

$\alpha $ | 1 |

$\beta $ | 5 |

$\rho $ | 0.2 |

${\tau}_{0}$ | 0.0001 |

$\xi $ | 1.3 |

memory | $\u230a2\sqrt{R+C}\u230b$ |

L | $\u230a3\sqrt{R\times C}\u230b$ |

**Table 2.**The metrics correspond to the optimal F-measure of edge images obtained by FACAFCV with different fractional differential orders.

v | Threshold | Recall | Precision | F-Measure |
---|---|---|---|---|

−0.8 | 0.03 | 0.8761 | 0.9471 | 0.9102 |

−0.6 | 0.04 | 0.8733 | 0.9426 | 0.9066 |

−0.4 | 0.07 | 0.8612 | 0.9453 | 0.9013 |

−0.2 | 0.10 | 0.8513 | 0.9380 | 0.8925 |

0 | 0.13 | 0.8418 | 0.9280 | 0.8828 |

0.2 | 0.17 | 0.8330 | 0.9142 | 0.8717 |

0.4 | 0.20 | 0.8215 | 0.8884 | 0.8536 |

0.6 | 0.27 | 0.8011 | 0.8644 | 0.8315 |

0.8 | 0.44 | 0.7916 | 0.8355 | 0.8129 |

Low Level of Noise | Medium Level of Noise | High Level of Noise | ||||
---|---|---|---|---|---|---|

FACAFCV | FACACV | FACAFCV | FACACV | FACAFCV | FACACV | |

Recall | 0.9923 | 0.8776 | 0.9286 | 0.4923 | 0.5689 | 0.2857 |

Precision | 1.0000 | 0.8982 | 0.8771 | 0.3333 | 0.5247 | 0.1532 |

F-Measure | 0.9962 | 0.8877 | 0.9021 | 0.3975 | 0.5459 | 0.1995 |

Threshold | 0.25 | 0.43 | 0.25 | 0.44 | 0.26 | 0.57 |

Low Level of Noise | Medium Level of Noise | High Level of Noise | ||||
---|---|---|---|---|---|---|

FACAFCV | FACACV | FACAFCV | FACACV | FACAFCV | FACACV | |

Recall | 0.8362 | 0.8278 | 0.8236 | 0.7587 | 0.7943 | 0.6799 |

Precision | 0.9535 | 0.9260 | 0.9551 | 0.9138 | 0.9408 | 0.7886 |

F-Measure | 0.8910 | 0.8741 | 0.8845 | 0.8291 | 0.8614 | 0.7302 |

Threshold | 0.02 | 0.1 | 0.04 | 0.24 | 0.08 | 0.35 |

ODS | OIS | Average Precision | |
---|---|---|---|

Human | 0.803 | 0.803 | - |

Canny | 0.611 | 0.676 | 0.520 |

FACAFCV | 0.589 | 0.608 | 0.533 |

FACACV | 0.558 | 0.579 | 0.487 |

IACACV | 0.552 | 0.571 | 0.497 |

Sobel | 0.539 | 0.575 | 0.498 |

Roberts | 0.483 | 0.513 | 0.413 |

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**MDPI and ACS Style**

Liu, X.; Pu, Y.-F.
Image Edge Detection Based on Fractional-Order Ant Colony Algorithm. *Fractal Fract.* **2023**, *7*, 420.
https://doi.org/10.3390/fractalfract7060420

**AMA Style**

Liu X, Pu Y-F.
Image Edge Detection Based on Fractional-Order Ant Colony Algorithm. *Fractal and Fractional*. 2023; 7(6):420.
https://doi.org/10.3390/fractalfract7060420

**Chicago/Turabian Style**

Liu, Xinyu, and Yi-Fei Pu.
2023. "Image Edge Detection Based on Fractional-Order Ant Colony Algorithm" *Fractal and Fractional* 7, no. 6: 420.
https://doi.org/10.3390/fractalfract7060420