International Journal of Engineering

International Journal of Engineering

Adaptive Fractional-order Differentiation for Enhanced Image Contrast Utilizing Caputo Masks

Document Type : Original Article

Authors
Z. Mortezaie 1
Z. Amiri 2
M. Mortezaee 3
1 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
2 Department of Computer Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran
3 Faculty of Computer Engineering, Shahrood University of Technology, Shahrood, Iran
10.5829/ije.2026.39.05b.19
Abstract
Image enhancement remains a cornerstone in digital image processing, aiming to improve visual clarity through various methods. Spatial domain techniques include integer-order and fractional-order differentiation. Although widely used, traditional integer-order differentiation techniques suffer from limitations such as indiscriminate spatial frequency treatment and noise amplification, leading to degraded image quality. This paper proposes an adaptive fractional-order differentiation approach employing Caputo fractional differential masks to selectively enhance image details. This approach uses image gradient information to determine the appropriate fractional order. By dynamically adjusting the fractional order based on specific image requirements, the method achieves superior contrast improvement while preserving fine details and minimizing noise. Experimental results, evaluated using metrics such as Pratt's Figure of Merit (FOM), Structural Similarity Index (SSIM), and Peak Signal-to-Noise Ratio (PSNR), demonstrate that this approach outperforms comparable techniques, highlighting its effectiveness in image enhancement.

Graphical Abstract

Adaptive Fractional-order Differentiation for Enhanced Image Contrast Utilizing Caputo Masks
Keywords
Fractional-order Differentiation
Caputo Fractional Differential Mask
Image Enhancement
Image Processing
Blur Images

Subjects

  1. Dyke RM, Hormann K. Histogram equalization using a selective filter. The visual computer. 2023;39(12):6221-35. https://doi.org/10.1007/s00371-022-02723-8
  2. Archana R, Jeevaraj PE. Deep learning models for digital image processing: a review. Artificial Intelligence Review. 2024;57(1):11. https://doi.org/10.1007/s10462-023-10631-z
  3. J. Y. Advancements in Spatial Domain Image Steganography: Techniques, Applications, and Future Outlook. Applied and Computational Engineering. 2024;94:6-19. https://doi.org/10.54254/2755-2721/94/2024MELB0058
  4. Chaudhary G. Optimizing fast fourier transform (FFT) image compression using intelligent water drop (IWD) algorithm. International Journal of Interactive Multimedia and Artificial Intelligence. 2022. https://doi.org/10.9781/ijimai.2022.01.004
  5. Gonzalez R, Woods R. Digital image processing, addison-wesle y longman publishing co. Inc, Boston, MA, USA. 2001.
  6. Khidse S, Nagori M. A comparative study of image enhancement techniques. Int J Comput Appl. 2013;81(15):28-32. https://doi.org/10.5120/14201-2421
  7. Ding L, Goshtasby A. On the Canny edge detector. Pattern recognition. 2001;34(3):721-5. https://doi.org/10.1016/S0031-3203(00)00023-6
  8. Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on pattern analysis and machine intelligence. 2002;12(7):629-39. https://doi.org/10.1109/34.56205
  9. Huang G, Qin H-y, Chen Q, Shi Z, Jiang S, Huang C. Research on application of fractional calculus operator in image underlying processing. Fractal and Fractional. 2024;8(1):37. https://doi.org/10.3390/fractalfract8010037
  10. Muresan CI, Birs IR, Dulf EH, Copot D, Miclea L. A review of recent advances in fractional-order sensing and filtering techniques. Sensors. 2021;21(17):5920. https://doi.org/10.3390/s21175920
  11. Sánchez-Rivero M, Duarte-Mermoud MA, Travieso-Torres JC, Orchard ME, Ceballos-Benavides G. Analysis of Fractional Order-Adaptive Systems Represented by Error Model 1 Using a Fractional-Order Gradient Approach. Mathematics. 2024;12(20):3212. https://doi.org/10.3390/math12203212
  12. Almeida R. On the variable-order fractional derivatives with respect to another function. Aequationes mathematicae. 2024:1-18. https://doi.org/10.1007/s00010-024-01082-0
  13. Motłoch S, Sarwas G, Dzieliński A. Fractional derivatives application to image fusion problems. Sensors. 2022;22(3):1049. https://doi.org/10.3390/s22031049
  14. Leonenko N, Podlubny I. Monte Carlo method for fractional-order differentiation. Fractional Calculus and Applied Analysis. 2022;25(2):346-61. https://doi.org/10.1007/s13540-022-00017-3
  15. Nchama GAM, Alfonso LDL, Morales RR, Aneke EN. Asumu Fractional Derivative Applied to Edge Detection on SARS‐COV2 Images. Journal of Applied Mathematics. 2022;2022(1):1131831. https://doi.org/10.1155/2022/1131831
  16. Jalab HA, Ibrahim RW. Texture Enhancement Based on the Savitzky‐Golay Fractional Differential Operator. Mathematical Problems in Engineering. 2013;2013(1):149289. https://doi.org/10.1155/2013/149289
  17. Zakaria M, Moujahid A, Ikhouba M. A new fractional derivative operator and applications. International Journal of Nonlinear Analysis and Applications. 2023;14(1):1277-82. https://doi.org/10.22075/ijnaa.2022.26841.3423
  18. Ben Makhlouf A, Benjemaa M, Boucenna D, Mchiri L, Rhaima M. On generalized proportional fractional order derivatives and Darboux problem for partial differential equations. Discrete Dynamics in Nature and Society. 2023;2023(1):6648524. https://doi.org/10.1155/2023/6648524
  19. Caputo M. Linear models of dissipation whose Q is almost frequency independent—II. Geophysical journal international. 1967;13(5):529-39. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
  20. Mishra SK, Singh KK, Dixit R, Bajpai MK. Design of Fractional Calculus based differentiator for edge detection in color images. Multimedia Tools and Applications. 2021;80(19):29965-83. https://doi.org/10.1007/s11042-021-11187-2
  21. Gonzalez-Lee M, Vazquez-Leal H, Garcia-Martinez JR, Pale-Ramon EG, Morales-Mendoza LJ, Nakano-Miyatake M, et al. A new class of edge filter based on a cross-correlation-like equation derived from fractional calculus principles. Applied Sciences. 2024;14(13):5428. https://doi.org/10.3390/app14135428
  22. Ananthi V, Thangaraj C, Easwaramoorthy D. Multifractal dimensions and fractional differentiation in automated edge detection on intuitionistic fuzzy enhanced image. Frontiers of Fractal Analysis: CRC Press; 2022. p. 153-71.
  23. Mortazavi M, Gachpazan M, Amintoosi M. Improving Canny edge detection algorithm using fractional-order derivatives. Journal of Mathematical Modeling. 2022;10(4):495-514. https://doi.org/10.22124/jmm.2022.21875.1921
  24. Chen M. Fractional‐Order Adaptive P‐Laplace Equation‐Based Art Image Edge Detection. Advances in Mathematical Physics. 2021;2021(1):2337712. https://doi.org/10.1155/2021/2337712
  25. Elgezouli DDE, Abdoon M, Belhaouari SB, Almutairi DK. A Novel Fractional Edge Detector Based on Generalized Fractional Operator. European Journal of Pure and Applied Mathematics. 2024;17(2):1009-28. https://doi.org/10.29020/nybg.ejpam.v17i2.5141
  26. Ismail SM, Said LA, Madian AH, Radwan AG. Fractional-order edge detection masks for diabetic retinopathy diagnosis as a case study. Computers. 2021;10(3):30. https://doi.org/10.3390/computers10030030
  27. Balochian S, Baloochian H. Edge detection on noisy images using Prewitt operator and fractional order differentiation. Multimedia Tools and Applications. 2022;81(7):9759-70. https://doi.org/10.1007/s11042-022-12011-1
  28. Babu NR, Sanjay K, Balasubramaniam P. EED: enhanced edge detection algorithm via generalized integer and fractional-order operators. Circuits, Systems, and Signal Processing. 2022;41(10):5492-534. https://doi.org/10.1007/s00034-022-02028-0
  29. Wang W, Jia Y, Wang Q, Xu P. An Image Enhancement Algorithm Based on Fractional‐Order Phase Stretch Transform and Relative Total Variation. Computational Intelligence and Neuroscience. 2021;2021(1):8818331. https://doi.org/10.1155/2021/8818331
  30. Asghari MH, Jalali B. Edge detection in digital images using dispersive phase stretch transform. International journal of biomedical imaging. 2015;2015(1):687819. https://doi.org/10.1155/2015/687819
  31. Xu L, Yan Q, Xia Y, Jia J. Structure extraction from texture via relative total variation. ACM transactions on graphics (TOG). 2012;31(6):1-10. https://doi.org/10.1145/2366145.2366158
  32. Huang T, Wang X, Xie D, Wang C, Liu X. Depth image enhancement algorithm based on fractional differentiation. Fractal and Fractional. 2023;7(5):394. https://doi.org/10.3390/fractalfract7050394
  33. Ruiyin T, Bo L. Application of Fractional Differential Model in Image Enhancement of Strong Reflection Surface. Mathematics. 2023;11(2):444. https://doi.org/10.3390/math11020444
  34. Land EH. The retinex theory of color vision. Scientific american. 1977;237(6):108-29.
  35. Musa P, Al Rafi F, Lamsani M, editors. A Review: Contrast-Limited Adaptive Histogram Equalization (CLAHE) methods to help the application of face recognition. 2018 third international conference on informatics and computing (ICIC); 2018: IEEE.
  36. Chen D, Chen Y, Xue D. Digital fractional order Savitzky-Golay differentiator. IEEE Transactions on Circuits and Systems II: Express Briefs. 2011;58(11):758-62. https://doi.org/10.1109/TCSII.2011.2168022
  37. Mcbride AC. Univalent functions, fractional calculus, and their applications, edited by HM Srivastava and S. Owa. Pp 404.£ 39· 95. 1989. ISBN 0-7458-0701-1 (Ellis Horwood). The Mathematical Gazette. 1990;74(469):326-7. https://doi.org/10.2307/3619871
  38. Kaur K, Jindal N, Singh K. Fractional Fourier Transform based Riesz fractional derivative approach for edge detection and its application in image enhancement. Signal Processing. 2021;180:107852. https://doi.org/10.1016/j.sigpro.2020.107852
  39. Bouzeffour F. The Orthogonal Riesz Fractional Derivative. Axioms. 2024;13(10):715. https://doi.org/10.3390/axioms13100715
  40. Mortazavi M, Gachpazan M, Amintoosi M, Salahshour S. Fractional derivative approach to sparse super-resolution. The Visual Computer. 2023;39(7):3011-28. https://doi.org/10.1007/s00371-022-02509-y
  41. Azarang A, Ghassemian H. Application of fractional-order differentiation in multispectral image fusion. Remote sensing letters. 2018;9(1):91-100. https://doi.org/10.1080/2150704X.2017.1395963
  42. Nchama GM, Alfonso L, Mecıas A, Richard M. Construction of Caputo-Fabrizio fractional differential mask for image enhancement. Progress in Fractional Differentiation and Application. 2020. https://doi.org/10.18576/pfda/070203
  43. Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Progress in fractional differentiation & applications. 2015;1(2):73-85. http://dx.doi.org/10.12785/pfda/010201
  44. Jalab HA, Ibrahim RW, Hasan AM, Karim FK, Al-Shamasneh AaR, Baleanu D. A new medical image enhancement algorithm based on fractional calculus. 2021. https://doi.org/10.32604/cmc.2021.016047
  45. Gamini S, Kumar SS. Homomorphic filtering for the image enhancement based on fractional-order derivative and genetic algorithm. Computers and Electrical Engineering. 2023;106:108566. https://doi.org/10.1016/j.compeleceng.2022.108566
  46. Karim FK, Jalab HA, Ibrahim RW, Al-Shamasneh AaR. Mathematical model based on fractional trace operator for COVID-19 image enhancement. Journal of King Saud University-Science. 2022;34(7):102254. https://doi.org/10.1016/j.jksus.2022.102254
  47. Al-Shamasneh AaR, Ibrahim RW. Image denoising based on quantum calculus of local fractional entropy. Symmetry. 2023;15(2):396. https://doi.org/10.3390/sym15020396
  48. Huang T, Wang C, Liu X. Depth image denoising algorithm based on fractional calculus. Electronics. 2022;11(12):1910. https://doi.org/10.3390/electronics11121910
  49. Tanriover E, Kiris A, Tunga B, Tunga MA. A novel image denoising technique with Caputo type space–time fractional operators. Nonlinear Dynamics. 2024;112(21):19487-513. https://doi.org/10.1007/s11071-024-10087-y
  50. Chen H, Qiao H, Wei W, Li J. Time fractional diffusion equation based on caputo fractional derivative for image denoising. Optics & Laser Technology. 2024;168:109855. https://doi.org/10.1016/j.optlastec.2023.109855
  51. Diwakar M, Singh P, Garg D. Edge-guided filtering based CT image denoising using fractional order total variation. Biomedical Signal Processing and Control. 2024;92:106072. https://doi.org/10.1016/j.bspc.2024.106072
  52. Ben-Loghfyry A, Charkaoui A. Regularized Perona & Malik model involving Caputo time-fractional derivative with application to image denoising. Chaos, Solitons & Fractals. 2023;175:113925. https://doi.org/10.1016/j.chaos.2023.113925
  53. Atlas A, Bendahmane M, Karami F, Meskine D, Oubbih O. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete and Continuous Dynamical Systems-B. 2021;26(9):4963-98. https://doi.org/10.3934/dcdsb.2020321
  54. Khan A, Gaur S, Suthar D. Generalized Caputo–Fabrizio fractional operator: an application in image denoising. Applied Mathematics in Science and Engineering. 2024;32(1):2434002. https://doi.org/10.1080/27690911.2024.2434002
  55. Scherer R, Kalla SL, Tang Y, Huang J. The Grünwald–Letnikov method for fractional differential equations. Computers & Mathematics with Applications. 2011;62(3):902-17. https://doi.org/10.1016/j.camwa.2011.03.054
  56. Heymans N, Podlubny I. Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheologica Acta. 2006;45(5):765-71. https://doi.org/10.1007/s00397-005-0043-5
  57. Pratt WK. Digital image processing: PIKS Scientific inside: Wiley Online Library; 2007.
  58. Mortezaie Z, Hassanpour H, Asadi Amiri S. An adaptive block based un-sharp masking for image quality enhancement. Multimedia Tools and Applications. 2019;78(16):23521-34. https://doi.org/10.1007/s11042-019-7594-4
  59. Wang Z, Bovik AC. A universal image quality index. IEEE signal processing letters. 2002;9(3):81-4. https://doi.org/10.1109/97.995823
  60. Mortezaie Z, Hassanpour H, Asadi Amiri S. Image enhancement using an adaptive un-sharp masking method considering the gradient variation. International Journal of Engineering Transactions B: Applications. 2017;30(8):1118-25. https://doi.org/10.5829/idosi.ije.2017.30.08b.02
  61. Poobathy D, Chezian RM. Edge detection operators: Peak signal to noise ratio based comparison. IJ Image, Graphics and Signal Processing. 2014;10:55-61. https://doi.org/10.5815/ijigsp.2014.10.07
  62. Larson EC, Chandler DM. Most apparent distortion: full-reference image quality assessment and the role of strategy. Journal of electronic imaging. 2010;19(1):011006--21. https://doi.org/10.1117/1.3267105
  63. Dey A, Biswas S. Shot-ViT: cricket batting shots classification with vision transformer network. International Journal of Engineering Transactions C: Aspects. 2024;37(12):10.5829. https://doi.org/10.5829/ije.2024.37.12c.04
  64. Dey A, Biswas S, Abualigah L. Umpire’s signal recognition in cricket using an attention based DC-GRU network. International Journal of Engineering Transactions A: Basics. 2024;37(4):662-74. https://doi.org/10.5829/ije.2024.37.04a.08
  65. Dorrani Z. Traffic scene analysis and classification using deep learning. International Journal of Engineering, Transactions C: Aspects. 2024;37(3):496-502. https://doi.org/10.5829/IJE.2024.37.03C.06
  66. Farsi H NM, SM, Barati A, Mohamadzadeh, S. . Development of a Deep Learning Model Inspired by Transformer Networks for Multi-class Skin Lesion Classification. International Journal of Engineering Transactions A: Basics. 2026;39(1):135-47. https://doi.org/10.5829/ije.2026.39.01a.11
  67. Mortezaie Z, Hassanpour H, Beghdadi A. People re-identification under occlusion and crowded background. Multimedia Tools and Applications. 2022;81(16):22549-69. https://doi.org/10.1007/s11042-021-11868-y
International Journal of Engineering
Volume 39, Issue 5
TRANSACTIONS B: Applications
May 2026
Pages 1275-1292