Máscaras diferenciales fraccionarias de Caputo y Caputo-Fabrizio para la mejora de imágenes
https://doi.org/10.5281/zenodo.5728127
Resumen
La mejora de imágenes es una de las tareas más importantes en el campo del procesamiento de imágenes. Con la ayuda de lenguajes informáticos y de programación, se han implementado muchos métodos matemáticos para mejorar la calidad visual de una imagen. Uno de los métodos más eficaces para este propósito es la ecualización del histograma. También se ha propuesto la construcción de máscaras diferenciales fraccionarias para la mejora de imágenes. En este artı́culo, se propone una nueva forma de construcción de máscara diferencial fraccional basada en las derivadas de Caputo y Caputo-Fabrizio. La eficacia de los métodos propuestos se ha comparado con el método de ecualización del histograma y la multiplicación de cada pı́xel de una imagen por una constante. Los resultados de los experimentos han demostrado la superioridad de los métodos propuestos, con una mejor calidad visual y valores de matriz de co-ocurrencia de nivel de gris más altos en cuatro direcciones.
Citas
Ajou, A.; Oqielat, M. N.; Al-Zhour, Z.; Kumar, S. and Momani, S.; Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative, Chaos (2019), 29 (2019), 093102.
Al-Zhour, Z.; Barfeie, M.; Soleymani, F. and Tohidi, E.; A computational method to price with transaction costs under the nonlinear Black-Scholes model, Chaos, Solitons & Fractals, 127 (2019), 291--301.
Andrade, A. M. F.; Lima, E. G. and Dartora, C. A.; An introduction to fractional calculus and its Applications in Electric Circuits, Revista Brasileira de Ensino de F'isica, 40 (2018), e3314.
Atangana, A. and Baleanu, D.; New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, arXiv preprint, arXiv:1602.03408 (2016).
Baleanu, D.; Diethelm, K.; Escalas, E.; and Trujillo, J. J.; Fractional Calculus: Models and numerical methods, Series on Complexity, Nonlinearity and Chaos, 3 (2012).
Baleanu, D.; Jajarmi, A.; Sajjadi, S. S. and Mozyrska, D.; A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos, 29 (2019), 083127.
Caputo, M. and Fabrizio, M.; A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2005), 73--85.
Concezzi, M. and Spigler, R.; Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data, Fractal Fract., 2(1) (2014), 14.
El-Ajou, A.; Al-Zhour, Z.; Oqielat, M.; Momani, S. and Hayat, T.; Series Solutions of Nonlinear Conformable Fractional KdV-Burgers Equation with Some Applications, The European Physical Journal Plus, 134 (2019), 402.
El-Ajou, A.; Oqielat, M.; Al-Zhour, Z. and Momani, S.; Analytical Numerical Solutions of the Fractional Multi-Pantograph System: Two Attractive Methods and Comparisons, Results in Physics, 14 (2019), 102500.
El-Ajou, A.; Oqielat, M.; Al-Zhour, Z.; Kumar, S. and Momani, S.; Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative, Chaos, 29 (2019), 093102.
Garg, P. and Jain, T.; A comparative study on histogram equalization and cumulative histogram equalization. International Journal of technology and research, 3(9) (2017), 41--43.
Goufo, E. F. D. and Mugisha, S. B.; Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Solitons & Fractals, 130 (2020), 10946.
Jajarmi, A.; Arshad, S. and Baleanu, D.; A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A., 535 (2019), 122524.
Jajarmi, A.; Baleanu, D.; Sajjadi, S. S. and Asad, J. H.; A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019), 00196.
Jajarmi, A.; Ghanbari, B. and Baleanu, Dumitru. A new and efficient numerical method for the fractional modelling and optimal control of diabetes and tuberculosis co-existence, Chaos, 29 (2019), 093111.
Kilbas, A. A.; Srivastava, H. M. and Trujillo, J. J.; Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, 204 (2006).
Kumar, S.; A new fractional modeling arising in engineering sciences and its analytical approximate solution, Alexandria Engineering Journal, 52 (2013), 813--819.
Kumar, S.; Kumar, A.; Abbas, S.; Al Qurashi, M. and Baleanu, D.; A modified analytical approach with existence and uniqueness for fractional Cauchy reaction-diffusion equations, Advances in Difference Equations, 2020(1) (2020), 1--18.
Kumar, S.; Kumar, A. and Nisar, K. S.; Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two dimensional systems, Advances in Difference Equations, 413 (2019).
Kumar, R.; Kumar, S.; Singh, J. and Al-Zhour, Z. A comparative study for fractional chemical kinetics and carbon dioxide $CO_{2}$ absorbed into phenyl glycidyl ether problems, Mathematics, 5(4) (2020), 3201--3222.
Kumar, S.; Nisar, K. S.; Kumar, R.; Cattani, C. and Samet, B.; A new Rabotnov fractional-exponential function based fractional derivative for diffusion equation under external force, Mathematical Methods in Applied Science, 43 (2020), 4460--4471.
Liang, X.; Gao, F.; Zhou, C. B.; Wang, Z. and Yang, X. J.; An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type, Advances in Difference Equations, 18(1) (2018), 25.
Losada, J. and Nieto, J.; Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 87--92.
Mboro Nchama, G. A.; Alfonso, L. L.; León Mecías, A. M. and Rodríguez Richard, M.; Construction of Caputo-Fabrizio fractional differential mask for image enhancement, Progress in Fractional Differentiation and Application (2020).
Mboro Nchama, G. A.; León Mecías, A. M. and Rodríguez Ricard, M.; Perona-Malik model with diffusion coefficient depending on fractional gradient via Caputo-Fabrizio derivative, Abstract and Applied Analysis (2020), 2020 (2020), 15 pages.
Mboro-Nchama, G. A.; Mecías, A. L. and Ricard, M. R.; The Caputo-Fabrizio fractional integral to generate some new inequalities, Information Sciences Letters, 8 (2019), 73--80.
Mboro-Nchama, G. A.; Mecías, A. L. and Ricard, M. R.; Properties of the Caputo-Fabrizio fractional derivative, Applied Mathematics & Information Sciences, 14 (2020), 1--10.
Mboro Nchama, G. A.; New fractional integral inequalities via Caputo-Fabrizio operator and an open problem concerning an integral inequality, New trends in Mathematical Sciences (2020), 8(2) (2020), 9--21.
Mboro Nchama, G. A.; On open problems concerning Riemann-Liouville fractional integral inequality, Mediterranean Journal of Modeling and Simulation, 11 (2019), 001--008.
Mboro-Nchama, G. A.; Properties of Caputo-Fabrizio fractional operators, New Trends in Mathematical Sciences, 8 (2020), 1--25.
Morales-Delgado, V. F.; Gómez-Aguilar, J. F. and Taneco-Hernández, M. A.; Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, International Journal of Electronics and Communications, 85 (2018), 108--117.
Nandal, A.; Gamboa-Rosales, H.; Dhaka, A.; Celaya-Padilla, J. M.; Galvan-Tejada, J. I.; Galvan-Tejada, C. E.; Martinez-Ruiz, F. J. and Guzman-Valdivia, C.; Image edge detection using fractional calculus with feature and contrast enhancement. Circuits, Systems, and Signal Processing, 37(9) (2018), 3946--3972.
Odham, K. B. and Spanier, J.; The Fractional Calculus, Academic Press (1984), New York.
Oqielat, M.; El-Ajou, A.; Al-Zhour, Z.; Alkhasawneh, R. and Alrabaiah, H.; Series solutions for nonlinear time-fractional Schrödinger equations: Comparisons between conformable and Caputo derivatives, Alexandria Engineering Journal (2020).
Polubny, I.; Fractional Differential Equations, Academic Press (1999), New York.
Pu, Y.; Application of fractional differential approach to digital image processing. Journal of Sichuan University, 39(3) (2007), 124--132.
Pu, Y.; Wang, W.; Zhou, J. et al. Fractional differential approach to detecting texture features of digital image and its fractional differential filter implementation, Sci. China Ser. F, Inf. Sci.; 38(12) (2008), 2252--2272.
Qing-li, C.; H. Guo, H. and Xiu-qiong, Z.; A Fractional Differential Approach to Low Contrast Image Enhancement, International Journal of Knowledge and Language Processing (2012), 3(2) (2012), 20--27.
Ramírez, C.; Astorga, V.; Nuñez, H.; Jaques, A. and Simpson, R.; Anomalous diffusion based on fractional calculus approach applied to drying analysis of apple slices: The effects of relative humidity and temperature, Food Process Engineering (2017), 40(5) (2017), e12549.
Rchid, M.; Ammi, S. and Jamiai, I.; Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration, Discrete and Continuous Dynamical Systems Series, 11 (2018), 103--117.
Sabatier, J.; Agrawal, O. P. and Machado, J. A. T.; Advanced in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer (2007).
Sikora, R.; Fractional derivatives in electrical circuit theory critical remarks, Archives of Electrical Engineering, 66 (2017), 155--163.
Surya Prasath, V. B.; Image denoising by anisotropic diffusion with inter-scale information fusion, Pattern Recognition and Image Analysis, 27 (2017), 748--753.
Yirenkyi, P. A.; Appati, J. K. and Dontwi, I. K.; A new construction of a fractional derivative mask for image edge analysis based on Riemann-Liouville fractional derivative. Advances in Difference Equations (2016), DOI 10.1186/s13662-016-0946-8, pp. 1-21.
Yu, J.; Zhai, R.; Zhou, S. and Tan, L.; Image Denoising Based on Adaptive Fractional Order with Improved PM Model, Mathematical Problems in Engineering (2018), Article ID 9620754.
Zhang, Y.; Pu, Y. and Zhou, J.; Construction of Fractional differential Masks Based on Riemann-Liouville Definition. Journal of Computational Information Systems, 6 (2010), 3191--3198.
