Image Denoising of Adaptive Fractional Operator Based on Atangana–Baleanu Derivatives

A fractional integral operator can preserve an image edge and texture details as a denoising filter. Recently, a newly defined fractional-order integral, Atangana–Baleanu derivatives (ABC), has been used successfully in image denoising. However, determining the appropriate order requires numerous experiments, and different image regions using the same order may cause too much smoothing or insufficient denoising. -us, we propose an adaptive fractional integral operator based on the Atangana–Baleanu derivatives. Edge intensity, global entropy, local entropy, and local variance weights are used to construct an adaptive order function that can adapt to changes in different regions of an image.-en, we use the adaptive order function to improve the masks based on the Grumwald–Letnikov scheme (GL_ABC) and Toufik–Atangana scheme (TA_ABC), namely, Ada_GL_ABC and Ada_TA_ABC, respectively. Finally, multiple evaluation indicators are used to assess the proposed masks. -e experimental results demonstrate that the proposed adaptive operator can better preserve texture details when denoising than other similar operators. Furthermore, the image processed by the Ada_TA_ABC operator has less noise and more detail, which means the proposed adaptive function has universality.


Introduction
e theory of fractional-order derivatives has been applied in many fields, such as physics, fluid mechanics, physiology, medical science, and epidemic diseases [1][2][3][4]. With the development of information science, fractional operators have gained incomparable advantages over integral operators in many fields. A fractional derivative recurrent neural network can effectively improve estimation accuracy in parameter identification [5]. Complex behaviors in fractional-order financial systems can provide theoretical basis for the government [6]. Fractional-order control systems perform more accurately and elegantly than traditional systems [7]. In signal processing, the characteristics of fractional differential operators, such as "nonlocality," "memorability," and "weak derivatives," are also applied [8][9][10].
ese properties can improve the high frequency of an image while preserving the performance of the low and medium frequencies. In other words, methods based on fractional calculus for enhanced images can enhance the texture details while preserving the texture details of the smooth region in images [11,12]. erefore, many scholars are engaged in research on the application of fractional operators in image enhancement and denoising. e most representative scholar is Y.F. Pu, who, with his team, constructed image enhancement and denoising operators by fractional calculus [10,13]. Based on the Grunwald-Letnikov (GL) approximation, a medical image enhancement method was proposed by Guan et al. [14]. An adaptive image enhancement operator based on fractionaldifferential and image gradient feature was proposed by Lan [15]. Arian Azarang inferred different structure mask to image fusion [16]. An adaptive fractional-order integral filter was presented for echocardiographic image denoising [17]. e basic theory of abovementioned fractional operators is mainly the definitions of GL and Riemann-Liouville (RL).
e Caputo derivative is another definition of fractional order that is widely studied and applied; it includes the numerical solutions of fractional equations and the properties of systems [18,19]. New fractional derivatives and applications based on the frame of the Caputo derivative have received much attention from experts. e existence and stability of Belouso-Zhabotinskii reaction systems with Atangana-Baleanu fractional-order derivatives are discussed in [20]. In [21], the locally and globally asymptotically stable of symbiosis system modelling by the Atangana-Baleanu derivative are analyzed. With the development of research, fractional-differential operators with nonlocal and nonsingular kernels are used to image filters [22][23][24]. Furthermore, an AB-fractional differential mask based on the Gaussian kernel has been introduced to detect blood vessels in retinal images [25]. Behzad Ghanbari and Abdon Atangana designed an ABC-fractional derivatives mask that is used for image denoising. e ABC-fractional derivatives mask is computationally efficient and has excellent performance in the denoising of nosy images [26]. In the process of denoising, many experiments are required to determine the order of the mask. Moreover, because using a fixed order may lead to excess or deficiency for denoising effect, an adaptive fractional operator based on Atangana-Baleanu derivatives is proposed in this paper, which is called Ada_GL_ABC. We consider the gradient of the image, local entropy, global entropy, and local variance weights to construct a function for solving the adaptive order. e starting point of this idea is removing the image noise while preserving the edge and texture details of the image as much as possible. e adaptive function proposed by us is different from that of other studies. We consider both global and local information, the adaptive function contains more comprehensive information when determining the order, and the order used for denoising is more appropriate. And, the adaptive function designed by us has a certain generality. e function can be applied not only to GL_ABC mask but also to TA_ABC mask, which is rarely seen in the previous literature. e remainder of this paper is organized as follows: the basic definitions of fractional derivatives and the structure of fractional-masks are introduced in Section 2. In Section 3, the function of the adaptive fractional-order integral operator based on Atangana-Baleanu derivatives is described. e performance of the proposed adaptive operator is discussed in Section 4. In Section 5, the conclusions are elaborated.

Definitions of the Fractional Derivatives.
Many basic definitions of fractional derivatives exist [27]. Recently, the Mittag-Leffler function was introduced to compute fractional derivatives. is new definition is named the Atangana-Baleanu fractional derivative; it is based on the definition of Liouville-Caputo (ABC) and can be defined as follows [28]: (1) e Atangana-Baleanu fractional integral with order β can be depicted as where A( * ) is a normalization function, and this function satisfies A(0) � A(1) � 1. It can be described by e ABC derivative inherits the memory of the Mittag-Leffler function, which, with index c, is denoted as e GL definition is one of the best-known definitions of discrete fractional calculus and is widely applied to image processing. Details of the GL definition are expounded in Definition 1.

Definition 1.
e GL definition of fractional calculus formula with α-order of [29] is described as where h is the step, [ * ] represents the rounded operation, β j � ((Γ(β + 1))/(j!Γ(β − j + 1))), and Γ(β) is Gamma function. Equation (5) can be further decomposed as follows: We know that equation (6) is the fractional derivative operator with α > 0, and it takes a part of the fractional integral operator with α < 0. When α > 0, we set c � − α, and the integral GL of order c using equation (6) is described as From the above discussion, the ABC-fractional integral can be described by equation (7): In [26], the newly defined fractional order integral is mentioned. It can be approximated to the following as t � t n : e function f(τ) can be described by a two-step Lagrange polynomial interpolation as follows: Using equation (10), equation (9) can be discretized as follows:

e Mask Based on Grunwald-Letnikov (GL_ABC) and
Toufik-Atangana (TA-ABC). In an image, the distance between two pixels can be assumed to be 1. is distance is the same as h in equation (7). erefore, the GL integral with fractional-order in the x and y directions [30] are described by equations (12) and (13): Journal of Mathematics e TA_ABC integral with fractional-order [31] in the x and y directions is presented as equations (14) and (15): with equations (12)-(15), the 5 * 5 fractional integral mask can be constructed as follows: is 5 * 5 mask is used in image denoising as the filter. e mask is rotation-invariant mainly because it is obtained by superimposition of fractional integral in eight directions. us, we can use different fractional-order integrals for airspace filtering to denoise images. erefore, the coefficients of GL_ABC and TA_ABC mask are described by equations (16) and (17), respectively (Table 1). , ,

Adaptive Fractional Operators Based on Atangana-Baleanu Derivatives
For an image with noise of different intensities and in different regions, one fixed order in the fractional integral operator is insufficient to achieve a good denoising effect. erefore, this paper proposes an adaptive fractional operator for image denoising. e edge intensity coefficient, image entropy, local entropy, and local variance weight are used to construct the expression of the adaptive fractional order.
e image gradient represents the image edge intensity information. In this paper, e Kirsch algorithm is applied to calculate the image edge intensity. However, the Kirsch algorithm can suppress image noise [32].
P is the probability that an image pixel will appear. e local variance weight can not only measure the local gray change of an image but also reflect the importance of the image local change rate in the whole image. e larger the difference in partial pixel values is, the greater the local variance weight. Conversely, the smaller the changes are, the smaller the value of the local variance weight [33].
where Num represents the number of image pixels, I is the image to be processed, h is the local pixel, h' is the local pixel of the current window, and σ 2 I (h ′ ) is the variance in the pixel value in the current window. e function established in this paper takes the global entropy of the image as a measure of the overall image characteristics. e order value should be small to maintain the texture details. We consider taking the product of three measures of frequency information, to ensure that the fractional-order is inversely proportional to high frequency information such as edges and texture details. e adaptive order function is as follows: where E t represents the entropy of the whole image, ε is the coefficient of E t and take 0.22 in the experiment, and G * St * E l is the product of the local information entropy, local gradient, and local variance. en, the entropy of the global image is subtracted from the product. e results are small in the region of the edge and texture and high in the region of smoothness. According to this equation, the orders of different local textures of the image vary. As shown in Figure 1, the order used in the edge and texture details is relatively small, while the order used in the smooth area is larger. In this way, the obtained orders are reduced in the edge and texture detail region and enhanced in the smooth region so that the edge and texture detail information can be preserved as much as possible while denoising.

Numerical Examples
In this paper, the peak signal-to-noise ratio (PSNR), entropy, and structural similarity index measurement (SSIM) are used to assess the performance of the proposed operator. e PSNR is the most popular assessment criterion to evaluate the performance of denoising algorithms. In general, the value of the PSNR is higher when the image quality is better. e PSNR is defined as follows [34]: where M and N are the size dimensions of the original image. I(j, k) and I ′ (j, k) are the original and denoised images, respectively. e SSIM is also a well-known criterion among image quality assessment metrics [35] defined as where p and q represent different images; φ p and φ q represent the mean of images p and q; σ 2 p and σ 2 b represent the variances of p and q, respectively; σ pq is the covariance of p and q; and ρ 1 and ρ 2 are constants added to maintain stability. e value of the SSIM represents how similar two images are. When the SSIM value is higher, the pixel values of the two images are closer. e range of this index is [0, 1]. If the value of this index is closer to one, the two images are more approximate.
In the experiments, we employed five grayscale images to test the proposed mask: "Lena," "Elaine," "Goldhill," "Peppers," and "Cameraman," with 512 * 512 pixels each. We use the proposed adaptive function to improve the TA_ABC and GL_ABC mask. e improved mask "Ada_-TA_ABC" and "Ada_GL_ABC" compare the "GL_ABC mask" [31], "TA_ABC mask" [30], and the method proposed in [36]. e orders c in "GL_ABC mask" and "TA_ABC mask" are from literature [30]. In the experiments, we added noise with different variances σ ∈ 15, 20, 25 { } to the test images, respectively. Figures 2-16 show the results for image denoising by the different methods. Tables 2-7 show the PSNR, SSIM and entropy of these test images for the different algorithms. From Figures 1-16, we find that the test images lost image details when TA_ABC mask was applied. e method proposed by [36] has a poor denoising ability. e Ada_TA_ABC mask performs better than the TA_ABC mask, the order of which is determined by our proposed Journal of Mathematics method, as depicted by equation (15). e test images processed by the GL_ABC mask and Ada_GL_ABC mask contain less noise and more details. e Ada_GL_ABC mask is better than the GL_ABC mask. From another perspective, this result shows that the adaptive function proposed by us has certain universality. is result is verified in Tables 2-6. e PNSR of images proposed by the Ada_GL_ABC mask is higher than that of the other methods. is outcome means that the quality of images processed by the Ada_GL_ABC mask is better than that delivered by other methods. Meanwhile, the images processed by the Ada_GL_ABC mask are closer to the original images. is conclusion can be confirmed by the higher SSIM, which indicates the similarity between two images. Additionally, we calculated          Table 7. e entropy of images processed by the Ada_GL_ABC mask is closer to that of the original images than that got by other masks, but it is also higher than that of the original images. e results demonstrate that the detailed information of images is preserved while denoising. For Figures 2-16, (e) and (f ) are clearer than the others by visual evaluation. TA_ABC and GL_ABC have been improved by the proposed adaptive function. According to the quantitative indicators shown in the tables, the Ada_GL_ABC mask has better denoising and detail-preserving ability than other masks. Furthermore, Ada_GL_ABC and Ada_TA_ABC masks are robust for different intensity noise by the analysis of the results. e effectiveness of our proposed adaptive operator can be proved from the two aspects of vision and evaluation index.

Conclusions
In this paper, the adaptive denoising mask is proposed based on Atangana-Baleanu derivatives. e key to this method is the calculation of order. e order is determined by the intensity of the gradient, global entropy, local entropy, and local variance. ese variables represent the whole and local information of the image. To protect the texture details, we design the adaptive order integral operator considering global and local information. is operator can produce smaller orders in the image edge and texture details while larger orders in the smooth region. e proposed function is used to improve the GL_ABC mask and TA_ABC mask operator. We test the effectiveness of our proposed algorithm on multiple images. From a visual point of view, the denoising ability of Ada_TA_ABC and Ada_GL_ABC are reliable. Compared with other operators by the evaluation indicators, the Ada_GL_ABC operator works better. And, the PSNR and the SSIM are all higher under different intensities of noise. e information entropy index of the image processed by Ada_TA_ABC and Ada_GL_ABC operators is closer to the original images. e entropy of image filtered by Ada_GL_ABC mask is slightly larger, which indicates that Ada_GL_ABC mask can preserve texture details.
ese experimental results confirm that GL_ABC and TA_ABC have all been improved. And, the proposed adaptive function has a certain degree of universality.

Data Availability
e test images used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.