Images Enhancement Based on a New Multi-Dimensional Fractal Created by Rectangular Function

applications. Several academics have created and applied a strategy for minimizing noise in features multiplicatively throughout the last several years. The outcomes reveal that the proposed strategy is successful. The method is based on the definition of the rectangular function (the elementary component of all digital signals, videos, and images), where this function indicates a rectangular-formed rhythm that is concentrated at the origin. For example, the suggested process received a rate of 97% for PNSR and 95% for RMSD.


Introduction
The use of fractional and fractal notions has grown in popularity over the last twenty years due to their benefits. Almost every study found that reordering these two principles resulted in a significant increase in image processing. Image noise is a significant issue in all image investigations, such as segmentation, texture examination, and image removal. Denoising is the most common stage in image processing. Reduced noise in image multiplicative (DNM) has recently been suggested. The exploration of Commonsensical Imaging Assemblies (CIA) [1,2]. Employing synthetic aperture radar, laser, and ultrasound is being driven by DNM imitators. The normal stabilizer noise assembly is insufficient for catching such images because of the commonsense setup of these image success operations. DNM models provide excellent CIA score explanations. DNM formed using partial and ordinary diffusion equations, variation techniques, slope operational methodology, fractional entropy type Tsallis entropy, and updated fractional calculus, including conformable fractional calculus, have all shown improved performance [3][4][5][6].
In all areas, including computer sciences, the fractal and fractional operators (differential and integral) are concisely proposed in image processing [7][8][9][10][11]. It's for medical MRI improvement [12] and is suggested in feature treatment. Furthermore, noise refers to any unwanted signal (note that multiplicative noise is defined as an unrequited arbitrary signal multiplied into a specific connected signal during processing) that perverts digital images, primarily through feature realization, which is the process of converting an optical image into continuous electrical signals. To clarify the challenges in constructing an X-ray feature, some research related to numerical image processing and statistical analysis of X-ray images is suggested [13]. In [14,15], the authors suggest a well-organized differential box-counting approach for calculating the fractal dimension of feature enhancement. The study of polymer images [16] has seen more applications of FD. Furthermore, a sort of fractal known as the quantum fractal (also known as the Jackson fractal) has recently been utilized to assess medical photographs [17,18,19].
Researching consistent imaging arrangements such as sonar, ultrasound, laser, and radar requires multiplicative noise images. In light of the systematic Gaussian preservative noise issue, these photos explain two additional films of difficulty: first, the noise is multiplied by the original image; second, the noise is not Gaussian [20]. In [21], the researchers investigated a theoretical investigation for non-autonomous stochastic fractional equations generated by multiplicative noise, with the fractal dimension bounding in the unit interval . Researchers in [22,23] provide fractal analyses and surface organization in authentication and identification protocols. A medicinal picture verification technique is accomplished in the researchers' attempt in [24] related to wavelet reestablishment and fractal dimension analysis. The images are eliminated, and a fractal property is formed as the verification image of the pictures when rendering to the singularity of the fractal dimension of the block data [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
In this work, a new approach based on a central kind of MDF and the rectangular function was proposed in this investigation. MDF's advantages include using a multi-dimensional formula in its creation, which may be applied to analyzing complicated systems such as images. Four masks are proposed for and orders intended to predict the predicted structures of MDF windows. For capacity, different strainers are used. The investigation revealed that skilled filtering has a higher score than several current fractional and fractal filters.
Following are the stages: The approach to be employed in the recommended strategy is discussed in Section 2. The new MDF, generic window, and fractional mask are all part of it. Section 3 discusses the use of the recommended approach MDF in image enhancement. Finally, section 4 will include the conclusions, last comments, and future works.

Related Works
This section presents a list of related works using the method of rectangle function in image processing. The smooth rectangle function is considered in the work of Boukharouba [40] to design a filter in image processing. Yang et al. [41] utilized image processing approaches to investigate hydrodynamic features in a rectangular function. Training data is also investigated by Bargsten and Schlaefer [42] with the help of the rectangle function. He et al. [43] improved the edge extraction algorithm via the rectangle function. Meng et al. [44] designed a mathematical model for heterogeneous material based on digital image processing by employing the rectangle function. Ferreira et al. [45] approximated the rectangle function to model color image processing. The multi-dimensional fractal approach in this study by using the rectangle function was modified. The new technology is used to improve images as a software application. Recently, the multi-dimensional fractal has been indicated in image processing covering many factors in this area, such as the remote sensing images [46], the similarity of images [47], enhanced images [48], and classification of images [49] and optical images [50].

The Proposed Method
The following subsections are included in this section:

Multi-Dimensional Fractal (Mdf)
Dimension (size) box-totaling: The box-counting mod [25] for fractal dimension is the most casual and often used method in nonfiction. Using a binarized image of the retinal tree as a starting point, the image was covered with a network of sidelength boxes and calculated the number of boxes, which included a portion of the tree. Now minimizing the casing, more information about the tree was extracted. The structure formulates the box count as a function of the box-totaling size by letting be the box-count as a function of the box-totaling size [26,27].
Where denotes the grid cell size (a place-designated neuron) and the number of cells. The technique for determining the box-totaling dimension is computed similarly in information theory (entropy theory) and by employing values of the item, as follows: [28,29] ( ∑ ) ( where is the entropy, is the overall sum of boxes that contains a portion of the tree and is the quantity of retinal tree enclosed in the -the box; the total of pixels in -the box is , and the sum of pixels in the tree is . Finally, the correlation integral, approximated by the summation formal, is used to build a correlation structure [30]. ∑ Where is the correlation, indicates the Heaviside function, and the number of tree pixel pairs with the distance between them is less than . Finally, the extended structure (multi-fractal structures) is considered by the formal for a real number [31].
where ∑ Confirm that all of the above MDF fulfill the inequality below, which is a highly useful pattern for computing fractal dimensions in rehearsal [32,33] Each of these MDFs was used to explain the overall dynamical, topological, and geometric properties of a given arrangement, including the higher order extensions . However, understanding how these assets evolve at various dimensions is crucial [34].

MDF Defining By RF
The rectangular function is formulated by the structure: { A limit of a rational function can also be used to represent the rectangle function.
New formulate MDF utilizing RF was proceeded. The correlation structure (6) implies that: Where: (∑ ) ∑ Using the accumulated formula to get the generalized structure: Where: ∑ Where:

Image Denoising Method
In digital images, the Image Denoising Method (IDM) has various effects, some of which are almost invisible. Image denoising aims to yield a fresh feature with less noise, i.e., a feature closer to noise-free. Pixel-based picture filtering and patch-based filtering are the two basic ways that IDMs use to recognize images. The first technique uses a contiguity functional (juxtaposition) that operates on a single pixel and depends on its three-dimensional neighboring pixels inside a kernel. The second category employs blocks of identical patches that are then operated definitively to transmit an estimate of the main pixel values based on comparable bits within an observed window. This approach focuses on the work image's termination, and similarity among the most common portions see Figure1 and Figure 2.
The problem of image denoising is designed mathematically as follows: (9) where presents the figured noisy image, indicates the novel (main) image, and is the additive white Gaussian noise (AWGN) with standard deviation. Many methods, including median absolute deviation [36], block-based approximation [37], and conventional module analysis cited3, are used to calculate AWGN in applied presentations. The noise reduction goal is to minimize the loss of creative elements while improving the signal-to-noise ratio (SNR). According to polls [39], the basic standards for image denoising are: smooth areas must remain planar, edges must be threatened without blurring, textures must be preserved, and new objects must not be stopped.

Algorithm
Eq.s (7) and (8) are used in this method. Eq.(7) shows a 1-D parametric conclusion with the parameter , whereas Eq. (8) shows a 2D one with the parameters and . To improve the noisy image will use both of these equations to get the indicated improved picture by using Eq.s (7) and (8) together (10) and (11) respectively, where is the convolution product and represents the window mask of size . In this study, different masks are organized as follows, using (7) [ Similarly, by using (8) and then the following structures are obtained: [ ] [ ] [ ] [ ] The algorithm is structured as follows: Algorithm 1: The systematic algorithm utilizing the new formula of MDF Input: Noise Images; Output: estimate the value of b with the higher PNSR; begin; Input noise images; Convoluted with the window; Estimate the value of b with the higher PNSR; end At this point, we should mention that PNSR (peak signal-to-noise ratio) is a trade term for the ratio between a signal's utmost probable power and the strength of degrading noise, indicating its representation's accuracy. PSNR is usually expressed as a logarithmic number using the decibel scale since many images have a wide dynamic range. The structure has been formulated as follows: where is the maximum value of the pixel and is the mean error

∑ ∑
Where is the noise image. Furthermore, the root-mean-square error (RMSE) is a commonly utilized ratio of the variations between standards predicted by a design, an estimator, and the principles discovered. The square root of the second model moment of the discrepancies between projected tenets and experimental ethics, or the quadratic mean of these alterations, is referred to as the RMSD. The following structure is used to identify it.
Each imaging unit (pixel) has a color value that can change when a photograph is compressed and subsequently uncompressed. Because signals have a wide dynamic range, PSNR (RMSD) is usually expressed in decibels, a logarithmic scale. The analysis is presented between PNSR (RMSD) and the probability of the pixel since the value of the pixel can be charged by its probability, as in this technique, which is based on the probability of the pixel (7) and (8).

Results and Discussion
In this part, the specified material from the previous part was used in this section to demonstrate the findings. Mathematica 11.2 was used to carry out the performance calculations. The two picture collections included two gray-scale images and two color images. The proposed method's windows mask designed to work with a 3 3 pixel window see  Both PSNR and RMSD measurements were used to plan the MDF calculation presentations. PSNR values of in (7) and for (8) are described in the intervals when (0,1) and (0,1). The optimum value of PSNR is identified for the second approach, (8), for all proposed data, as shown in Table 1. The findings of the RMSD test are the same as those in Table 2. The author of the reference citemm6 utilized a one-dimensional fractal differential operator, which is compared to the recent technique. When compared to the findings of [6], it is evident that this solution provides a tremendous improvement. For example, in the earth image, the enhancement rate given by the PNSR test in [6] is equal to 89.99%, while the proposed method (7) yields 97.2% and (8) implies 92%.
Another description of MDF, based on specific special functions, might be proposed for further research. This approach may be used in various fields of computer science and can also be used to research different sorts of image processing. A range of applications or modifications is indicated using parametric mathematical frameworks. The effort of Ibrahim [6] is compared in the manner shown in Tables 1 and 2.

Conclusions
A new method of image enhancement based on Multi-Dimensional Fractal (MDF) as a pre-processing step for lowcontrast images is planned in this investigation. Different images with low gray-level have been enhanced by applying the suggested image enhancement model, which estimates the possibility of every image pixel. Regarding testing consequences on a diversity of images, the suggested approach outperforms existing approaches in the broad application of image development. The main restraint of this investigation is that the presentation of the suggested model extremely declines as the complexity of the input images grows. Future efforts may modify the existing model to detailed applications to optimize the elevation's aids. The improvement brought a 97% rate in PNSR in the earth image and a 95% rate in RMSD in the Lena image.

Author contribution
All authors contributed equally to this work.

Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.