Texture enhancement of skin lesion images via Hankel determinants of $ \lambda $-generalized Sakaguchi type functions in symmetric domain

Bushra Kanwal; Kashaf Fatima; Dalal Alhwikem; Sheza El-Deeb; Bushra Kanwal; Kashaf Fatima; Dalal Alhwikem; Sheza El-Deeb

doi:10.3934/math.2026690

AIMS Mathematics

2026, Volume 11, Issue 6: 16811-16836. doi: 10.3934/math.2026690

Previous Article Next Article

Research article

Texture enhancement of skin lesion images via Hankel determinants of $ \lambda $-generalized Sakaguchi type functions in symmetric domain

1.
Department of Mathematical Sciences, Fatima Jinnah Women University, The Mall Rawalpindi, Pakistan
2.
Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Damietta University, New Damietta, 34517, Egypt

Received: 13 March 2026 Revised: 14 May 2026 Accepted: 03 June 2026 Published: 11 June 2026
MSC : 30C45, 30C50

Digital image processing is essential in fields such as medical imaging, satellite imagery, and autonomous vehicles. Besides its applications, texture enhancement is vital for improving image visibility while preserving geometric structure. To improve visual clarity and image texture, we introduced a novel algorithm for texture enhancement. Initially, we determined coefficient inequalities of $ \mathcal{S}^{*}{(\varphi _H)} $ which is a subclass of $ \lambda $- generalized Sakaguchi type functions, and examined upper bounds of second and third order Hankel determinants and obtained sharp results. We proposed a texture enhancement algorithm that utilized convolution masks derived from Hankel determinants with the pixels of segmented image. This approach provided a mathematical tool for computer-aided dermatological analysis, which was used to enhance lesion boundaries and structural visibility in dermascopic images. Image quality was evaluated using different quality metrics like contrast, correlation, energy, homogeneity, and entropy. The experimental results demonstrated uniform texture enhancement and improved edge preservation in all directions. Comparative analysis showed the efficacy of our proposed algorithm compared to existing methods reported in this study, proving it suitable to enhance image quality.
- $ \lambda $-generalized Sakaguchi type functions,
- image edge detection,
- Hankel determinant,
- algorithms,
- texture enhancement,
- medical imaging
Citation: Bushra Kanwal, Kashaf Fatima, Dalal Alhwikem, Sheza El-Deeb. Texture enhancement of skin lesion images via Hankel determinants of $ \lambda $-generalized Sakaguchi type functions in symmetric domain[J]. AIMS Mathematics, 2026, 11(6): 16811-16836. doi: 10.3934/math.2026690

Related Papers:

Abstract

Digital image processing is essential in fields such as medical imaging, satellite imagery, and autonomous vehicles. Besides its applications, texture enhancement is vital for improving image visibility while preserving geometric structure. To improve visual clarity and image texture, we introduced a novel algorithm for texture enhancement. Initially, we determined coefficient inequalities of $ \mathcal{S}^{*}{(\varphi _H)} $ which is a subclass of $ \lambda $- generalized Sakaguchi type functions, and examined upper bounds of second and third order Hankel determinants and obtained sharp results. We proposed a texture enhancement algorithm that utilized convolution masks derived from Hankel determinants with the pixels of segmented image. This approach provided a mathematical tool for computer-aided dermatological analysis, which was used to enhance lesion boundaries and structural visibility in dermascopic images. Image quality was evaluated using different quality metrics like contrast, correlation, energy, homogeneity, and entropy. The experimental results demonstrated uniform texture enhancement and improved edge preservation in all directions. Comparative analysis showed the efficacy of our proposed algorithm compared to existing methods reported in this study, proving it suitable to enhance image quality.

References

[1]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[2]	C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika, 14 (1967), 108–112. https://doi.org/10.1112/S002557930000807X doi: 10.1112/S002557930000807X
[3]	J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean $p$-valent functions, Trans. Amer. Math. Soc., 223 (1976), 337–346. https://doi.org/10.2307/1997533 doi: 10.2307/1997533
[4]	W. Hu, J. Deng, Hankel determinants, Fekete–Szegö inequality, and estimates of initial coefficients for certain subclasses of analytic functions, AIMS Mathematics, 9 (2024), 6445–6467. https://doi.org/10.3934/math.2024314 doi: 10.3934/math.2024314
[5]	A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl., 155 (1991), 364–370. https://doi.org/10.1016/0022-247X(91)90006-L doi: 10.1016/0022-247X(91)90006-L
[6]	E. Lindelöf, Mémoire sur certaines inégalités dans la théorie des fonctions monogènes et sur quelques propriétés nouvelles de ces fonctions dans le voisinage d'un point singulier essentiel, Acta. Soc. Sci. Fenn., 35 (1909), 1–35.
[7]	W. Rogosinski, G. Szegö, Über die Abschnitte von Potenzreihen, die in einem Kreise beschränkt bleiben, Math. Z., 28 (1928), 73–94. https://doi.org/10.1007/BF01181146 doi: 10.1007/BF01181146
[8]	A. W. Goodman, Univalent functions, Tampa, FL: Mariner Publishing Company, 1983.
[9]	P. Sunthrayuth, I. Aldawish, M. Arif, M. Abbas, S. El-Deeb, Estimation of the second-order Hankel determinant of logarithmic coefficients for two subclasses of starlike functions, Symmetry, 14 (2022), 2039. https://doi.org/10.3390/sym14102039 doi: 10.3390/sym14102039
[10]	A. A. Lupaş, A. S. Tayyah, J. Sokół, Sharp bounds on Hankel determinants for starlike functions defined by symmetry with respect to symmetric domains, Symmetry, 17 (2025), 1244. https://doi.org/10.3390/sym17081244 doi: 10.3390/sym17081244
[11]	L. Shi, M. Arif, J. Iqbal, K. Ullah, S. M. Ghufran, Sharp bounds of Hankel determinant on logarithmic coefficients for functions starlike with exponential function, Fractal Fract., 6 (2022), 645. https://doi.org/10.3390/fractalfract6110645 doi: 10.3390/fractalfract6110645
[12]	Q. A. Shakir, A. S. Tayyah, D. Breaz, L. I. Cotîrlă, E. Rapeanu, F. M. Sakar, Upper bounds of the third Hankel determinant for bi-univalent functions in crescent-shaped domains, Symmetry, 16 (2024), 1281. https://doi.org/10.3390/sym16101281 doi: 10.3390/sym16101281
[13]	M. F. Khan, M. Abaoud, Coefficient inequalities and Hankel determinant for a new subclass of $q$-starlike functions, J. Inequal. Appl., 2025 (2025), 95. https://doi.org/10.1186/s13660-025-03337-z doi: 10.1186/s13660-025-03337-z
[14]	A. Riaz, M. Raza, D. K. Thomas, Hankel determinants for starlike and convex functions associated with sigmoid functions, Forum Math., 34 (2021), 137–156. https://doi.org/10.1515/forum-2021-0188 doi: 10.1515/forum-2021-0188
[15]	B. Khan, Z. Orouji, A. Ebadian, Some results related to Booth lemniscate and integral operators, Fractal Fract., 9 (2025), 271. https://doi.org/10.3390/fractalfract9050271 doi: 10.3390/fractalfract9050271
[16]	K. S. Sundari, B. S. Keerthi, Enhancing the quality of low-light images via the coefficient bounds derived for a subclass of Sakaguchi-type function, J. Image Video Proc., 2025 (2025), 4. https://doi.org/10.1186/s13640-025-00663-6 doi: 10.1186/s13640-025-00663-6
[17]	P. Hari, B. Sruthakeerthi, Digital image processing using coefficient of Sakaguchi kind functions defined in Limacon-shaped domain with power law transformation, Contemp. Math., 5 (2024), 2680–2692. https://doi.org/10.37256/cm.5320244489 doi: 10.37256/cm.5320244489
[18]	R. Kamali, S. Prema, A. S. Ajay Shrikaanth, V. Govindan, M. Donganont, Image edge detection enhancement using coefficient estimates for classes of quasi-subordination: Fekete–Szegö problems, Eur. J. Pure Appl. Math., 18 (2025), 6632. https://doi.org/10.29020/nybg.ejpam.v18i4.6632 doi: 10.29020/nybg.ejpam.v18i4.6632
[19]	B. Kanwal, A. Iman, S. Kanwal, A. K. Alkhalifa, Estimation of Hankel inequalities of symmetric starlike functions in crescent-shaped domain and their application in image processing, Sci. Rep., 15 (2025), 27402. https://doi.org/10.1038/s41598-025-12935-2 doi: 10.1038/s41598-025-12935-2
[20]	F. Aldi, Sumijan, Extraction of shape and texture features of dermoscopy image for skin cancer identification, Sinkron: J. Penelitian Teknik Inform., 8 (2024), 650–660. https://doi.org/10.33395/sinkron.v8i2.13557 doi: 10.33395/sinkron.v8i2.13557
[21]	P. Sunthrayuth, Y. Jawarneh, M. Naeem, N. Iqbal, J. Kafle, Some sharp results on coefficient estimate problems for four-leaf-type bounded turning functions, J. Funct. Spaces, 2022 (2022), 8356125. https://doi.org/10.1155/2022/8356125 doi: 10.1155/2022/8356125
[22]	R. J. Libera, E. J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87 (1983), 251–257. https://doi.org/10.1090/S0002-9939-1983-0681830-8 doi: 10.1090/S0002-9939-1983-0681830-8
[23]	C. Pommerenke, A. Vasil'ev, Angular derivatives of bounded univalent functions and extremal partitions of the unit disk, Pacific J. Math., 206 (2002), 425–450. https://doi.org/10.2140/pjm.2002.206.425 doi: 10.2140/pjm.2002.206.425
[24]	O. S. Kwon, A. Lecko, Y. J. Sim, On the fourth coefficient of functions in the Carathéodory class, Comput. Methods Funct. Theory, 18 (2018), 307–314. https://doi.org/10.1007/s40315-017-0229-8 doi: 10.1007/s40315-017-0229-8
[25]	R. J. Libera, E. J. Złotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc., 85 (1982), 225–230. https://doi.org/10.1090/S0002-9939-1982-0652447-5 doi: 10.1090/S0002-9939-1982-0652447-5
[26]	V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math., 353 (2015), 505–510. https://doi.org/10.1016/j.crma.2015.03.003 doi: 10.1016/j.crma.2015.03.003
[27]	L. S. G. Kovásznay, H. M. Joseph, Image processing, Proc. IRE, 43 (1955), 560–570. https://doi.org/10.1109/JRPROC.1955.278100 doi: 10.1109/JRPROC.1955.278100
[28]	F. Hossain, M. R. Alsharif, Image enhancement based on logarithmic transform coefficient and adaptive histogram equalization, In: 2007 International conference on convergence information technology, 2007. https://doi.org/10.1109/ICCIT.2007.258
[29]	K. Panetta, J. Xia, S. Agaian, Color image enhancement based on the discrete cosine transform coefficient histogram, J. Electron. Imaging, 21 (2012), 021117. https://doi.org/10.1117/1.JEI.21.2.021117 doi: 10.1117/1.JEI.21.2.021117
[30]	L. Lu, Y. Zhou, K. Panetta, S. Agaian, Comparative study of histogram equalization algorithms for image enhancement, Proc. SPIE, 7708 (2010), 770811. https://doi.org/10.1117/12.853502 doi: 10.1117/12.853502
[31]	J. Xia, K. Panetta, S. Agaian, Color image enhancement algorithm based on logarithmic transform coefficient histogram, Proc. SPIE, 7870 (2011), 78700Y. https://doi.org/10.1117/12.872418 doi: 10.1117/12.872418
[32]	A. Katanskiy, Skin Cancer ISIC, Kaggle, 2019. Available from: https://www.kaggle.com/datasets/nodoubttome/skin-cancer9-classesisic

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)