doi: 10.3934/naco.2021022

A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration

1. 

KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

2. 

Faculty of Science, Energy and Environment, King Mongkut's University of Technology North Bangkok, 19 Moo 11, Tambon Nonglalok, Amphur Bankhai, Rayong 21120 Thailand

3. 

Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano. Kano, Nigeria, Department of Mathematics and Applied Mathematics

4. 

Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa

5. 

Department of Mathematics, Usmanu Danfodiyo University, Sokoto State, Nigeria, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria, Medunsa-0204, South Africa

* Corresponding author: Jitsupa Deepho

Received  February 2021 Revised  May 2021 Published  June 2021

Fund Project: This research is funded by King Mongkut’s University of Technology North Bangkok, Contract no. KMUTNB-63-KNOW-016

We present an iterative method for solving the convex constraint nonlinear equation problem. The method incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method by Yang et al. for solving the unconstrained optimization problem. The proposed method does not require the Jacobian information, nor does it require to store any matrix at each iteration. Thus, it has the potential to solve large-scale non-smooth problems. Under some standard assumptions, the convergence analysis of the method is established. Finally, to show the applicability of the proposed method, the proposed method is used to solve the $ \ell_1 $-norm regularized problems to restore blurred and noisy images. The numerical experiment indicates that our result is a significant improvement compared with the related methods for solving the convex constraint nonlinear equation problem.

Citation: Abdulkarim Hassan Ibrahim, Jitsupa Deepho, Auwal Bala Abubakar, Kazeem Olalekan Aremu. A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021022
References:
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A. B. AbubakarJ. RilwanS. E. YimerA. B. Ibrahim and I. Ahmed, Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints, Thai Journal of Mathematics, 18 (2020), 501-517.   Google Scholar

[3]

A. B. AbubakarA. H. IbrahimA. B Muhammad and C. Tammer, A modified descent Dai-Yuan conjugate gradient method for constraint nonlinear monotone operator equations, Applied Analysis and Optimization, 4 (2020), 1-24.   Google Scholar

[4]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Computational and Applied Mathematics, 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.  Google Scholar

[5]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numerical Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.  Google Scholar

[6]

A. B. Abubakar, P. Kumam, A. H. Ibrahim and J. Rilwan, Derivative-free HS-DY-type method for solving nonlinear equations and image restoration, Heliyon, 6 (2020), e05400. Google Scholar

[7]

A. B. AbubakarP. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Computational and Applied Mathematics, 39 (2020), 1-35.  doi: 10.1007/s40314-020-01151-5.  Google Scholar

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A. B. Abubakar, P. Kumam, H. Mohammad and A. H. Ibrahim, PRP-like algorithm for monotone operator equations, Japan Journal of Industrial and Applied Mathematics, (2021), 1–18. Google Scholar

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A. B. Abubakar, K. Muangchoo, A. H. Ibrahim, J. Abubakar and S. A. Rano, FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations, Arabian Journal of Mathematics, (2021), 1–10., Google Scholar

[10]

A. B. AbubakarK. MuangchooA. H. IbrahimA. B. MuhammadL. O. Jolaoso and K. O. Aremu, A new three-term Hestenes-Stiefel type method for nonlinear monotone operator equations and image restoration, IEEE Access, 9 (2021), 18262-18277.   Google Scholar

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Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM Journal on Optimization, 1 (1991), 206-221.  doi: 10.1137/0801015.  Google Scholar

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E ChidumeAbubakar Adamu and Lois C Okereke, Iterative algorithms for solutions of hammerstein equations in real banach spaces, Fixed Point Theory and Applications, 2020 (2020), 1-23.  doi: 10.1186/s13663-020-0670-7.  Google Scholar

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E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

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A. H. IbrahimA. I. GarbaH. UsmanJ. Abubakar and A. B. Abubakar, Derivative-free RMIL conjugate gradient method for convex constrained equations, Thai Journal of Mathematics, 18 (2019), 212-232.   Google Scholar

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A. H. Ibrahim and P. Kumam, Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints, Ain Shams Engineering Journal, 2021. Google Scholar

[21]

A. H. Ibrahim, P. Kumam, A. B. Abubakar, J. Abubakar and A. B. Muhammad, Least-square-based three-term conjugate gradient projection method for $\ell_1$-norm problems with application to compressed sensing, Mathematics, 8 (2020), 602. Google Scholar

[22]

A. H. Ibrahim, P. Kumam, A. B. Abubakar, W. Jirakitpuwapat and J. Abubakar, A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing, Heliyon, 6 (2020), e03466. Google Scholar

[23]

A. H. IbrahimP. KumamA. B. AbubakarU. B. Yusuf and J. Rilwan, Derivative-free conjugate residual algorithms for convex constraints nonlinear monotone equations and signal recovery, Journal of Nonlinear and Convex Analysis, 21 (2020), 1959-1972.   Google Scholar

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A. H. IbrahimP. KumamA. B. AbubakarU. B. YusufS. E. Yimer and K. O. Aremu, An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration, Aims Mathematics, 6 (2020), 235-260.  doi: 10.3934/math.2021016.  Google Scholar

[25]

A. H. IbrahimP. Kumam and W. Kumam, A family of derivative-free conjugate gradient methods for constrained nonlinear equations and image restoration, IEEE Access, 8 (2020), 162714-162729.   Google Scholar

[26]

A. H. IbrahimK. MuangchooA. B. AbubakarA. D. Adedokun and H. Mohammed, Spectral conjugate gradient like method for signal reconstruction, Thai Journal of Mathematics, 18 (2020), 2013-2022.   Google Scholar

[27]

A. H. IbrahimaK. MuangchooN. S. Mohamed and A. B. Abubakard, Derivative-free SMR conjugate gradient method for con-straint nonlinear equations, Journal of Mathematics and Computer Science, 24 (2022), 147-164.   Google Scholar

[28]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numerical Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

[29]

W. La CruzJ. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[30]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

[31]

S. M. Lajevardi, Structural similarity classifier for facial expression recognition, Signal, Image and Video Processing, 8 (2014), 1103-1110.   Google Scholar

[32]

D. Li and M. Fukushima, A globally and superlinearly convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal on Numerical Analysis, 37 (1999), 152-172.  doi: 10.1137/S0036142998335704.  Google Scholar

[33]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, (2018), 1–18. doi: 10.1007/s11075-018-0603-2.  Google Scholar

[34]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Applied Mathematics and Computation, 22 (1987), 333-361.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[35]

G. J. Minty, Monotone networks, Proceedings of the Royal Society of London, Series A. Mathematical and Physical Sciences, 257 (1960), 194-212.  doi: 10.1098/rspa.1960.0144.  Google Scholar

[36]

H. Mohammad, Barzilai-borwein-like method for solving large-scale non-linear systems of equations, Journal of the Nigerian Mathematical Society, 36 (2017), 71-83.   Google Scholar

[37]

H. Mohammad and A. B. Abubakar, A descent derivative-free algorithm for nonlinear monotone equations with convex constraints, RAIRO-Operations Research, 54 (2020), 489-505.  doi: 10.1051/ro/2020008.  Google Scholar

[38]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.   Google Scholar

[39]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM Journal on Control and Optimization, 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar

[40]

A. J. Wood, B. F. Wollenberg and G. B. Sheblé, Power Generation, Operation, and Control, John Wiley & Sons, 2013. Google Scholar

[41]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, Journal of Mathematical Analysis and Applications, 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar

[42]

X. YangZ. Luo and and X. Dai, A global convergence of LS-CD hybrid conjugate gradient method, Adv. Numerical Analysis, 2013 (2013), 517452-1.  doi: 10.1155/2013/517452.  Google Scholar

[43]

Z. YuJ. LinJ. SunY. H. XiaoL. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Applied Numerical Mathematics, 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[44]

L. Zhang and W. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, 196 (2006), 478-484.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

[45]

G. Zhou and K. Toh, Superlinear convergence of a Newton-type algorithm for monotone equations, Journal of Optimization Theory and Applications, 125 (2005), 205-221.  doi: 10.1007/s10957-004-1721-7.  Google Scholar

[46]

W. Zhou and D. Li, A globally convergent BFGS method for nonlinear monotone equations without any merit functions, Mathematics of Computation, 77 (2008), 2231-2240.  doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

show all references

References:
[1]

A. B. Abubakar, P. Kumam, H. Mohammad, A. M. Awwal and S. Kanokwan, A modified Fletcher–Reeves conjugate gradient method for monotone nonlinear equations with some applications, Mathematics, 7 (2019), 745. doi: 10.1016/j.apnum.2019.05.012.  Google Scholar

[2]

A. B. AbubakarJ. RilwanS. E. YimerA. B. Ibrahim and I. Ahmed, Spectral three-term conjugate descent method for solving nonlinear monotone equations with convex constraints, Thai Journal of Mathematics, 18 (2020), 501-517.   Google Scholar

[3]

A. B. AbubakarA. H. IbrahimA. B Muhammad and C. Tammer, A modified descent Dai-Yuan conjugate gradient method for constraint nonlinear monotone operator equations, Applied Analysis and Optimization, 4 (2020), 1-24.   Google Scholar

[4]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Computational and Applied Mathematics, 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.  Google Scholar

[5]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numerical Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.  Google Scholar

[6]

A. B. Abubakar, P. Kumam, A. H. Ibrahim and J. Rilwan, Derivative-free HS-DY-type method for solving nonlinear equations and image restoration, Heliyon, 6 (2020), e05400. Google Scholar

[7]

A. B. AbubakarP. Kumam and H. Mohammad, A note on the spectral gradient projection method for nonlinear monotone equations with applications, Computational and Applied Mathematics, 39 (2020), 1-35.  doi: 10.1007/s40314-020-01151-5.  Google Scholar

[8]

A. B. Abubakar, P. Kumam, H. Mohammad and A. H. Ibrahim, PRP-like algorithm for monotone operator equations, Japan Journal of Industrial and Applied Mathematics, (2021), 1–18. Google Scholar

[9]

A. B. Abubakar, K. Muangchoo, A. H. Ibrahim, J. Abubakar and S. A. Rano, FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations, Arabian Journal of Mathematics, (2021), 1–10., Google Scholar

[10]

A. B. AbubakarK. MuangchooA. H. IbrahimA. B. MuhammadL. O. Jolaoso and K. O. Aremu, A new three-term Hestenes-Stiefel type method for nonlinear monotone operator equations and image restoration, IEEE Access, 9 (2021), 18262-18277.   Google Scholar

[11]

Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM Journal on Optimization, 1 (1991), 206-221.  doi: 10.1137/0801015.  Google Scholar

[12] Alan C Bovik, Handbook of Image and Video Processing, Academic press, 2010.   Google Scholar
[13]

W. Cheng, A PRP type method for systems of monotone equations, Mathematical and Computer Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.  Google Scholar

[14]

E ChidumeAbubakar Adamu and Lois C Okereke, Iterative algorithms for solutions of hammerstein equations in real banach spaces, Fixed Point Theory and Applications, 2020 (2020), 1-23.  doi: 10.1186/s13663-020-0670-7.  Google Scholar

[15]

J. E. Dennis and J. J. Moré, A characterization of superlinear convergence and its application to quasi-newton methods, Mathematics of Computation, 28 (1974), 549-560.  doi: 10.2307/2005926.  Google Scholar

[16]

Y. DingY. Xiao and and J. Li, A class of conjugate gradient methods for convex constrained monotone equations, Optimization, 66 (2017), 2309-2328.  doi: 10.1080/02331934.2017.1372438.  Google Scholar

[17]

S. P. Dirkse and M. C. Ferris, Mcplib: A collection of nonlinear mixed complementarity problems, Optimization Methods and Software, 5 (1995), 319-345.   Google Scholar

[18]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[19]

A. H. IbrahimA. I. GarbaH. UsmanJ. Abubakar and A. B. Abubakar, Derivative-free RMIL conjugate gradient method for convex constrained equations, Thai Journal of Mathematics, 18 (2019), 212-232.   Google Scholar

[20]

A. H. Ibrahim and P. Kumam, Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints, Ain Shams Engineering Journal, 2021. Google Scholar

[21]

A. H. Ibrahim, P. Kumam, A. B. Abubakar, J. Abubakar and A. B. Muhammad, Least-square-based three-term conjugate gradient projection method for $\ell_1$-norm problems with application to compressed sensing, Mathematics, 8 (2020), 602. Google Scholar

[22]

A. H. Ibrahim, P. Kumam, A. B. Abubakar, W. Jirakitpuwapat and J. Abubakar, A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing, Heliyon, 6 (2020), e03466. Google Scholar

[23]

A. H. IbrahimP. KumamA. B. AbubakarU. B. Yusuf and J. Rilwan, Derivative-free conjugate residual algorithms for convex constraints nonlinear monotone equations and signal recovery, Journal of Nonlinear and Convex Analysis, 21 (2020), 1959-1972.   Google Scholar

[24]

A. H. IbrahimP. KumamA. B. AbubakarU. B. YusufS. E. Yimer and K. O. Aremu, An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration, Aims Mathematics, 6 (2020), 235-260.  doi: 10.3934/math.2021016.  Google Scholar

[25]

A. H. IbrahimP. Kumam and W. Kumam, A family of derivative-free conjugate gradient methods for constrained nonlinear equations and image restoration, IEEE Access, 8 (2020), 162714-162729.   Google Scholar

[26]

A. H. IbrahimK. MuangchooA. B. AbubakarA. D. Adedokun and H. Mohammed, Spectral conjugate gradient like method for signal reconstruction, Thai Journal of Mathematics, 18 (2020), 2013-2022.   Google Scholar

[27]

A. H. IbrahimaK. MuangchooN. S. Mohamed and A. B. Abubakard, Derivative-free SMR conjugate gradient method for con-straint nonlinear equations, Journal of Mathematics and Computer Science, 24 (2022), 147-164.   Google Scholar

[28]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numerical Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

[29]

W. La CruzJ. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Mathematics of Computation, 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[30]

W. La Cruz and M. Raydan, Nonmonotone spectral methods for large-scale nonlinear systems, Optimization Methods and Software, 18 (2003), 583-599.  doi: 10.1080/10556780310001610493.  Google Scholar

[31]

S. M. Lajevardi, Structural similarity classifier for facial expression recognition, Signal, Image and Video Processing, 8 (2014), 1103-1110.   Google Scholar

[32]

D. Li and M. Fukushima, A globally and superlinearly convergent Gauss–Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal on Numerical Analysis, 37 (1999), 152-172.  doi: 10.1137/S0036142998335704.  Google Scholar

[33]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, (2018), 1–18. doi: 10.1007/s11075-018-0603-2.  Google Scholar

[34]

K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems, Applied Mathematics and Computation, 22 (1987), 333-361.  doi: 10.1016/0096-3003(87)90076-2.  Google Scholar

[35]

G. J. Minty, Monotone networks, Proceedings of the Royal Society of London, Series A. Mathematical and Physical Sciences, 257 (1960), 194-212.  doi: 10.1098/rspa.1960.0144.  Google Scholar

[36]

H. Mohammad, Barzilai-borwein-like method for solving large-scale non-linear systems of equations, Journal of the Nigerian Mathematical Society, 36 (2017), 71-83.   Google Scholar

[37]

H. Mohammad and A. B. Abubakar, A descent derivative-free algorithm for nonlinear monotone equations with convex constraints, RAIRO-Operations Research, 54 (2020), 489-505.  doi: 10.1051/ro/2020008.  Google Scholar

[38]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Computational Mathematics and Mathematical Physics, 9 (1969), 94-112.   Google Scholar

[39]

M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM Journal on Control and Optimization, 37 (1999), 765-776.  doi: 10.1137/S0363012997317475.  Google Scholar

[40]

A. J. Wood, B. F. Wollenberg and G. B. Sheblé, Power Generation, Operation, and Control, John Wiley & Sons, 2013. Google Scholar

[41]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, Journal of Mathematical Analysis and Applications, 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar

[42]

X. YangZ. Luo and and X. Dai, A global convergence of LS-CD hybrid conjugate gradient method, Adv. Numerical Analysis, 2013 (2013), 517452-1.  doi: 10.1155/2013/517452.  Google Scholar

[43]

Z. YuJ. LinJ. SunY. H. XiaoL. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Applied Numerical Mathematics, 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[44]

L. Zhang and W. Zhou, Spectral gradient projection method for solving nonlinear monotone equations, Journal of Computational and Applied Mathematics, 196 (2006), 478-484.  doi: 10.1016/j.cam.2005.10.002.  Google Scholar

[45]

G. Zhou and K. Toh, Superlinear convergence of a Newton-type algorithm for monotone equations, Journal of Optimization Theory and Applications, 125 (2005), 205-221.  doi: 10.1007/s10957-004-1721-7.  Google Scholar

[46]

W. Zhou and D. Li, A globally convergent BFGS method for nonlinear monotone equations without any merit functions, Mathematics of Computation, 77 (2008), 2231-2240.  doi: 10.1090/S0025-5718-08-02121-2.  Google Scholar

Figure 1.  Performance profiles based on number of iterations
Figure 2.  Performance profiles based on number of function evaluations
Figure 3.  Performance profiles based on CPU time (in seconds)
Figure 4.  Image restoration: The original images (First column), Blurred and noisy images (Second Column), restored images by MLSCD (Third column), MFRM (Fouth column) and restored images by ELSFR (Fifth column)
Table 1.  Numerical result for the image restoration problem
MLSCD MFRM ELSFR
Images SNR PSNR SSIM SNR PSNR SSIM SNR PSNR SSIM
Lena 20.43 25.76 0.943 18.02 23.35 0.924 19.84 25.18 0.941
Tiffany 24.27 26.11 0.933 22.13 23.96 0.905 23.75 25.59 0.935
Barbara 16.03 22.45 0.706 14.22 20.64 0.645 15.65 22.07 0.701
MLSCD MFRM ELSFR
Images SNR PSNR SSIM SNR PSNR SSIM SNR PSNR SSIM
Lena 20.43 25.76 0.943 18.02 23.35 0.924 19.84 25.18 0.941
Tiffany 24.27 26.11 0.933 22.13 23.96 0.905 23.75 25.59 0.935
Barbara 16.03 22.45 0.706 14.22 20.64 0.645 15.65 22.07 0.701
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