# Novel Low-Pass Two-Dimensional Mittag–Leffler Filter and Its Application in Image Processing

## Abstract

**:**

## 1. Introduction

- •
- Low-pass filters, commonly called smoothing filters, effectively eliminate high spatial frequency noise from a digital image. Using a moving window operator, these filters systematically process individual pixels within the image by modifying their values according to a predefined function that incorporates the neighboring region of pixels.
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- High-pass filters impart a sharper visual perception to an image (edge detection and sharpening). In direct contrast to low-pass filters, high-pass filters accentuate intricate details within the image. Like low-pass filtering, high-pass filtering operates using a convolution kernel, distinct from low-pass filters.

- •
- Gaussian filter—this variant of a low-pass filter effectively smooths images, reduces noise, and performs image blurring while simultaneously preserving significant image attributes, such as edges and corners.
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- Laplacian filter—a high-pass filter predominantly utilized for enhancing edges and detecting notable features such as edges and corners.
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- Median filter—as a non-linear filter, it specifically targets the removal of salt-and-pepper noise while striving to preserve image edges.
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- Bilateral filter—an edge-preserving filter that facilitates image smoothing while preserving edges and details.
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- Anisotropic filter—an edge-preserving filter capable of smoothing images while preserving edges oriented in a specific direction.
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- Homomorphic filter—a frequency-domain filter used to rectify non-uniform illumination and augment image contrast.
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- Sobel filter—this filter is commonly employed for detecting edges within an image.
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- Wavelet filter—a multi-scale filter that finds utility in image compression, feature extraction, and image denoising.
- •
- Wiener filter—an adaptive filter designed for removing noise from images.

- Definition of new two-dimensional Mittag–Leffler distribution.
- The novel two-dimensional Mittag–Leffler filter with three adjustable parameters and its practical implementation in Matlab are defined.
- We provide evidence using performance indicators (peak signal-to-noise ratio) that the proposed filter is more flexible and gives better results than a classical filter with similar structure and properties.

## 2. Preliminaries

#### 2.1. Gaussian Distribution

#### 2.2. Gaussian Filter

#### 2.3. Mittag–Leffler Distribution

## 3. Methods

#### 3.1. Novel Filter Definition

#### 3.2. Implementation Notes

`mlf(alpha,beta,Z,P)`. Moreover, using the mentioned algorithm, the proposed filter (11) can be easily implemented as a Matlab function. The Matlab function of the suggested Mittag–Leffler filter (11) was also created, and it is freely available together with demo on the MathWorks, Inc. website. It has the following header [24]:

`[img_filt]=ML_filter_2D(img,sigma,alpha,beta)`.

#### 3.3. Performance Indicator

`psnr()`in Matlab) and has precise physical meanings [25]. Higher PSNR values correspond to better quality of reconstructed or filtered image. This indicator can also be used as a cost function for optimal tuning of the filter parameters.

## 4. Results

## 5. Discussion

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Burger, W.; Burge, M.J. Digital Image Processing: An Algorithmic Introduction; Springer: Cham, Switzerland, 2022. [Google Scholar]
- Matei, R. A Class of directional zero-phase 2D filters designed using analytical approach. IEEE Trans. Circuits Syst. I Regul. Pap.
**2022**, 69, 1629–1640. [Google Scholar] [CrossRef] - Mafi, M.; Martin, H.; Cabrerizo, M.; Andrian, J.; Barreto, A.; Adjouadi, M. A comprehensive survey on impulse and Gaussian denoising filters for digital images. Signal Process.
**2019**, 157, 236–260. [Google Scholar] [CrossRef] - Sumiya, Y.; Otsuka, T.; Maeda, Y.; Fukushima, N. Gaussian Fourier pyramid for local Laplacian filter. IEEE Signal Process. Lett.
**2022**, 29, 11–15. [Google Scholar] [CrossRef] - Li, K.; Príncipe, J.C. Functional Bayesian filter. IEEE Trans. Signal Process.
**2022**, 70, 57–71. [Google Scholar] [CrossRef] - Deng, G.; Cahill, L.W. An adaptive Gaussian filter for noise reduction and edge detection. In Proceedings of the 1993 IEEE Conference Record Nuclear Science Symposium and Medical Imaging Conference, San Francisco, CA, USA, 31 October–6 November 1993; Volume 3, pp. 1615–1619. [Google Scholar] [CrossRef]
- Chang, S.Y.; Wu, H.C. Tensor Wiener Filter. IEEE Trans. Signal Process.
**2022**, 70, 410–422. [Google Scholar] [CrossRef] - Koranga, P.; Singh, G.; Verma, D.; Chaube, S.; Kumar, A.; Pant, S. Image denoising based on wavelet transform using visu thresholding technique. Int. J. Math. Eng. Manag. Sci.
**2018**, 3, 444–449. [Google Scholar] [CrossRef] - Seddik, H. A new family of Gaussian filters with adaptive lobe location and smoothing strength for efficient image restoration. EURASIP J. Adv. Signal Process.
**2014**, 2014, 25. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Chen, D.; Chen, Y.Q.; Xue, D. Digital fractional order Savitzky-Golay differentiator. IEEE Trans. Circuits Syst. II Express Briefs
**2011**, 58, 758–762. [Google Scholar] [CrossRef] - Sheng, H.; Chen, Y.Q.; Qiu, T.S. Fractional Processes and Fractional-Order Signal Processing; Springer: London, UK, 2012. [Google Scholar]
- Yang, Q.; Chen, D.; Zhao, T.; Chen, Y.Q. Fractional calculus in image processing: A review. Fract. Calc. Appl. Anal.
**2016**, 19, 1222–1249. [Google Scholar] [CrossRef] - Gupta, A.; Kumar, S. Design of Mittag–Leffler kernel-based fractional-order digital filter using fractional delay interpolation. Circuits Syst. Signal Process.
**2022**, 41, 3415–3445. [Google Scholar] [CrossRef] - Agahi, H.; Alipour, M. Mittag–Leffler–Gaussian distribution: Theory and application to real data. Math. Comput. Simul.
**2019**, 156, 227–235. [Google Scholar] [CrossRef] - Pillai, R.N. On Mittag–Leffler functions and related distributions. Ann. Inst. Stat. Math.
**1990**, 42, 157–161. [Google Scholar] [CrossRef] - Huillet, T.E. On Mittag–Leffler distributions and related stochastic processes. J. Comput. Appl. Math.
**2016**, 296, 181–211. [Google Scholar] [CrossRef] - Albrecher, H.; Bladt, M.; Bladt, M. Matrix Mittag–Leffler distributions and modeling heavy-tailed risks. Extremes
**2020**, 23, 425–450. [Google Scholar] [CrossRef] - Leonenko, N.; Podlubny, I. Monte Carlo method for fractional-order differentiation extended to higher orders. Fract. Calc. Appl. Anal.
**2022**, 25, 841–857. [Google Scholar] [CrossRef] - Petráš, I. Novel generalized low-pass filter with adjustable parameters of exponential-type forgetting and its application to ECG signal. Sensors
**2022**, 22, 8740. [Google Scholar] [CrossRef] [PubMed] - Rau, R.; McClellan, J.H. Efficient approximation of Gaussian filters. IEEE Trans. Signal Process.
**1997**, 45, 468–471. [Google Scholar] [CrossRef] - Wells, W.M. Efficient synthesis of Gaussian filters by cascaded uniform filters. IEEE Trans. Pattern Anal. Mach. Intell.
**1986**, PAMI-8, 234–239. [Google Scholar] [CrossRef] [PubMed] - Podlubny, I.; Kacenak, M. Mittag–Leffler Function, MATLAB Central File Exchange. Available online: https://www.mathworks.com/matlabcentral/fileexchange/8738 (accessed on 31 July 2022).
- Petráš, I. Mittag–Leffler 2D Filter for Image Processing, MATLAB Central File Exchange. Available online: https://www.mathworks.com/matlabcentral/fileexchange/131039 (accessed on 13 June 2023).
- Wang, Z.; Bovik, A.C.; Sheikh, H.R.; Simoncelli, E.P. Image quality assessment: From error visibility to structural similarity. IEEE Trans. Image Process.
**2004**, 13, 600–612. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Behavior of the Mittag–Leffler function ${E}_{\alpha ,1}(-{x}^{2})$ for various parameter $\alpha $ within interval $\alpha \in (0;2]$ and fixed $\beta =1$.

**Figure 2.**Mittag–Leffler two-dimensional distribution function (9) for parameters $\alpha =0.5$, $\beta =0.5$, and $\sigma =3$.

**Figure 3.**Results of image blur after applying the Mittag–Leffler filter (11): (

**a**) Original image without using a filter. (

**b**) Filter parameters $\alpha =1.0$, $\beta =1.0$, and $\sigma =3$, i.e., the Gaussian filter (3). (

**c**) Filter with parameters $\alpha =0.5$, $\beta =1.0$, and $\sigma =3$. (

**d**) Filter with parameters $\alpha =\beta =0.5$, and $\sigma =3$.

**Figure 4.**Result of image denoising with added Gaussian noise of zero mean and variance equal 0.05: (

**a**) Noised image without filtering. (

**b**) Mittag–Leffler filter (11) with parameters $\alpha =\beta =0.90$, $\sigma =1.10$, and PSNR = 24.20 dB. (

**c**) Wiener filter with neighborhoods of size $3\times 3$, and PSNR = 20.93 dB. (

**d**) Gaussian filter (3) with parameter $\sigma =1.10$, and PSNR = 22.10 dB.

**Figure 5.**Result of image denoising with added salt-and-pepper noise with the density 0.02: (

**a**) Noised image without filtering. (

**b**) Mittag–Leffler filter (11) with parameters $\alpha =\beta =0.95$, $\sigma =1.15$, and PSNR = 29.23 dB. (

**c**) Wiener filter with neighborhoods of size $3\times 3$, and PSNR = 23.83 dB. (

**d**) Gaussian filter (3) with parameter $\sigma =1.15$, and PSNR = 27.16 dB.

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**MDPI and ACS Style**

Petráš, I.
Novel Low-Pass Two-Dimensional Mittag–Leffler Filter and Its Application in Image Processing. *Fractal Fract.* **2023**, *7*, 881.
https://doi.org/10.3390/fractalfract7120881

**AMA Style**

Petráš I.
Novel Low-Pass Two-Dimensional Mittag–Leffler Filter and Its Application in Image Processing. *Fractal and Fractional*. 2023; 7(12):881.
https://doi.org/10.3390/fractalfract7120881

**Chicago/Turabian Style**

Petráš, Ivo.
2023. "Novel Low-Pass Two-Dimensional Mittag–Leffler Filter and Its Application in Image Processing" *Fractal and Fractional* 7, no. 12: 881.
https://doi.org/10.3390/fractalfract7120881