Research Article

Year 2023,
Volume: 5 Issue: 3, 198 - 206, 30.11.2023
### Abstract

### References

- Abirami, A., P. Prakash, and Y.-K. Ma, 2021 Variable-Order Fractional Diffusion Model-Based Medical Image Denoising. Mathematical Problems in Engineering 2021: 1–10.
- Al-Dhabyani, W., M. Gomaa, H. Khaled, and A. Fahmy, 2020 Dataset of breast ultrasound images. Data in Brief 28: 104863.
- Alvarez, L. and J. Esclarin, 1997 Image quantization using reactiondiffusion equations. SIAM Journal on Applied Mathematics 57: 153–175.
- Álvarez, L., P.-L. Lions, and J.-M. Morel, 1992 Image selective smoothing and edge detection by nonlinear diffusion. ii. SIAM Journal on Numerical Analysis 29: 845–866.
- Aubert, G. and J.-F. Aujol, 2008 A variational approach to removing multiplicative noise. SIAM Journal on Applied Mathematics 68: 925–946.
- Bai, J. and X.-C. Feng, 2007 Fractional-order anisotropic diffusion for image denoising. IEEE Transactions on Image Processing 16: 2492–2502.
- Barbu, T., 2014 Robust anisotropic diffusion scheme for image noise removal. Procedia Computer Science 35: 522–530, Knowledge- Based and Intelligent Information and Engineering Systems 18th Annual Conference, KES-2014 Gdynia, Poland, September 2014 Proceedings.
- Barbu, T., V. Barbu, V. Biga, and D. Coca, 2009 A pde variational approach to image denoising and restoration. Nonlinear Analysis: Real World Applications 10: 1351–1361.
- Butera, S. and M. D. Paola, 2014 A physically based connection between fractional calculus and fractal geometry. Annals of Physics 350: 146–158.
- Catté, F., P. Lions, J. Morel, and T. Coll, 1992 Image selective smoothing and edge detection by nonlinear diffusion*. SIAM J. Numer. Anal. 29: 182–193.
- Chan, T. F., G. H. Golub, and P. Mulet, 1999 A nonlinear primaldual method for total variation-based image restoration. SIAM J. Sci. Comput. 20: 1964–1977.
- Chang, Q. and I.-L. Chern, 2003 Acceleration methods for total variation-based image denoising. SIAM Journal on Scientific Computing 25: 982–994.
- Charbonnier, P., L. Blanc-Féraud, G. Aubert, and M. Barlaud, 1994 Two deterministic half-quadratic regularization algorithms for computed imaging. Proceedings of 1st International Conference on Image Processing 2: 168–172 vol.2.
- Chen, B., S. Huang, Z. Liang,W. Chen, and B. Pan, 2019 A fractional order derivative based active contour model for inhomogeneous image segmentation. Applied Mathematical Modelling 65: 120– 136.
- Chen, D., S. Sun, C. Zhang, Y. Chen, and D. Xue, 2013 Fractionalorder TV-L2 model for image denoising. Central European Journal of Physics 11: 1414–1422.
- Contreras, A. O., J. Rosales, L. M. Jimenez, and J. M. Cruz-Duarte, 2018 Analysis of projectile motion in view of conformable derivative. Open Physics 16: 581–587.
- Crandall, M. G., H. Ishii, and P.-L. Lions, 1992 Users guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27: 1–67.
- Cresson, J., 2010 Inverse problem of fractional calculus of variations for partial differential equations. Communications in Nonlinear Science and Numerical Simulation 15: 987–996.
- Evans, L. C. and J. Spruck, 1991 Motion of level sets by mean curvature. I. Journal of Differential Geometry 33: 635–681.
- Fan, L., F. Zhang, H. Fan, and C. Zhang, 2019 Brief review of image denoising techniques. Vis. Comput. Ind. Biomed. Art 2.
- Fang, Z.-W., H.-W. Sun, and H. Wang, 2020 A fast method for variable-order caputo fractional derivative with applications to time-fractional diffusion equations. Computers and Mathematics with Applications 80: 1443–1458.
- Giga, Y., H. Mitake, and S. Sato, 2022 On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation. Journal of Differential Equations 316: 364– 386.
- Gilboa, G., N. Sochen, and Y. Zeevi, 2006 Variational denoising of partly textured images by spatially varying constraints. IEEE Transactions on Image Processing 15: 2281–2289.
- Hammad, I. A. and R. Khalil, 2014a Fractional fourier series with applications. American Journal of Computational and Applied Mathematics 4: 187–191.
- Hammad, M. A. and R. Khalil, 2014b Conformable fractional heat differential equation. International journal of pure and applied mathematics 94: 215–221.
- Herrmann, R., 2011 Fractional Calculus: An Introduction for Physicists. World Scientific.
- Hilfer, R., 2000 Applications of Fractional Calculus in Physics.World Scientific.
- Ibrahim,W. R., 2020 A new image denoising model utilizing the conformable fractional calculus for multiplicative noise. SN Applied Sciences 2: 120–136.
- Janev, M., S. Pilipoviaea, T. Atanackoviaea, R. Obradoviaea, and N. Raleviaea, 2011 Fully fractional anisotropic diffusion for image denoising. Mathematical and Computer Modelling 54: 729– 741.
- Karaca, Y. and D. Baleanu, 2022 Chapter 9 - computational fractional-order calculus and classical calculus ai for comparative differentiability prediction analyses of complex-systemsgrounded paradigm. In Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, edited by Y. Karaca, D. Baleanu, Y.-D. Zhang, O. Gervasi, and M. Moonis, pp. 149–168, Academic Press.
- Khalil, R., M. Al Horani, A. Yousef, and M. Sababheh, 2014 A new definition of fractional derivative. Journal of Computational and Applied Mathematics 264: 65–70.
- Kumar, S. and M. Ahmad, 2014 A time dependent model for image denoising. Journal of Signal and Information Processing 6: 28–38.
- Kumar, S., M. Sarfaraz, and M. Ahmad, 2016 An efficient pdebased nonlinear anisotropic diffusion model for image denoising. Neural, Parallel and Scientific Computations 24: 305–315.
- Lapidus, L. and G. F. Pinder, 1983 Numerical solution of partial differential equations in science and engineering. SIAM Review 25: 581–582.
- Mazloum, B. and H. Siahkal-Mahalle, 2022 A time-splitting local meshfree approach for time-fractional anisotropic diffusion equation: application in image denoising. Adv Cont Discr Mod 56.
- Othman, M. I. A. and S. Shaw, 2021 On the concept of a conformable fractional differential equation. Journal of Engineering and Thermal Sciences 1: 17–29.
- Perona, P. and J. Malik, 1990 Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12: 629–639.
- Prasath, V. B. S. and D. Vorotnikov, 2014 Weighted and wellbalanced anisotropic diffusion scheme for image denoising and restoration. Nonlinear Analysis-real World Applications 17: 33– 46.
- Shi, Y. and Q. Chang, 2006 New time dependent model for image restoration. Applied Mathematics and Computation 179: 121– 134.
- Strong, D., 1997 Adaptive total variation minimizing image restoration. Wang, Q., J. Ma, S. Yu, and L. Tan, 2020 Noise detection and image denoising based on fractional calculus. Chaos, Solitons and Fractals 131: 109463.
- Weickert, J., 1997 A review of nonlinear diffusion filtering. In Scale- Space Theory in Computer Vision, edited by B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever, pp. 1–28, Berlin, Heidelberg, Springer Berlin Heidelberg.
- Welk, M., D. Theis, T. Brox, and J. Weickert, 2005 Pde-based deconvolution with forward-backward diffusivities and diffusion tensors. In Scale Space and PDE Methods in Computer Vision, edited by R. Kimmel, N. A. Sochen, and J.Weickert, pp. 585–597, Berlin, Heidelberg, Springer Berlin Heidelberg.
- Witkin, A. P., 1983 Scale-space filtering. In International Joint Conference on Artificial Intelligence.
- Yin, X., S. Zhou, and M. A. Siddique, 2015 Fractional nonlinear anisotropic diffusion with p-laplace variation method for image restoration. Multimedia Tools and Applications 75: 4505 – 4526.
- Zhang, Z., Q. Liu, and T. Gao, 2021 A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising.
- Zhao, D. and M. kang Luo, 2017 General conformable fractional derivative and its physical interpretation. Calcolo 54: 903–917.
- Zhou, H., S. Yang, and S. Zhang, 2018 Conformable derivative approach to anomalous diffusion. Physica A: Statistical Mechanics and its Applications 491: 1001–1013.

Medical imaging, the process of visual representation of different organs and tissues of the human body, is employed for monitoring the normal as well as abnormal anatomy and physiology of the body. Imaging which can provide healthcare solutions ensuring a regular measurement of various complex diseases plays a critical role in the diagnosis and management of many complex diseases and medical conditions, and the quality of a medical image, which is not a single factor but a composite of contrast, artifacts, distortion, noise, blur, and so forth, depends on several factors such as the characteristics of the equipment, the imaging method in question as well as the imaging variables chosen by the operator. The medical images (ultrasound image, X-rays, CT scans, MRIs, etc.) may lose significant features and become degraded due to the emergence of noise as a result of which the process of improvement pertaining to medical images has become a thought-provoking area of inquiry with challenges related to detecting the speckle noise in the images and finding the applicable solution in a timely manner. The partial differential equations (PDEs), in this sense, can be used extensively in different aspects with regard to image processing ranging from filtering to restoration, segmentation to edge enhancement and detection, denoising in particular, among the other ones. In this research paper, we present a conformable fractional derivative-based anisotropic diffusion model for removing speckle noise in ultrasound images. The proposed model providing to be efficient in reducing noise by preserving the essential image features like edges, corners and other sharp structures for ultrasound images in comparison to the classical anisotropic diffusion model. Furthermore, we aim at proving the viscosity solution of the fractional diffusion model. The finite difference method is used to discretize the fractional diffusion model and classical diffusion models. The peak signal-to-noise ratio (PSNR) is used for the quality of the smooth images. The comparative experimental results corroborate that the proposed, developed and extended mathematical model is capable of denoising and preserving the significant features in ultrasound towards better accuracy, precision and examination within the framework of biomedical imaging and other related medical, clinical, and image-signal related applied as well as computational processes.

Anisotropic diffusion model Nonlinear mathematical diffusion model Fractional diffusion model Fractional order derivatives Biomedical imaging Denoising Chaotic signals and noise Image smoothing Viscosity solution Explicit scheme Multiplicative noise Conformable fractional derivative Partial differential equations (PDEs)

- Abirami, A., P. Prakash, and Y.-K. Ma, 2021 Variable-Order Fractional Diffusion Model-Based Medical Image Denoising. Mathematical Problems in Engineering 2021: 1–10.
- Al-Dhabyani, W., M. Gomaa, H. Khaled, and A. Fahmy, 2020 Dataset of breast ultrasound images. Data in Brief 28: 104863.
- Alvarez, L. and J. Esclarin, 1997 Image quantization using reactiondiffusion equations. SIAM Journal on Applied Mathematics 57: 153–175.
- Álvarez, L., P.-L. Lions, and J.-M. Morel, 1992 Image selective smoothing and edge detection by nonlinear diffusion. ii. SIAM Journal on Numerical Analysis 29: 845–866.
- Aubert, G. and J.-F. Aujol, 2008 A variational approach to removing multiplicative noise. SIAM Journal on Applied Mathematics 68: 925–946.
- Bai, J. and X.-C. Feng, 2007 Fractional-order anisotropic diffusion for image denoising. IEEE Transactions on Image Processing 16: 2492–2502.
- Barbu, T., 2014 Robust anisotropic diffusion scheme for image noise removal. Procedia Computer Science 35: 522–530, Knowledge- Based and Intelligent Information and Engineering Systems 18th Annual Conference, KES-2014 Gdynia, Poland, September 2014 Proceedings.
- Barbu, T., V. Barbu, V. Biga, and D. Coca, 2009 A pde variational approach to image denoising and restoration. Nonlinear Analysis: Real World Applications 10: 1351–1361.
- Butera, S. and M. D. Paola, 2014 A physically based connection between fractional calculus and fractal geometry. Annals of Physics 350: 146–158.
- Catté, F., P. Lions, J. Morel, and T. Coll, 1992 Image selective smoothing and edge detection by nonlinear diffusion*. SIAM J. Numer. Anal. 29: 182–193.
- Chan, T. F., G. H. Golub, and P. Mulet, 1999 A nonlinear primaldual method for total variation-based image restoration. SIAM J. Sci. Comput. 20: 1964–1977.
- Chang, Q. and I.-L. Chern, 2003 Acceleration methods for total variation-based image denoising. SIAM Journal on Scientific Computing 25: 982–994.
- Charbonnier, P., L. Blanc-Féraud, G. Aubert, and M. Barlaud, 1994 Two deterministic half-quadratic regularization algorithms for computed imaging. Proceedings of 1st International Conference on Image Processing 2: 168–172 vol.2.
- Chen, B., S. Huang, Z. Liang,W. Chen, and B. Pan, 2019 A fractional order derivative based active contour model for inhomogeneous image segmentation. Applied Mathematical Modelling 65: 120– 136.
- Chen, D., S. Sun, C. Zhang, Y. Chen, and D. Xue, 2013 Fractionalorder TV-L2 model for image denoising. Central European Journal of Physics 11: 1414–1422.
- Contreras, A. O., J. Rosales, L. M. Jimenez, and J. M. Cruz-Duarte, 2018 Analysis of projectile motion in view of conformable derivative. Open Physics 16: 581–587.
- Crandall, M. G., H. Ishii, and P.-L. Lions, 1992 Users guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society 27: 1–67.
- Cresson, J., 2010 Inverse problem of fractional calculus of variations for partial differential equations. Communications in Nonlinear Science and Numerical Simulation 15: 987–996.
- Evans, L. C. and J. Spruck, 1991 Motion of level sets by mean curvature. I. Journal of Differential Geometry 33: 635–681.
- Fan, L., F. Zhang, H. Fan, and C. Zhang, 2019 Brief review of image denoising techniques. Vis. Comput. Ind. Biomed. Art 2.
- Fang, Z.-W., H.-W. Sun, and H. Wang, 2020 A fast method for variable-order caputo fractional derivative with applications to time-fractional diffusion equations. Computers and Mathematics with Applications 80: 1443–1458.
- Giga, Y., H. Mitake, and S. Sato, 2022 On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation. Journal of Differential Equations 316: 364– 386.
- Gilboa, G., N. Sochen, and Y. Zeevi, 2006 Variational denoising of partly textured images by spatially varying constraints. IEEE Transactions on Image Processing 15: 2281–2289.
- Hammad, I. A. and R. Khalil, 2014a Fractional fourier series with applications. American Journal of Computational and Applied Mathematics 4: 187–191.
- Hammad, M. A. and R. Khalil, 2014b Conformable fractional heat differential equation. International journal of pure and applied mathematics 94: 215–221.
- Herrmann, R., 2011 Fractional Calculus: An Introduction for Physicists. World Scientific.
- Hilfer, R., 2000 Applications of Fractional Calculus in Physics.World Scientific.
- Ibrahim,W. R., 2020 A new image denoising model utilizing the conformable fractional calculus for multiplicative noise. SN Applied Sciences 2: 120–136.
- Janev, M., S. Pilipoviaea, T. Atanackoviaea, R. Obradoviaea, and N. Raleviaea, 2011 Fully fractional anisotropic diffusion for image denoising. Mathematical and Computer Modelling 54: 729– 741.
- Karaca, Y. and D. Baleanu, 2022 Chapter 9 - computational fractional-order calculus and classical calculus ai for comparative differentiability prediction analyses of complex-systemsgrounded paradigm. In Multi-Chaos, Fractal and Multi-Fractional Artificial Intelligence of Different Complex Systems, edited by Y. Karaca, D. Baleanu, Y.-D. Zhang, O. Gervasi, and M. Moonis, pp. 149–168, Academic Press.
- Khalil, R., M. Al Horani, A. Yousef, and M. Sababheh, 2014 A new definition of fractional derivative. Journal of Computational and Applied Mathematics 264: 65–70.
- Kumar, S. and M. Ahmad, 2014 A time dependent model for image denoising. Journal of Signal and Information Processing 6: 28–38.
- Kumar, S., M. Sarfaraz, and M. Ahmad, 2016 An efficient pdebased nonlinear anisotropic diffusion model for image denoising. Neural, Parallel and Scientific Computations 24: 305–315.
- Lapidus, L. and G. F. Pinder, 1983 Numerical solution of partial differential equations in science and engineering. SIAM Review 25: 581–582.
- Mazloum, B. and H. Siahkal-Mahalle, 2022 A time-splitting local meshfree approach for time-fractional anisotropic diffusion equation: application in image denoising. Adv Cont Discr Mod 56.
- Othman, M. I. A. and S. Shaw, 2021 On the concept of a conformable fractional differential equation. Journal of Engineering and Thermal Sciences 1: 17–29.
- Perona, P. and J. Malik, 1990 Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12: 629–639.
- Prasath, V. B. S. and D. Vorotnikov, 2014 Weighted and wellbalanced anisotropic diffusion scheme for image denoising and restoration. Nonlinear Analysis-real World Applications 17: 33– 46.
- Shi, Y. and Q. Chang, 2006 New time dependent model for image restoration. Applied Mathematics and Computation 179: 121– 134.
- Strong, D., 1997 Adaptive total variation minimizing image restoration. Wang, Q., J. Ma, S. Yu, and L. Tan, 2020 Noise detection and image denoising based on fractional calculus. Chaos, Solitons and Fractals 131: 109463.
- Weickert, J., 1997 A review of nonlinear diffusion filtering. In Scale- Space Theory in Computer Vision, edited by B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever, pp. 1–28, Berlin, Heidelberg, Springer Berlin Heidelberg.
- Welk, M., D. Theis, T. Brox, and J. Weickert, 2005 Pde-based deconvolution with forward-backward diffusivities and diffusion tensors. In Scale Space and PDE Methods in Computer Vision, edited by R. Kimmel, N. A. Sochen, and J.Weickert, pp. 585–597, Berlin, Heidelberg, Springer Berlin Heidelberg.
- Witkin, A. P., 1983 Scale-space filtering. In International Joint Conference on Artificial Intelligence.
- Yin, X., S. Zhou, and M. A. Siddique, 2015 Fractional nonlinear anisotropic diffusion with p-laplace variation method for image restoration. Multimedia Tools and Applications 75: 4505 – 4526.
- Zhang, Z., Q. Liu, and T. Gao, 2021 A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising.
- Zhao, D. and M. kang Luo, 2017 General conformable fractional derivative and its physical interpretation. Calcolo 54: 903–917.
- Zhou, H., S. Yang, and S. Zhang, 2018 Conformable derivative approach to anomalous diffusion. Physica A: Statistical Mechanics and its Applications 491: 1001–1013.

There are 47 citations in total.

Primary Language | English |
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Subjects | Applied Mathematics (Other) |

Journal Section | Research Articles |

Authors | |

Publication Date | November 30, 2023 |

Published in Issue | Year 2023 Volume: 5 Issue: 3 |

Chaos Theory and Applications in Applied Sciences and Engineering: An interdisciplinary journal of nonlinear science

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