# American Institute of Mathematical Sciences

December  2021, 15(6): 1451-1469. doi: 10.3934/ipi.2021018

## A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising

 1 College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China 2 College of Mathematics and Computational Science, Shenzhen University, Shenzhen, 518060, China

* Corresponding author: Tianling Gao

Received  November 2019 Revised  October 2020 Published  December 2021 Early access  February 2021

Fund Project: The first author is supported by Young Foundation of Three Gorges University(19QN09). The last two authors are supported by NSF of Guandong grant 2018A030310454, 2020A1515010554.

In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estimated. In particular, we obtain the stability condition of the iteration by the spectrum analysis method. Through implementing the fast explicit format iteration algorithm with periodic change of time step size, the efficiency of the algorithm is greatly improved. At last, we show some numerical results on denoising tasks. Many experimental results confirm that the algorithm can more quickly achieve satisfactory denoising results.

Citation: Zhiguang Zhang, Qiang Liu, Tianling Gao. A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising. Inverse Problems & Imaging, 2021, 15 (6) : 1451-1469. doi: 10.3934/ipi.2021018
##### References:
 [1] G. Acosta and J. P. Borthagaray, A fractional laplace equation: regularity of solutions and finite element approximations, SIAM Journal on Numerical Analysis, 55 (2017), 472-495.  doi: 10.1137/15M1033952.  Google Scholar [2] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.  Google Scholar [3] R. S. Anderssen, Richardson's Non-stationary Matrix Iterative Procedure, Technical Report, STAN-CS-72-304, Computer Science Department, Stanford University, 1972. Google Scholar [4] F. Andreu, J. M. Mzaón, J. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl, 90 (2008), 201-227.  doi: 10.1016/j.matpur.2008.04.003.  Google Scholar [5] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs AMS, (2010). doi: 10.1090/surv/165.  Google Scholar [6] J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.  Google Scholar [7] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Modeling and Simulation, 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar [8] A. Buades, B. Coll and J. M. Morel, Image denoising methods, a new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar [9] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal of Numerical Analysis, 29 (1992), 182-193.  doi: 10.1137/0729012.  Google Scholar [10] D. L. Chen, S. S. Sun, C. R. Zhang, Y. Q. Chen and D. Y. Xue, Fractional order TV-$L^2$ model for image denoising, Central European Journal of Physics, 11 (2013), 1414-1422.   Google Scholar [11] Y. Chzhao-Din, Some Difference Schemes for the Solution of the First Boundary Value Problem for Linear Differential Equations with Partial Derivatives, PhD Thesis, Moscow State University (in Russian), 1958. Google Scholar [12] F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Problems and Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.  Google Scholar [13] W. Gentzsch and A. Schluter, Über ein Einschrittverfahren mit zyklischer Schrittweiten anderung zur Losung parabolischer Differentialgleichungen (German), Zeitschrift fur Angewandte Mathematik und Mechanik, 58 (1987), 415-416.   Google Scholar [14] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling and Simulation, 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar [15] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar [16] S. Grewenig, J. Weickert and A. Bruhn, From box filtering to fast explicit diffusion, in Pattern Recognition, Lecture Notes in Comput. Sci., Springer, Berlin, 2010,533–542. doi: 10.1007/978-3-642-15986-2_54.  Google Scholar [17] C. Jin, G. Qian and X. Y. Wang, Image denoising based on adaptive fractional partial differential equations, in 2013 6th International Congress on Image and Signal Processing, (2013), 288–292. Google Scholar [18] Q. Ma, F. Dong and D. Kong, A fractional differential fidelity-based PDE model for image denoising, Machine Vision and Applications, 28 (2017), 635-647.  doi: 10.1007/s00138-017-0857-z.  Google Scholar [19] H.-K. Pang and H.-W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.  doi: 10.1016/j.jcp.2011.10.005.  Google Scholar [20] M. Pérez-Llanos and J. D. Rossi, Numerical approximations for a nonlocal evolution equation, SIAM Journal on Numerical Analysis, 49 (2011), 2103-2123.  doi: 10.1137/110823559.  Google Scholar [21] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar [22] E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun Nonlinear Sci Numer Simul, 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.  Google Scholar [23] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci.Engrg. 198, Academic Press, San Diego, CA, 1999.  Google Scholar [24] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen and B. M. V. Jara, Matrix approach to discrete fractional calculus ii: Partial fractional differential equations, Journal of Computational Physics, 228 (2009), 3137-3153.  doi: 10.1016/j.jcp.2009.01.014.  Google Scholar [25] L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Transactions of the Royal Society of London Series A, 210 (1910), 307-357.  doi: 10.1098/rsta.1911.0009.  Google Scholar [26] P. D. Romero and V. F. Candela, Blind deconvolution models regularized by fractional powers of the Laplacian, J. Math. Imaging Vision, 32 (2008), 181-191.  doi: 10.1007/s10851-008-0093-2.  Google Scholar [27] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [28] V. K. Saul'ev, Integration of Equations of Parabolic Type by the Method of Nets, International Series of Monographs in Pure and Applied Mathematics, Vol. 54 Pergamon Press, London-Edinburgh-New York 1960.  Google Scholar [29] D. Tian, D. Xue and D. Wang, A fractional-order adaptive regularization primal-dual algorithm for image denoising, Inf. Sci., 296 (2015), 147-159.  doi: 10.1016/j.ins.2014.10.050.  Google Scholar [30] R. S. Varga, Matrix Iterative Analysis, Englewood Cliffs, NJ, USA: Prentice-Hall, 1962.  Google Scholar [31] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar [32] H. Wang and N. Du, Fast solution methods for space-fractional diffusion equations, J. Comput. Appl.Math., 255 (2014), 376-383.  doi: 10.1016/j.cam.2013.06.002.  Google Scholar [33] J. Weickert, S. Grewenig, C. Schroers and A. Bruhn, Cyclic schemes for PDE-based image analysis, International Journal of Computer Vision, 118 (2016), 275-299. doi: 10.1007/s11263-015-0874-1.  Google Scholar [34] Q. Yang, D. Chen, T. Zhao and Y. Chen, Fractional calculus in image processing: A review, Fractional Calculus and Applied Analysis, 19 (2016), 1222-1249.  doi: 10.1515/fca-2016-0063.  Google Scholar [35] D. Young, On Richardson's method for solving linear systems with positive definite matrices, Journal of Mathematics and Physics, 32 (1954), 243-255.   Google Scholar [36] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.  doi: 10.1109/83.869184.  Google Scholar [37] J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, Siam Journal on Imaging Sciences, 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar [38] Y. Zhang, H. D. Cheng, J. Tian, J. Huang and X. Tang, Fractional subpixel diffusion and fuzzy logic approach for ultrasound speckle reduction, Pattern Recognition, 43 (2010), 2962-2970.  doi: 10.1016/j.patcog.2010.02.014.  Google Scholar [39] Y. Zhang, Y.-F. Pu, J.-R. Hu and J.-L. Zhou, A class of fractional-order variational image inpainting models, Applied Mathematics and Information Sciences, 6 (2012), 299-306.   Google Scholar [40] J. Zhang and Z. Wei, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, Appl. Math. Model., 35 (2011), 2516-2528.  doi: 10.1016/j.apm.2010.11.049.  Google Scholar

show all references

##### References:
 [1] G. Acosta and J. P. Borthagaray, A fractional laplace equation: regularity of solutions and finite element approximations, SIAM Journal on Numerical Analysis, 55 (2017), 472-495.  doi: 10.1137/15M1033952.  Google Scholar [2] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.  Google Scholar [3] R. S. Anderssen, Richardson's Non-stationary Matrix Iterative Procedure, Technical Report, STAN-CS-72-304, Computer Science Department, Stanford University, 1972. Google Scholar [4] F. Andreu, J. M. Mzaón, J. D. Rossi and J. Toledo, A nonlocal $p$-Laplacian evolution equation with Neumann boundary conditions, J. Math. Pures Appl, 90 (2008), 201-227.  doi: 10.1016/j.matpur.2008.04.003.  Google Scholar [5] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs AMS, (2010). doi: 10.1090/surv/165.  Google Scholar [6] J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process, 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.  Google Scholar [7] A. Buades, B. Coll and J. M. Morel, A review of image denoising algorithms, with a new one, Multiscale Modeling and Simulation, 4 (2005), 490-530.  doi: 10.1137/040616024.  Google Scholar [8] A. Buades, B. Coll and J. M. Morel, Image denoising methods, a new nonlocal principle, SIAM Review, 52 (2010), 113-147.  doi: 10.1137/090773908.  Google Scholar [9] F. Catte, P. L. Lions, J. M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM Journal of Numerical Analysis, 29 (1992), 182-193.  doi: 10.1137/0729012.  Google Scholar [10] D. L. Chen, S. S. Sun, C. R. Zhang, Y. Q. Chen and D. Y. Xue, Fractional order TV-$L^2$ model for image denoising, Central European Journal of Physics, 11 (2013), 1414-1422.   Google Scholar [11] Y. Chzhao-Din, Some Difference Schemes for the Solution of the First Boundary Value Problem for Linear Differential Equations with Partial Derivatives, PhD Thesis, Moscow State University (in Russian), 1958. Google Scholar [12] F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Problems and Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.  Google Scholar [13] W. Gentzsch and A. Schluter, Über ein Einschrittverfahren mit zyklischer Schrittweiten anderung zur Losung parabolischer Differentialgleichungen (German), Zeitschrift fur Angewandte Mathematik und Mechanik, 58 (1987), 415-416.   Google Scholar [14] G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Modeling and Simulation, 6 (2007), 595-630.  doi: 10.1137/060669358.  Google Scholar [15] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar [16] S. Grewenig, J. Weickert and A. Bruhn, From box filtering to fast explicit diffusion, in Pattern Recognition, Lecture Notes in Comput. Sci., Springer, Berlin, 2010,533–542. doi: 10.1007/978-3-642-15986-2_54.  Google Scholar [17] C. Jin, G. Qian and X. Y. Wang, Image denoising based on adaptive fractional partial differential equations, in 2013 6th International Congress on Image and Signal Processing, (2013), 288–292. Google Scholar [18] Q. Ma, F. Dong and D. Kong, A fractional differential fidelity-based PDE model for image denoising, Machine Vision and Applications, 28 (2017), 635-647.  doi: 10.1007/s00138-017-0857-z.  Google Scholar [19] H.-K. Pang and H.-W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.  doi: 10.1016/j.jcp.2011.10.005.  Google Scholar [20] M. Pérez-Llanos and J. D. Rossi, Numerical approximations for a nonlocal evolution equation, SIAM Journal on Numerical Analysis, 49 (2011), 2103-2123.  doi: 10.1137/110823559.  Google Scholar [21] P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205.  Google Scholar [22] E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun Nonlinear Sci Numer Simul, 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.  Google Scholar [23] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci.Engrg. 198, Academic Press, San Diego, CA, 1999.  Google Scholar [24] I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen and B. M. V. Jara, Matrix approach to discrete fractional calculus ii: Partial fractional differential equations, Journal of Computational Physics, 228 (2009), 3137-3153.  doi: 10.1016/j.jcp.2009.01.014.  Google Scholar [25] L. F. Richardson, The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam, Transactions of the Royal Society of London Series A, 210 (1910), 307-357.  doi: 10.1098/rsta.1911.0009.  Google Scholar [26] P. D. Romero and V. F. Candela, Blind deconvolution models regularized by fractional powers of the Laplacian, J. Math. Imaging Vision, 32 (2008), 181-191.  doi: 10.1007/s10851-008-0093-2.  Google Scholar [27] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [28] V. K. Saul'ev, Integration of Equations of Parabolic Type by the Method of Nets, International Series of Monographs in Pure and Applied Mathematics, Vol. 54 Pergamon Press, London-Edinburgh-New York 1960.  Google Scholar [29] D. Tian, D. Xue and D. Wang, A fractional-order adaptive regularization primal-dual algorithm for image denoising, Inf. Sci., 296 (2015), 147-159.  doi: 10.1016/j.ins.2014.10.050.  Google Scholar [30] R. S. Varga, Matrix Iterative Analysis, Englewood Cliffs, NJ, USA: Prentice-Hall, 1962.  Google Scholar [31] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar [32] H. Wang and N. Du, Fast solution methods for space-fractional diffusion equations, J. Comput. Appl.Math., 255 (2014), 376-383.  doi: 10.1016/j.cam.2013.06.002.  Google Scholar [33] J. Weickert, S. Grewenig, C. Schroers and A. Bruhn, Cyclic schemes for PDE-based image analysis, International Journal of Computer Vision, 118 (2016), 275-299. doi: 10.1007/s11263-015-0874-1.  Google Scholar [34] Q. Yang, D. Chen, T. Zhao and Y. Chen, Fractional calculus in image processing: A review, Fractional Calculus and Applied Analysis, 19 (2016), 1222-1249.  doi: 10.1515/fca-2016-0063.  Google Scholar [35] D. Young, On Richardson's method for solving linear systems with positive definite matrices, Journal of Mathematics and Physics, 32 (1954), 243-255.   Google Scholar [36] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.  doi: 10.1109/83.869184.  Google Scholar [37] J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, Siam Journal on Imaging Sciences, 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar [38] Y. Zhang, H. D. Cheng, J. Tian, J. Huang and X. Tang, Fractional subpixel diffusion and fuzzy logic approach for ultrasound speckle reduction, Pattern Recognition, 43 (2010), 2962-2970.  doi: 10.1016/j.patcog.2010.02.014.  Google Scholar [39] Y. Zhang, Y.-F. Pu, J.-R. Hu and J.-L. Zhou, A class of fractional-order variational image inpainting models, Applied Mathematics and Information Sciences, 6 (2012), 299-306.   Google Scholar [40] J. Zhang and Z. Wei, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, Appl. Math. Model., 35 (2011), 2516-2528.  doi: 10.1016/j.apm.2010.11.049.  Google Scholar
The truncation error of the truncated polynomial ${{\rho _N}\left( t \right)}$ at $z = 1$ with truncated lengths $t$ from 10 to 50 and fractional-orders $\alpha$ from 1.1 to 1.9
First row to last row: comparing the convergence rate of three algorithms for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40
], respectively">Figure 3.  First column to last column: Origin images, noisy images with $\sigma {\rm{ = }}10$, and the restored images by the FOFED algorithm, FOFED$_f$ algorithm and FOFFT algorithm in [6], respectively
], respectively">Figure 4.  First column to last column: Origin images, noisy images with $\sigma {\rm{ = }}20$, and the restored images by the FOFED algorithm, FOFED$_f$ algorithm and FOFFT algorithm in [6], respectively
], respectively">Figure 5.  First column to last column: Origin images, noisy images with $\sigma {\rm{ = }}40$, and the restored images by the FOFED algorithm, FOFED$_f$ algorithm and FOFFT algorithm in [6], respectively
The truncation error of the truncated polynomial ${{\rho _N}\left( t \right)}$ at $z = 1$ with truncated lengths $t$ from 10 to 50 (rows) and fractional-orders $\alpha$ from 1.1 to 1.9 (columns)
 t $\alpha = 1.1$ $\alpha = 1.2$ $\alpha = 1.3$ $\alpha = 1.4$ $\alpha = 1.5$ $\alpha = 1.6$ $\alpha = 1.7$ $\alpha = 1.8$ $\alpha = 1.9$ 10 0.0084 0.0125 0.0136 0.0128 0.0109 0.0085 0.0060 0.0036 0.0016 20 0.0037 0.0051 0.0051 0.0044 0.0035 0.0025 0.0016 0.0009 0.0004 30 0.0023 0.0030 0.0029 0.0024 0.0018 0.0013 0.0008 0.0004 0.0002 40 0.0017 0.0021 0.0020 0.0016 0.0012 0.0008 0.0005 0.0002 0.00009 50 0.0013 0.0016 0.0015 0.0012 0.0008 0.0005 0.0003 0.00016 0.00005
 t $\alpha = 1.1$ $\alpha = 1.2$ $\alpha = 1.3$ $\alpha = 1.4$ $\alpha = 1.5$ $\alpha = 1.6$ $\alpha = 1.7$ $\alpha = 1.8$ $\alpha = 1.9$ 10 0.0084 0.0125 0.0136 0.0128 0.0109 0.0085 0.0060 0.0036 0.0016 20 0.0037 0.0051 0.0051 0.0044 0.0035 0.0025 0.0016 0.0009 0.0004 30 0.0023 0.0030 0.0029 0.0024 0.0018 0.0013 0.0008 0.0004 0.0002 40 0.0017 0.0021 0.0020 0.0016 0.0012 0.0008 0.0005 0.0002 0.00009 50 0.0013 0.0016 0.0015 0.0012 0.0008 0.0005 0.0003 0.00016 0.00005
PSNR, MAE, and SSIM results for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The PSNR, MAE and SSIM values are similar
 PSNR MAE SSIM $\sigma$ 10 20 40 10 20 40 10 20 40 Cameraman image ($1024 \times 1024$) FOFED 39.4084 35.4073 33.7938 2.0434 3.2504 5.1143 0.5978 0.4621 0.3333 FOFED$_{f}$ 39.6005 35.6151 33.9866 1.9789 3.1111 4.8185 0.6030 0.4671 0.3374 FOFFT 39.4881 35.5515 33.8737 1.9958 3.1240 4.8238 0.6021 0.4665 0.3370 Lena image ($512 \times 512$) FOFED 34.2268 30.9968 28.5389 3.7184 5.2972 7.3706 0.6056 0.4904 0.3856 FOFED$_{f}$ 34.2882 31.0802 28.6008 3.6588 5.0912 7.1460 0.6082 0.4957 0.3908 FOFFT 34.2816 31.0847 28.5937 3.6564 5.0856 7.1361 0.6095 0.4962 0.3910 Butterfly image ($256 \times 256$) FOFED 32.8641 28.5347 26.6302 4.2297 6.8386 10.5351 0.7961 0.6960 0.5777 FOFED$_{f}$ 32.8944 28.5595 26.6618 4.1910 6.7488 10.3155 0.8126 0.7198 0.5960 FOFFT 32.8459 28.5339 26.6119 4.2067 6.7536 10.2976 0.8134 0.7209 0.5973 House image ($128 \times 128$) FOFED 31.9328 27.8935 27.0384 4.7282 7.3484 11.1483 0.5607 0.4463 0.3439 FOFED$_{f}$ 31.9543 27.9203 27.0600 4.6964 7.2673 10.6816 0.5610 0.4470 0.3446 FOFFT 31.9849 27.9562 27.0898 4.6672 7.2156 10.6107 0.5624 0.4476 0.3453
 PSNR MAE SSIM $\sigma$ 10 20 40 10 20 40 10 20 40 Cameraman image ($1024 \times 1024$) FOFED 39.4084 35.4073 33.7938 2.0434 3.2504 5.1143 0.5978 0.4621 0.3333 FOFED$_{f}$ 39.6005 35.6151 33.9866 1.9789 3.1111 4.8185 0.6030 0.4671 0.3374 FOFFT 39.4881 35.5515 33.8737 1.9958 3.1240 4.8238 0.6021 0.4665 0.3370 Lena image ($512 \times 512$) FOFED 34.2268 30.9968 28.5389 3.7184 5.2972 7.3706 0.6056 0.4904 0.3856 FOFED$_{f}$ 34.2882 31.0802 28.6008 3.6588 5.0912 7.1460 0.6082 0.4957 0.3908 FOFFT 34.2816 31.0847 28.5937 3.6564 5.0856 7.1361 0.6095 0.4962 0.3910 Butterfly image ($256 \times 256$) FOFED 32.8641 28.5347 26.6302 4.2297 6.8386 10.5351 0.7961 0.6960 0.5777 FOFED$_{f}$ 32.8944 28.5595 26.6618 4.1910 6.7488 10.3155 0.8126 0.7198 0.5960 FOFFT 32.8459 28.5339 26.6119 4.2067 6.7536 10.2976 0.8134 0.7209 0.5973 House image ($128 \times 128$) FOFED 31.9328 27.8935 27.0384 4.7282 7.3484 11.1483 0.5607 0.4463 0.3439 FOFED$_{f}$ 31.9543 27.9203 27.0600 4.6964 7.2673 10.6816 0.5610 0.4470 0.3446 FOFFT 31.9849 27.9562 27.0898 4.6672 7.2156 10.6107 0.5624 0.4476 0.3453
Number of Iteration and CPU time for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The best CPU times are in black
 Number of Iteration CPU time(s) $\sigma$ 10 20 40 10 20 40 Cameraman image ($1024 \times 1024$) FOFED 41 71 116 $12.0397$ $15.5452$ $25.8294$ FOFED$_{f}$ 100 230 553 32.8190 77.5531 187.0590 FOFFT 114 258 612 98.8958 227.8830 518.8290 Lena image ($512 \times 512$) FOFED 30 56 129 $1.9445$ $3.5040$ $8.1629$ FOFED$_{f}$ 58 157 420 4.6834 12.4698 32.5470 FOFFT 69 181 473 14.8598 39.0244 101.7675 Butterfly image ($256 \times 256$) FOFED 25 55 88 $0.3193$ $0.6546$ $1.0031$ FOFED$_{f}$ 47 112 289 0.6994 1.5716 4.0302 FOFFT 56 134 336 2.8438 6.7368 16.9259 House image ($128 \times 128$) FOFED 40 54 27.0384 $0.3604$ $0.4095$ $0.7060$ FOFED$_{f}$ 41 104 27.0600 0.3820 0.8661 2.2998 FOFFT 50 125 27.0898 0.7193 1.6759 3.9392
 Number of Iteration CPU time(s) $\sigma$ 10 20 40 10 20 40 Cameraman image ($1024 \times 1024$) FOFED 41 71 116 $12.0397$ $15.5452$ $25.8294$ FOFED$_{f}$ 100 230 553 32.8190 77.5531 187.0590 FOFFT 114 258 612 98.8958 227.8830 518.8290 Lena image ($512 \times 512$) FOFED 30 56 129 $1.9445$ $3.5040$ $8.1629$ FOFED$_{f}$ 58 157 420 4.6834 12.4698 32.5470 FOFFT 69 181 473 14.8598 39.0244 101.7675 Butterfly image ($256 \times 256$) FOFED 25 55 88 $0.3193$ $0.6546$ $1.0031$ FOFED$_{f}$ 47 112 289 0.6994 1.5716 4.0302 FOFFT 56 134 336 2.8438 6.7368 16.9259 House image ($128 \times 128$) FOFED 40 54 27.0384 $0.3604$ $0.4095$ $0.7060$ FOFED$_{f}$ 41 104 27.0600 0.3820 0.8661 2.2998 FOFFT 50 125 27.0898 0.7193 1.6759 3.9392
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