# Research on Application of Fractional Calculus Operator in Image Underlying Processing

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## Abstract

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## 1. Introduction

## 2. Basic Theory

#### 2.1. Fractional Calculus Theory

- The Grünwald–Letnikov approach to fractional calculus is defined as follows:$${}_{a}{}^{G}D{}_{t}^{v}f(x)=\underset{h\to 0}{\mathrm{lim}}\text{}{h}^{v}{\displaystyle \sum _{j=0}^{\frac{t-a}{h}}\frac{\mathsf{\Gamma}(v+j)}{j!\mathsf{\Gamma}(v)}f(x-jh)}\text{}v\in R$$
- The Riemann–Liouville definitions for fractional-order integration and differentiation are as follows:$$\left\{\begin{array}{l}{}_{a}{}^{R}D{}_{t}^{-v}f(x)=-\frac{1}{\mathsf{\Gamma}\left(v\right)}{\displaystyle {\int}_{a}^{t}\frac{f(y)}{{(x-y)}^{1-v}}dy}\\ {}_{a}{}^{R}D{}_{t}^{v}f(x)=-\frac{1}{\mathsf{\Gamma}(n-v)}\frac{{d}^{n}}{d{t}^{n}}{\displaystyle {\int}_{a}^{t}\frac{f(y)}{{(x-y)}^{1+v-n}}dy}\end{array}\right.$$
- Caputo’s definition of fractional calculus is outlined as follows:$${}_{a}{}^{C}D{}_{t}^{v}f(x)={}_{a}{}^{R}D{}_{t}^{v}f(x)-{\displaystyle \sum _{j=0}^{m-1}\frac{{f}^{(j)}(a)}{\mathsf{\Gamma}(j-v+1)}{(t-a)}^{k-v}}$$

#### 2.2. Amplitude–Frequency Characteristics of Fractional Calculus Operators

#### 2.2.1. Fractional-Order Differential Operator

#### 2.2.2. Fractional-Order Integral Operator

#### 2.3. Fractional-Order Calculus Processing and Analysis of Common Signals

## 3. Application of Fractional-Order Differential in Image Enhancement

#### 3.1. Amplitude–Frequency Characteristics of Fractional-Order Differential Image Enhancement Operators

#### 3.2. Image Enhancement Experiment and Analysis of Fractional-Order Differential Operator

## 4. Application of Fractional-Order Integral in Image Denoising

#### 4.1. Amplitude–Frequency Characteristics of Fractional-Order Integral Operator Image Denoising Operator

#### 4.2. Experiment and Analysis of Fractional-Order Integral Operator for Image Denoising

#### 4.2.1. Construction of Fractional-Order Integral Operators

#### 4.2.2. Evaluation Criterion

- Subjective EvaluationSubjective evaluation entails gauging enhanced image quality through direct human visual inspection, aiming to capture authentic human visual perceptions. This method proves particularly valuable as it involves firsthand interaction with the image using human vision [36,37]. Leveraging the human eye’s keen sensitivity to details such as texture and edges, we prioritize examining the edges and textural nuances to assess the overall visual impact of the denoised image.
- Objective Evaluation
- Objective evaluation, on the other hand, employs mathematical metrics tailored to mirror specific image qualities that align with human perception. The subsequent results are derived from certain image attributes based on the evaluation function. This study makes use of key metrics such as average gradient, edge preservation coefficient, and signal-to-noise ratio to critically compare the performance of different image-denoising operators [38].
- Average Gradient (AG)The average gradient (AG) in an image serves as an indicator of contrast variations, reflecting the image’s textural and detail transitions. This offers insights into the image’s overall sharpness. The formula to calculate the AG value is provided in Equation (20).$$AG=\frac{1}{M\ast N}{\displaystyle \sum _{i=1}^{row}{\displaystyle \sum _{j=1}^{col}\sqrt{{{\displaystyle \Delta}}_{\mathrm{horizontal}}f{(i,j)}^{2}+{{\displaystyle \Delta}}_{\mathrm{vertical}}f{(i,j)}^{2}}}}$$
- Edge Preservation Index (EPI)The edge preservation index gauges how effectively a filtering operator maintains the image’s horizontal or vertical edges. A higher EPI value signifies better edge preservation by the operator in question. The formula to compute this coefficient is outlined in Equation (21).$$EPI=\frac{{\displaystyle \sum _{i=1}^{\mathrm{row}}{\displaystyle \sum _{j=1}^{col}\left|{{\displaystyle \Delta}}_{\mathrm{horizontal}}{f}_{after}(i,j)+{{\displaystyle \Delta}}_{\mathrm{vertical}}{f}_{after}(i,j)\right|}}}{{\displaystyle \sum _{i=1}^{\mathrm{row}}{\displaystyle \sum _{j=1}^{col}\left|{{\displaystyle \Delta}}_{\mathrm{horizontal}}{f}_{befor}(i,j)+{{\displaystyle \Delta}}_{\mathrm{vertical}}{f}_{befor}(i,j)\right|}}}$$
- Contrast (C)Image contrast refers to the relationship between the black and white intensities within an image, serving as a gradient scale that transitions from black to white. A higher contrast ratio suggests a broader spectrum of gradient levels, enhancing the image’s textural details. The methodology for determining the image’s contrast is encapsulated in Equation (22). Here, the parameter $Number$ represents the logarithm of the differences in grayscale values among the image’s eight neighboring regions.$$C=\frac{\left|{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{row}}{\displaystyle \sum _{j=1}^{col}\Delta f(i,j}})\right|}{Number}$$
- Signal-to-Noise Ratio (SNR)Lastly, the signal-to-noise ratio (SNR) acts as a vital metric for assessing image quality. It quantifies the ratio between the magnitudes of the image signal and the noise, giving a numerical value to the image’s clarity. The expression for the SNR is elaborated upon in Equation (23).$$SNR=10\times lg\left(\frac{{\displaystyle \sum _{i=1}^{row}{\displaystyle \sum _{j=1}^{col}f{(i,j)}^{2}}}}{{\displaystyle \sum _{i=1}^{row}{\displaystyle \sum _{j=1}^{col}{\left|f(i,j)-{f}_{denoise}(i,j)\right|}^{2}}}}\right)$$

#### 4.2.3. Experimental Results and Comparative Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Amplitude–frequency characteristic curve of fractional calculus operator. (

**a**) Fractional-order differential; (

**b**) Fractional-order integral.

**Figure 2.**Fractional calculus amplitude–frequency diagram of common signals. (

**a**) Square wave; (

**b**) Triangular wave; (

**c**) Sine wave; (

**d**) Gaussian signal.

**Figure 3.**Amplitude–frequency characteristic surface of two-dimensional signal fractional-order differentiation.

**Figure 5.**Inverted images filtered by fractional-order differential operator of different orders. (

**a**) Original image; (

**b**) v = 0.1; (

**c**) v = 0.5; (

**d**) v = 0.8; (

**e**) v = 1.0; (

**f**) v = 2.0.

**Figure 6.**Enhanced images processed by fractional-order differential operators of different order. (

**a**) Original image; (

**b**) v = 0.1; (

**c**) v = 0.5; (

**d**) v = 0.8; (

**e**) v = 1.0; (

**f**) v = 2.0.

**Figure 7.**Enhanced images processed by differential image enhancement methods. (

**a**) Original image; (

**b**) v = 0.5; (

**c**) v = 0.8; (

**d**) Sobel; (

**e**) Prewitt; (

**f**) Laplacian; (

**g**) CS; (

**h**) HE.

**Figure 8.**Grayscale histogram. (

**a**) Original image; (

**b**) v = 0.5; (

**c**) v = 0.8; (

**d**) Sobel; (

**e**) Prewitt; (

**f**) Laplacian.

**Figure 9.**Amplitude–frequency characteristic surface of two-dimensional signal fractional-order integration.

**Figure 11.**Denoising images processed by fractional-order integration operators of different orders. (

**a**) Original image; (

**b**) Noisy image; (

**c**) v = 0.01; (

**d**) v = 0.1; (

**e**) v = 0.5; (

**f**) v = 1.0.

**Figure 12.**Denoising images processed by different denoising methods. (

**a**) Original image; (

**b**) Noisy image; (

**c**) Mean denoising; (

**d**) Gaussian denoising; (

**e**) Wiener denoising; (

**f**) Fractional-order integral denoising.

**Figure 13.**Residual images of denoised images using different denoising methods. (

**a**) Mean denoising; (

**b**) Gaussian denoising; (

**c**) Wiener denoising; (

**d**) Fractional-order integral denoising.

Methods | AG | Contrast | Entropy |
---|---|---|---|

v = 0.5 | 0.0493 | 0.0149 | 0.9777 |

v = 0.8 | 0.0466 | 0.0142 | 0.9945 |

Sobel | 0.0397 | 0.0088 | 0.9661 |

Prewitt | 0.0349 | 0.0067 | 0.9629 |

Laplacian | 0.0489 | 0.0139 | 0.9712 |

CS | 0.0411 | 0.0091 | 0.9667 |

HE | 0.0474 | 0.0130 | 0.9835 |

Methods | Average Gradient | Edge Retention Coefficient | Signal to Noise Ratio |
---|---|---|---|

Mean | 0.0138 | 0.3547 | 18.2964 |

Gaussian | 0.0186 | 0.5388 | 19.3706 |

Wiener | 0.0165 | 0.4356 | 19.2437 |

Fractional | 0.0208 | 0.7084 | 19.8679 |

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**MDPI and ACS Style**

Huang, G.; Qin, H.-y.; Chen, Q.; Shi, Z.; Jiang, S.; Huang, C.
Research on Application of Fractional Calculus Operator in Image Underlying Processing. *Fractal Fract.* **2024**, *8*, 37.
https://doi.org/10.3390/fractalfract8010037

**AMA Style**

Huang G, Qin H-y, Chen Q, Shi Z, Jiang S, Huang C.
Research on Application of Fractional Calculus Operator in Image Underlying Processing. *Fractal and Fractional*. 2024; 8(1):37.
https://doi.org/10.3390/fractalfract8010037

**Chicago/Turabian Style**

Huang, Guo, Hong-ying Qin, Qingli Chen, Zhanzhan Shi, Shan Jiang, and Chenying Huang.
2024. "Research on Application of Fractional Calculus Operator in Image Underlying Processing" *Fractal and Fractional* 8, no. 1: 37.
https://doi.org/10.3390/fractalfract8010037