Fractional Diffusion Equation for Medical Image Denoising using ADI Scheme
Main Article Content
Keywords
fractional order total variation; Grunwald Letnikov; Caputo derivative; ADI scheme; medical image denoising
Abstract
This paper proposes a fractional order total vari- ation model for additive noise removal which uses a different fractional order of the regularization term of the objective function. The denoising model based on space and time fractional derivatives on a finite domain is discretized with effective applications of Gru¨nwald-Letnikov(G-L) and Caputo derivatives. This model has been adopted to solve Alternative Direction Implicit (ADI) scheme to denoise medical images. The advantage of this model is for smooths the homogeneous regions and enhance edge information revealing more details of the image. The results show that the proposed model has desirable feedback for enhancing medical images, revealing more detailed information than ROF(Rudin, Osher and Fatemi), TV − L1 (Total Variation L1 space) and fourth order partial differential equation based models.
References
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2. Abirami, A new fractional order total variational model for multiplicative noise removal, Journal of Applied Science and Computations, 6(3) (2019), pp. 483–491.
3. Abirami and P. Prakash Survey on incorporating fractional derivatives in image denoising, Advances in Mathematics:Scientific Journal, 9(3) (2020), pp. 1367–1377.
4. Abirami P. Prakash and Y.K. Ma, Variable-Order Fractional Diffusion Model based Medical image denoising, Mathematical Problems in Engineering, Volume 2021 | Article ID 8050017 | https://doi.org/10.1155/2021/8050017.
5. J.F. Aujol, G. Gilboa, T. Chan, and S. Osher, Structure-texture image decomposition modeling, algorithms and parameter selection,International Journal of Computer Vision, 67(1) (2006), 111–136.
6. J. Bai and X.C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Transactions on Image Processing, 16(10) (2007), 2492–2502.
7. R. Campagna, S. Crisci, S. Cuomo, L. Marcellino and G. Toraldo, Modification of TV-ROF denoising model based on Split Bregman iterations, Applied Mathematics and Computation, 315 (2017), 453–467.
8. D. Chen, Y.Q. Chen and D. Xue, Fractional-order total variation image denoising based on proximity algorithm, Applied Mathematics and Computation, 257 (2015), 537–545.
9. Diana Andrushia, N. Anand and Prince Arulraj, Anisotropic diffusion based denoising on rete images and surface crack segmentation, International Journal of Structural Integrity, (2019), To print.
10. P. Guidotti and K. Longo, Two enhanced fourth order diffusion models for image denoising, Journal of Mathematical Imaging and Vision, 40(2) (2011), 188–198.
11. E. Isaacson and H.B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966.
12. M. Lysaker, A. Lundervold and X.C. Tai, Noise removal using fourth order partial differential equations with applications to medical resonance images in space and time, IEEE Transactions on Image Processing, 12 (2003), 1579–1590.
13. Matheieu, P. Melchior, A.Oustaliup and C. Ayral, Fractional differentiation for edge detection, Signal Processing, 11 (2003) 2421–2432.
14. M. Nikolova, A variational approach to remove outliers and impulse noise, Journal of Mathematical Image Vision, 20(1-2) (2004) 99–120.
15. Y.F. Pu, P. Siarry, J.Zhou, Y.G. Liu, N.Zhang, G. Huang and Y.Z. Liu, Fractional partial differential equation denoising models for texture image, Science China Information Sciences, 57(7) (2014), 1–19.
16. Y.F. Pu, N. Zhang, Y. Zhang and J.L. Zhou, A texture image denoising approach based on fractional developmental mathematics, Journal of Pattern Analysis and Applications, 19(2) (2016), 427–445.
17. L.Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica. D, 60 (1992), 259–268.
18. H.R. Shahdoosti and O. Khayat, Combination of annisotropic diffusion and nonsubsampled shearlet transform for image denoising, Journal of Intelligent Fuzzy Systems, 30(6), (2016), 3087–3098.
19. G. Sudha Priya, P. Prakash, J.J. Nieto and Z. Kayar, Higher-order numerical scheme for fractional heat equation with Dirichlet and Neumann boundary condition, Numerical Heat Transfer, Part-B, 63 (2013), 540–559.
20. Tadjeran and M.M. Meerschaert, A second order accurate numerical approximation for the two-dimensional fractional diffusion equation, Journal of Computational Physics, 220 (2007), 813–823.
21. M.J. Wang and S.X. Deng, Image restoration model of PDE variation, IEEE Information and Computer Sciences, 2 (2009), 184–187.
22. J. Wu and C. Tang, PDE-based random valued impulse noise removal based on new class of controlling function, IEEE Transactions on Image Processing, 20(9) (2011), 2428–2438.
23. A.A. Yahya, J. Tan and M. Hu, A blending method based on partial differential equations for image denoising, International Journal of Multimedia Tools and Applications, 73(3) (2014), 1843–1862.
24. Y.L. You and M. Kavesh, Four-order partial differential equations for noise removal, IEEE Transactions on Image Processing, 9(10) (2009), 1723–1730.
25. W. Zeng, X. Lu and X. Tan, A local structural adaptive partial differential equation for image denoising, International Journal of Multimedia Tools and Applications, 74(3) (2015), 743–757.
26. W. Zhang, J.Li and Y.Yang, Spatial fractional telegraph equation for image structure preserving
denoising, Signal Processing, 107 (2015), 368–377.
27. J. Zhang, Z. Wei and L. Xiao, Adaptive fractional order multi-scale method for image denoising, Journal of Mathematical Imaging Vision, 43 (2012), 39–49.
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Published
Jun 1, 2023
Article Details
How to Cite
A.Abirami, & P.Prakash. (2023). Fractional Diffusion Equation for Medical Image Denoising using ADI Scheme. Journal of Population Therapeutics and Clinical Pharmacology, 30(12), 52–63. https://doi.org/10.47750/jptcp.2023.30.12.008
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