This article is dedicated to the study of fractal operators and the review of their application for processing medical MRI images. The Atangana-Baleanu fractial operator is discussed in detail. Four numerical approximations of this operator are described, along with the process of finding the approximation coefficients. Based on the approximation operators, differently oriented mask are created for image denoising. These masks are implemented in the developed algorithm and software product, which allows improving image quality. A visual analysis of the results of the algorithms is performed, comparing the denoising image with original by using analytical params. Additionally, the results of the developed algorithms are compared with other algorithms, both based on fractial operators and classical image denoising methods. Conclusions regarding the use of the method based on the Atangana-Baleanu fractial operator for processing medical MRI images are presented.
Medical imaging is one of the tools used for diagnosing and monitoring patient conditions. It includes X-rays, ultrasound, computed tomography (CT), magnetic resonance imaging (MRI), and others. Each of these methods is used to visualize different body structures. MRI images provide detailed scans of internal organs and tissues, particularly the brain. Brain MRI is important for diagnosing diseases such as tumors, strokes, aneurysms, and other conditions that affect brain function. The quality of medical images can vary, which is especially important for brain MRI, where image clarity directly impacts diagnostic accuracy. Blurry, low-quality and noisy images can complicate data interpretation.
Image quality enhancement and denoising is possible through computer algorithms that help
remove noise and in the same time save structures on image. Possible method involves the use of
filter-based algorithms [
Fractial operators are used to model processes characterized by complex dynamics and irregular
structures. This applies to phenomena where traditional mathematical methods cannot adequately
describe system behavior, such as “memory” effects and self-organization in deformation-relaxation
processes [
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
nonlinear dynamics, chaotic behavior, and self-similar structures, allowing for a more accurate
description of their evolution and properties across different scales. Additionally, fractial operators
are used in various types of image processing, including edge detection, overall image enhancement,
noise reduction, and more [
Noise refers to unwanted signals or disturbances that disrupt or alter the main data, appearing at various frequencies and impacting the quality of signals or images. Fractional integrals are effective in reducing high-frequency noise while maintaining low-frequency components that carry important information. By adjusting the order of the integral, these methods facilitate efficient data smoothing and enhance complex textures without compromising their structure. This allows to improve image details without losing important information about the main structures, which is especially important when analyzing medical images.
Image enhancement using fractial operators, such as the Grunwald-Letnikov and
RiemannLiouville operators, is discussed in scientific research [
A new fractial operator is the Atangana-Baleanu operator [
The following section of the work involves the application of software-algorithmic tools to
reduce noise of medical MRI images using a fractial operators. The use of this method will allow to
improve quality of noised MRI images, contributing to more accurate diagnosis, better visual
representation, and enhanced automated systems for recognizing and classifying pathologies in MRI
images.
2. The main material is presentation
2.1. Fractial operators
Fractal operators based on fractional-order derivatives [
The fractional-order derivative based on the Grunwald-Letnikov [
− = lim →∞ Г( − − −1Г(
− ) ) =0 Г( + 1) ∗ − − ,
Formula (2) represents the fractial differential for the signal s(x), specifically the numerical approximation of the Grunwald-Letnikov fractial differential.
The Caputo [
1 Γ( − ) ( − ( )( ) ) − +1 , ( ) — the Caputo fractional differential of the function f(t) of order classical n-th derivative of f(t), t – the independent variable.
The Atangana-Beleanu derivative [
From formula (1) the fractial differential can be expressed as follows: values), - the Grunwald-Letnikov fractional differential operator, Г(·) - gamma function, (2) (3) (4) (5) (6) (7) to classical fractional derivatives, and makes it less sensitive to initial conditions.
The normalization function⋅ ℳ( )⋅can be expressed as:
Γ( ) The Atangana-Beleanu fractial integral has the following form: ℳ( ) = 1 − +
, In this section, the following approximations of the Atangana-Baleanu fractial operator are reviewed: approximation based on the Grunwald-Letnikov formula, the Toufiq-Atangana 퐴 ( ) =
' ( ) ( ) 1 − − ( − 1 − ) , 퐴
( )- the Atangana-Baleanu fractional derivative of a function f(t) of order σ, B(·) – the normalization constant,
-the generalized gamma function of Mittag-Leffler, f′(τ) – the first derivative of a function f(t), a - the lower limit of integration.
The Mittag-Leffler function can be expressed as: ( ) = ∞
=0 Γ( + 1) ,
> 0,
- the Mittag-Leffler kernel, which provides smoother behavior of the derivative compared
approximation [
The approximation based on the Grunwald-Letnikov formula (GL) Let’s take the Grunwald-Letnikov integral as the basis: ( − ( )
)1− = lim ℎ− ℎ→0 +⋯ + ( ) +
( − ℎ) + Γ( − ( − ! Γ( − − + 1)
By accepting the approximation step h = 1 for formula (8), and substituting it into formula (7), we obtain the approximation expression: ℐGL ( ) ≈
1 ℳ + 2 − 4 + ℳ
2 −3 + … −1 + 3 −
2 2ℳ( ) −2 +1 using the two(8) (9) (10) (11) (12)
Substituting (11) into (12) and performing the transformation with [
By expanding the expression under the sum and rewriting the polynom in the following form: ℐTA = (1 − )Γ(( +)Γ2() ++ 2ℎ) ( + 2) (14) + ℎ ( ( +)3Γ)(2 +− 22) − 4 −1 + ℎ ( + 4)3( −)Γ((2 ++25))2 + + 2 −2 + ℎ ( + 5)4 (− ()2Γ( ++6)23) + ( + 3)2 −3 + ⋯ The coefficients of the polynom (14) are equal to: 0 = (1 − )Γ(( )+Γ(2) ++ 2)( + 2) (15) 1 = 2 + 3( )2Γ(− 2+ 22)− 4 .
2 + 4 3 − 2 2 + 5 2 + 2 + 2 2 = ( )Γ( + 2) 2.2.3.
Approximation based on the Euler method (Eu) According to the Euler method described in [14], we approximate (7) at = ℐEu = ℳ1−( ) + ℳ( )Γℎ( + 1) −=10 , , as follows: where
, : ℐEu
, = ( − ) − ( − − 1) Substitute expression (17) into (16) and rewrite the polynom without the sign of the sum: = 1 −( ) + ( )Γℎ( + 1) −1 + (ℎ )Γ2( −+11) −2
+ ℎ( )Γ3( −+21) −3 + ⋯
The coefficients of the polynom (18) are equal to:
(16)
(17)
(18)
Based on the research results [
This formula can be extended to the two-dimensional space as follows: –signal value in particular , ≈ 0 , + S1 − 1, y + S2
− 2, y + …, , ≈ 0 , + S1 , − 1 + S2 , − 2 + …,
In article [
Figure 1 show mask corresponding to eight symmetric directions. The masks allow calculating the fractional differential in eight directions. The presence of eight masks creates anti-rotation properties, enabling the same processing result regardless of the texture's position in the image. 0, 1 … are the sequential coefficients from formulas (27) and (28). The application of matrices with approximation coefficients in the algorithm for image denoising is described next.
The algorithm consists of the following steps: 1. The original image is read in grayscale. The input image can be of any size and format. The color value is encoded in 1 byte, corresponding to 256 grayscale values. (25) (26) (27) (28) 2. Eight filters W1-W8 (matrices) are created to determine gradients in different directions (0°, 45°, 90°, 135°, …). Each filter highlights the edges or gradients of the image in a specific direction. 3. For each filter, a convolution operation is performed on the gradient-transformed grayscale image. This allow to extract different edge components of the image according to the filter directions. 4. The results of the convolutions from each filter are combined to create the final image, which represents the aggregate of the extracted gradients in various directions. The combination of gradients is achieved by calculating the arithmetic mean of the pixel values from the eight gradients. 5. The next step is to normalize the pixel values, as the color values may exceed or fall below the range [0; 255].
The algorithm was implemented in Python [15], using libraries for image processing such as OpenCV [16], also libraries NumPy and SciPy [17] were used.
Input data is a very important component of the research. Medical data is often hard to access because it contains confidential information. However, anonymized MRI images are available on certain platforms and can be used for scientific purposes. The dataset from the resource [18] contains hundreds of small-sized MRI images of the brain in .jpg format.
3.1. Visual comparison The first step in evaluating the algorithm is a visual comparison of the original image and the denoised images obtained using fractial approximations.
We will apply the approximation coefficients in the algorithm described above, using masks of size 5x5 and values in the range of [0.1; 0.9]. 3.1.1.
Results of the approximation (GL) algorithm based on the
In Figure 2 and Figure 3, the original image, noised and the denoise images using the GrunwaldLetnikov approximation are shown. It is noticeable that with an increase in the value of (σ), the noise become less visible. The best result achieved when σ = 0.9 for this approximation.
Result of the algorithm's operation for the approximation based on the Toufique-Atangana approach (T) The original image, noised and denoised image for σ = 0.3 are shown, while Figure 5 displays enhanced images for σ = {0.5, 0.7, 0.9} using the Taufik-Atangan approximation. It is noticeable that there is a progressive improvement in the image denoising, and for σ = 0.9 (Figure 4).
Result of the algorithm for approximation based on Euler's method (Eu) On Figure 6 and Figure 7, the original image, noised and approximations based on the Euler method are shown. This approximation denoise the noised image as the value of σ increases, visually σ=0.9 works best.
Result of the algorithm's operation for approximation based on the midpoint method. (MP) This approximation also shows gradual image denoising as the parameter σ increases.
To evaluate the quality of an image after applying enhancement techniques like denoising, it's important to look at two key measures: PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index).
PSNR helps quantify how much the processed image differs from the original, especially after noise reduction. A higher PSNR indicates fewer distortions, meaning the processed image retains better quality. Essentially, it tells you how close the filtered image is to the original in terms of pixel values.
On the other hand,SSIM goes beyond just pixel comparison and looks at the image's structure, contrast, and brightness. It’s more in line with how the human eye perceives image quality. SSIM values range from -1 to 1, with values closer to 1 indicating that the two images are highly similar in structure.
In summary, using both PSNR and SSIM together gives a more complete picture of image quality: PSNR provides a straightforward numerical measure of any loss, while SSIM offers insight into how similar the images look to the human eye.
PSNR can be calculated as: 푃 = 10 log10 푀퐴 푀 2
(μ 2 + μ 2 + 1)(σ 2 + σ 2 + 2) where, x and y are two images, μ and μ are the mean values (luminance) of x and y, σ 2 and σ 2 are the variances (contrast) of the x and y, σ is the covariance between the x and y, C1and C2 are constants to avoid instability when the denominators are close to zero.
From the graphs in Figure 10 all four approximation methods demonstrate an increasing trend in PSNR and SSIM values with higher σ values. The only exception is the method based on T, which shows a decline for small σ values.
The GL-based algorithm showed the worst image improvement among the observed parameters PSNR and SSIM. The PSNR starts at 26 and reaches around 29, indicating lower efficiency compared to other methods. SSIM values fluctuate between 0.38 and 0.5, indicating lower structural similarity of images compared to the other methods.
The T-based approximation algorithm shows better results than GL in the interval from 0.4. For lower values, the PSNR and SSIM values are the worst among the compared methods.
For the Euler (Eu) algorithm, PSNR values range from 26 to 31, which also indicates good quality, although lower than that of Eu. SSIM values increase from 0.4 to approximately 0.6. Although this method is less effective than Eu, it demonstrates satisfactory image quality while maintaining structure.
The MP-based algorithm starts with a PSNR of approximately 26 at σ=0 and reaches a maximum of about 32 at σ=0.6. This indicates high effectiveness in preserving image quality during processing. SSIM values start at 0.4 and rise to 0.65 at σ=0.8. The high SSIM value confirms that this method accurately reflects the structural similarity between the original and processed images.
In summary, based on the observations above, it can be stated that the best results are demonstrated by the Euler (Eu) approximation method in the range of σ=0.7 - 0.8.
Comparison of the result with other algorithms
We will conduct a visual and later an analytical comparison of the results of approximation
algorithms with other image denoising algorithms. First of all it is algorithm YiFePu3 [
Figure 11 shows the original image (a), noised original imgae (b), images denoised using the Atangana-Baleanu fractial operator approximations (c), (d), (e), (f), the YiFePu3 fractial operator (g), as well as those denoised by the Median filter algorithm (h).
Visually, the best results were achieved by the Atangana-Baleanu approximations (d),(e),(f) and YiFePu3 (g) algorithm. It is noticeable reduction in noise visibility and its smoothing in the denoised images. In addition, there is a preservation of texture in the image, as well as details in relatively smooth areas of the denoised images. For image (с) the noise is much more visible, but still enhancement is present. The median filter algorithm (h) has several drawbacks, one of which is that it can cause slight blurriness in images, particularly affecting fine details. As a result, both small details and the edges of textural elements appear blurred in the image.
According to Table 3, it can be stated that the highest values of PSNR is for the YiFePu3 algorithm, which aligns with the visual comparisons in Figure 11 (g). At the same time the hightest values of SSIM is for Median filter (h), hight SSIM indicates only similarity in texture but does not guarantee preservation of details.
For the algorithms based on the approximations of the Atangana-Baleanu integral, Table 1 shows that the maximum value of PSNR is achieved at the highest values of σ, except for the algorithm based on Eu approximation. For this algorithm, the maximum value of PSNR occurs at σ = 0.7, as for larger values of v PSNR do not become bigger. The SSIM values form Table 2 show almost the same behavior except that best SSIM for Eu method is for σ = 0.8 .
Thus, to summarize that in comparison, the YiFePu3 approximation (GL) and (Eu) AtanganaBaleanu show almost comparable results, while the approximations (A) and (Mp) show worst results, and GL the worst result. At the same time, algorithms based on fractial operators show significantly better results compared to standard algorithm of Median filter.
The algorithm based on the Atangana-Baleanu fractial operator can be effectively used to denoise image and improve the quality of MRI images, as confirmed by the results of the visual comparisons conducted. The analysis also showed that the enhanced images demonstrate hight values of PNSR and SSIM. High PSNR values indicate less distortion in the image and its proximity to the original, suggesting effective noise reduction. High SIMM values indicate a high structural similarity between the denoised and original images. Thus, the results of the visual comparison and numerical indicators confirm the effectiveness of the algorithm in denoising images, improving their quality and informativeness. Algorithm GL performed the worst for noise removal, while algorithm A showed better results. Algorithm Mp further improved the performance, but algorithm Eu achieved the best noise reduction outcomes.Each algorithm can be used with params σ = [0.1, 0.9].
When compared to other algorithms, it is evident that algorithms based on fractial operators (Atangana-Baleanu and YiFePu3 - Riemann-Liouville) show better results in image denoising compared to standard image denoising methods. Moreover, the algorithm based on (Eu) demonstrate better results than YiFePu3.
This study confirmed the properties of fractial operators to remove hight-frequency noise elements of images. Therefore, fractial operators are advisable for denoising of medical images, and further research in this direction is relevant.
During the preparation of this paper, the authors utilized Grammarly to verify spelling and grammar accuracy. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the publication’s content.
[14] C. Li, F. Zeng, The finite difference methods for fractional ordinary differential equations,
Numerical Functional Analysis and Optimization, vol. 34, no. 2, 2013, pp. 149–179. [15] G. Bradski, The OpenCV Library, Dr. Dobb's Journal of Software Tools, 2000. [16] G. Van Rossum, F.L. Drake Jr., Python Reference Manual, PythonLabs, 2001. [17] P. Virtanen, R. Gommers, T.E. Oliphant, et al., SciPy 1.0: Fundamental Algorithms for Scientific
Computing in Python, Nature Methods, vol. 17, 2020, pp. 261–272. [18] Kaggle, Brain MRI images for brain tumor detection, 2020. URL: https://www.kaggle.com/datasets/navoneel/brain-mri-images-for-brain-tumor-detection/data. [19] R.M. Haralick, K. Shanmugam, I. Dinstein, Textural Features for Image Classification, IEEE
Transactions on Systems, Man, and Cybernetics, vol. 3, no. 6, 1973, pp. 610–621. [20] H. Greenspan, C.H. Anderson, S. Akber, Image enhancement by nonlinear extrapolation in frequency space, IEEE Transactions on Image Processing, vol. 9, no. 6, Jun. 2000, pp. 1035–1048. [21] S. Dippel, M. Stahl, R. Wiemker, T. Blaffert, Multiscale contrast enhancement for radiographies: Laplacian pyramid versus fast wavelet transform, IEEE Transactions on Medical Imaging, vol. 21, no. 4, 2002, pp. 343–353. [22] C.E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, vol. 27, no. 3, 1948, pp. 379–423.