The interest in using fractal theory and its applications has grown in the field of image processing. Image enhancement is one of the feature processing tools, which aims to improve the details of an image. The enhancement of digital pictures is a challenging task due to the unforeseeable variation in the quality of the captured images. In this study, we present a mathematical model using a local conformable differential operator (LCDO). The proposed model is formulated by the theory of cantor fractal to generalize the definition of LCDO. The main advantage of utilizing LCDO for image enhancement is its capability to enhance the low contrast intensities using the coefficient estimate of LCDO. The proposed image enhancement algorithm is tested against different images with different qualities to show that it is robust and can withstand dramatic variations in quality. The quantitative results of Brisque, and Piqe were 30.38 and 35.53 respectively. The comparative consequences indicate that the proposed image enhancement model realizes the best image quality assessments. Overall, this model significantly improves the details of the given datasets, and can potentially help the medical staff during the diagnosis process. A MATLAB programming instrument utilized for application and valuation of the image quality measures. A comparison with other image techniques is illustrated regarding the visual review.

Kolwankar et al. [

Anderson et al. [

The image acquisition process sometimes generates poorly-illuminated images. To mitigate this issue, we suggest a new mathematical model that is based on combining the idea of fractal derivative and the conformable derivative to formulate the proposed LCDO for image enhancement. Applying the LCDO to enhance images based on estimating of the conformable differential operator. The coefficients are suggested by using the fractal sine function, which is a generalization of a fractal flame. We aim to provide an enhancement method to recover the interpretability or perception of data in pictures for human viewers, or to provide better input for other automated feature processing performances. The performance of the proposed image enhancement model was assessed using relevant image quality metrics, and compared with state-of-the art image enhancement techniques.

Medical imaging performances from time to time make pictures that: have objects, are low in difference, and/or do not obviously display the boundaries of the intuitive structures. To get these matters, we offer a novel mathematical modeling system, which is depended on the class of LCDO to improve the low contrast intensities of medical images. We aim to provide an enhancement technique for medical pictures so that physicians can offer scientific diagnoses earlier and additional positively.

The main contributions of this study are, as follows:

1-We present a unique low-light image enhancement method which can achieve better contrast enhancement on real low-light images.

2-We suggest a LCDO model using the coefficient estimation of a conformable differential operator for real low-light medical images enhancement.

3-The proposed LCDO can be applied as an efficient pre-processing step for any image processing approach.

Image enhancement methods are designed to enhance the visual appearance of an image such that the details of the image are significantly improved without altering the information of the image. In the literature, image enhancement can be categorized into spatial and frequency (regular) domain. Spatial domain image enhancement works on pixel values, while the image enhancement in frequency domain uses a transform approach of the images.

Recently, few image enhancement algorithms based on the concept of fractional calculus have been proposed. The fractional operators have the ability to keep the high frequency contour features, and to enhance the texture details. Roy et al. [

Despite the good results of this method, it was not shown to be able to preserve the fine details of medical images, which is required for the diagnosis process. Likewise, more image enhancement methods have been proposed deep learning approaches. Li et al. [

To overcome this problem, this investigation presents a new mathematical design based on LCDO for image enhancement, which is the main contribution of the study. The main advantage of LCDO is its ability to enhance the low contrast intensities through pixels’ illumination value based on a fractal conformable differential operator of the entire image. The proposed algorithm can enhance the image to a certain extent, and can display image information in a better way compared to the traditional image enhancement model.

This section briefly outlines the background of fractal theory and defines Fractal conformable differential operator. In terms of the LCDO, we introduce the fractal flame which is derived based on LCDO to get the enhanced image.

The conformable derivative of differential functions

which has applications in various scientific applications and engineering (see [

and it is not conformable at the limit

^{0} is the identity operator and ^{1} is the classical differential operator. Specifically,

Anderson et al. [_{p} is the proportional gain, _{d} is the derivative gain, and

_{0}, _{1}:[0, 1] × _{1}(_{0}(

_{1} and _{0}, one can assume that

We shall use the definition of the local fractional calculus that given in [_{0}| <

is finite and exists. We note that

For example

Thus, in general, for an analytic function

and fractal cosine function

The integral corresponds to the local fractional derivative operator is defined, as follows [

By combining the information in 3.1 and 3.2, we obtain the following LCDO:

such that_{1}(_{0}(

and

We have the following properties.

The integral operator corresponding to ^{(α)} is given by

Using the definition of LCDO, we have

Using the multiplication law of the fractal operator, we have

Finally, applying the division law of fractal derivative, we obtain

The last part can be obtained from Definition 3. This completes the proof.

Our class is based on the fractal flame component. Note that the individual function has the following form:_{n} is called the weight of the variation υ_{n}. Note that

υ_{0} = (_{1} = (

Using these components, the gamma correction is defined simply by

where υ_{in} is one of the component of the individual function,

In a generalization study, _{j}(

For example,

and

In the application part, we shall consider _{1}(_{0}(

Hence, we have the fractal conformable gamma corrections^{(α)}υ_{in} refers to the fractal conformable variation of the individual function. The fractal conformable operators showed a significant improvement in the application of image processing (see [

The following steps of the algorithm are presented to enhance the input image:

Consider the input image.

Experimentally obtain the fractal power (α) as tune image enhancement value.

Use the image to find the pixel probability value (x).

The values of α and x are used to enhance the input image as in

The quality of enhanced image is evaluated by using two no-reference image quality assessment metrics (Brisque, and Piqe).

In the proposed enhancement model, the fractal power α is the parameter for the fine detail enhancement, and it is fixed experimentally as shown in

The histogram analysis of the input and the enhanced images are shown in

The proposed image enhancement model LCDO is tested using the different images with different qualities. The code of the proposed image enhancement algorithm was developed using MATLAB 2020b. In this study, the brain MRI benchmark (BRATS) dataset [

To assess the suggested image enhancement design, we use two, no-reference image metrics, which are:

“The blind reference less image spatial quality evaluator (Brisque)”, which calculates the perceptual quality of images [

“Perception based Image Quality Evaluator (Piqe)”, which calculates the image quality affected by arbitrary distortion [

It is noted that lower scores of Brisque, and Piqe indicate better quality of the enhanced images.

To demonstrate that the proposed enhancement model is efficient as a medical image enhancement tool, we implemented the following existing methods for the comparative study: the fractional entropy based enhancement method of kidney images by Al-Shamasneh et al. [

The qualitative results of the proposed and the existing methods are illustrated in

Overall, the brightening caused by the proposed model makes the structures of the medical images, which usually represent boundaries, well defined and clear. This is accredited to the model's capability to capture high frequency details efficiently. The proposed method introduces fair visual results for the weakly illuminated images. This is the contribution of the fractional integral entropy in this study.

The achieved quantitative results of the proposed enhancement method and the existing image enhancement models are stated in

Methods | MRI images | |
---|---|---|

BRISQUE | PIQE | |

Input images | 36.50 | 41.00 |

Enhanced images by: | ||

Al-Shamasneh et al. 2018 [ |
36.46 | 39.83 |

Raghunandan et al. 2017 [ |
33.37 | 36.35 |

Z. Al-Ameen 2016 [ |
46.42 | 40.62 |

X. Fu et al. 2016 [ |
35.42 | 40.56 |

FPDEs [ |
42.35 | 40.77 |

30.38 | 35.53 |

In summary, it can be said that the proposed method achieves the best results compared with the mentioned methods due to its consistent results across the different datasets.

In this study, we presented a novel method for image enhancement utilizing fractal flame, which is based on LCDO. The proposed method embraces the local fractional theory and the fractal conformable differential operator. This model applied the LCDO, which dynamically enhanced the fine details of the medical images. The proposed enhancement model enhanced the fine details in images of the input low light images. The experimental results indicate that the proposed LCDO approach outperforms existing methods under the general application of image enhancement. This gives the proposed model the advantage of being more scalable for low light image enhancement than the existing methods. Future works may adapt the present model for specific applications to achieve maximum enhancement benefit.