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Deblurring Medical Images Using a New Grünwald-Letnikov Fractional Mask
Volume 35, Issue 4 (2024), pp. 817–836
Pub. online: 11 November 2024      Type: Research Article      Open accessOpen Access
Received
1 January 2024
Accepted
1 October 2024
Published
11 November 2024

Abstract

In this paper, we propose a novel image deblurring approach that utilizes a new mask based on the Grünwald-Letnikov fractional derivative. We employ the first five terms of the Grünwald-Letnikov fractional derivative to construct three masks corresponding to the horizontal, vertical, and diagonal directions. Using these matrices, we generate eight additional matrices of size $5\times 5$ for eight different orientations: $\frac{k\pi }{4}$, where $k=0,1,2,\dots ,7$. By combining these eight matrices, we construct a $9\times 9$ mask for image deblurring that relates to the order of the fractional derivative. We then categorize images into three distinct regions: smooth areas, textured regions, and edges, utilizing the Wakeby distribution for segmentation. Next, we determine an optimal fractional derivative value tailored to each image category to effectively construct masks for image deblurring. We applied the constructed mask to deblur eight brain images affected by blur. The effectiveness of our approach is demonstrated through evaluations using several metrics, including PSNR, AMBE, and Entropy. By comparing our results to those of other methods, we highlight the efficiency of our technique in image restoration.

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Biographies

Satvati Mohammad Amin
amints70@gmail.com

M.A. Satvati is a PhD student of applied mathematics at the University of Tabriz, Iran. He is also a mathematics instructor at University of Tabriz. His main research interests include image processing, and dynamical systems. He is a member of the Iranian Mathematical Association.

Lakestani Mehrdad
https://orcid.org/0000-0002-2752-0167
lakestani@tabrizu.ac.ir

M. Lakestani received his PhD in applied mathematics from Amirkabir University of Technology, Iran, in 2005. He is currently a professor at the University of Tabriz, Iran. His main research interests include optimal control, wavelets, and image processing.

Khamnei Hossein Jabbari
h_jabbari@tabrizu.ac.ir

H.J. Khamnei received his PhD degree in statistics from Panjab University, India. He is currently an associate professor at the University of Tabriz, Iran. His main research interests include reliability, stress and strength, distribution theory and quality control. He has authored several research articles in reputed international journals. He is a member of the Iranian Statistical Association and the Iranian Mathematical Association.

Allahviranloo Tofigh
tofigh.allahviranloo@istinye.edu.tr

T. Allahviranloo PhD, is a full professor of applied mathematics at Istinye University, Türkiye. He specializes in fuzzy applied mathematics, particularly fuzzy differential equations, and is actively involved in interdisciplinary research with applications in the biological sciences. Prof. Allahviranloo has published over 500 papers and 25 books. His work aims to innovate mathematical understanding of uncertainties in complex systems.


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Open access article under the CC BY license.

Keywords
Grünwald-Letnikov fractional derivatives gradient matrix Wakeby distribution

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