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An adaptive total variational despeckling model based on gray level indicator frame
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 |
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.
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. |
[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. |
[3] |
R. S. Anderssen, Richardson's Non-stationary Matrix Iterative Procedure, Technical Report, STAN-CS-72-304, Computer Science Department, Stanford University, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
R. S. Varga, Matrix Iterative Analysis, Englewood Cliffs, NJ, USA: Prentice-Hall, 1962. |
[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. |
[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. |
[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. |
[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. |
[35] |
D. Young,
On Richardson's method for solving linear systems with positive definite matrices, Journal of Mathematics and Physics, 32 (1954), 243-255.
|
[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. |
[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. |
[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. |
[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.
|
[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. |
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. |
[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. |
[3] |
R. S. Anderssen, Richardson's Non-stationary Matrix Iterative Procedure, Technical Report, STAN-CS-72-304, Computer Science Department, Stanford University, 1972. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
R. S. Varga, Matrix Iterative Analysis, Englewood Cliffs, NJ, USA: Prentice-Hall, 1962. |
[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. |
[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. |
[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. |
[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. |
[35] |
D. Young,
On Richardson's method for solving linear systems with positive definite matrices, Journal of Mathematics and Physics, 32 (1954), 243-255.
|
[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. |
[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. |
[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. |
[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.
|
[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. |


t | |||||||||
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 | |||||||||
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 | SSIM | |||||||
10 | 20 | 40 | 10 | 20 | 40 | 10 | 20 | 40 | |
Cameraman image ( |
|||||||||
FOFED | 39.4084 | 35.4073 | 33.7938 | 2.0434 | 3.2504 | 5.1143 | 0.5978 | 0.4621 | 0.3333 |
FOFED |
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 ( |
|||||||||
FOFED | 34.2268 | 30.9968 | 28.5389 | 3.7184 | 5.2972 | 7.3706 | 0.6056 | 0.4904 | 0.3856 |
FOFED |
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 ( |
|||||||||
FOFED | 32.8641 | 28.5347 | 26.6302 | 4.2297 | 6.8386 | 10.5351 | 0.7961 | 0.6960 | 0.5777 |
FOFED |
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 ( |
|||||||||
FOFED | 31.9328 | 27.8935 | 27.0384 | 4.7282 | 7.3484 | 11.1483 | 0.5607 | 0.4463 | 0.3439 |
FOFED |
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 | |||||||
10 | 20 | 40 | 10 | 20 | 40 | 10 | 20 | 40 | |
Cameraman image ( |
|||||||||
FOFED | 39.4084 | 35.4073 | 33.7938 | 2.0434 | 3.2504 | 5.1143 | 0.5978 | 0.4621 | 0.3333 |
FOFED |
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 ( |
|||||||||
FOFED | 34.2268 | 30.9968 | 28.5389 | 3.7184 | 5.2972 | 7.3706 | 0.6056 | 0.4904 | 0.3856 |
FOFED |
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 ( |
|||||||||
FOFED | 32.8641 | 28.5347 | 26.6302 | 4.2297 | 6.8386 | 10.5351 | 0.7961 | 0.6960 | 0.5777 |
FOFED |
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 ( |
|||||||||
FOFED | 31.9328 | 27.8935 | 27.0384 | 4.7282 | 7.3484 | 11.1483 | 0.5607 | 0.4463 | 0.3439 |
FOFED |
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 | CPU time(s) | |||||
10 | 20 | 40 | 10 | 20 | 40 | |
Cameraman image ( |
||||||
FOFED | 41 | 71 | 116 | |||
FOFED |
100 | 230 | 553 | 32.8190 | 77.5531 | 187.0590 |
FOFFT | 114 | 258 | 612 | 98.8958 | 227.8830 | 518.8290 |
Lena image ( |
||||||
FOFED | 30 | 56 | 129 | |||
FOFED |
58 | 157 | 420 | 4.6834 | 12.4698 | 32.5470 |
FOFFT | 69 | 181 | 473 | 14.8598 | 39.0244 | 101.7675 |
Butterfly image ( |
||||||
FOFED | 25 | 55 | 88 | |||
FOFED |
47 | 112 | 289 | 0.6994 | 1.5716 | 4.0302 |
FOFFT | 56 | 134 | 336 | 2.8438 | 6.7368 | 16.9259 |
House image ( |
||||||
FOFED | 40 | 54 | 27.0384 | |||
FOFED |
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) | |||||
10 | 20 | 40 | 10 | 20 | 40 | |
Cameraman image ( |
||||||
FOFED | 41 | 71 | 116 | |||
FOFED |
100 | 230 | 553 | 32.8190 | 77.5531 | 187.0590 |
FOFFT | 114 | 258 | 612 | 98.8958 | 227.8830 | 518.8290 |
Lena image ( |
||||||
FOFED | 30 | 56 | 129 | |||
FOFED |
58 | 157 | 420 | 4.6834 | 12.4698 | 32.5470 |
FOFFT | 69 | 181 | 473 | 14.8598 | 39.0244 | 101.7675 |
Butterfly image ( |
||||||
FOFED | 25 | 55 | 88 | |||
FOFED |
47 | 112 | 289 | 0.6994 | 1.5716 | 4.0302 |
FOFFT | 56 | 134 | 336 | 2.8438 | 6.7368 | 16.9259 |
House image ( |
||||||
FOFED | 40 | 54 | 27.0384 | |||
FOFED |
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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