doi: 10.3934/jimo.2021170
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Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions

Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

* Corresponding author: nurullahyilmaz@sdu.edu.tr

Received  April 2021 Revised  July 2021 Early access October 2021

In this study, we concentrate on the hyperbolic smoothing technique for some sub-classes of non-smooth functions and introduce a generalization of hyperbolic smoothing technique for non-Lipschitz functions. We present some useful properties of this generalization of hyperbolic smoothing technique. In order to illustrate the efficiency of the proposed smoothing technique, we consider the regularization problems of image restoration. The regularization problem is recast by considering the generalization of hyperbolic smoothing technique and a new algorithm is developed. Finally, the minimization algorithm is applied to image restoration problems and the numerical results are reported.

Citation: Nurullah Yilmaz, Ahmet Sahiner. Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021170
References:
[1]

A. M. BagirovA. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335.  Google Scholar

[2]

A. M. BagirovB. OrdinG. Ozturk and A. E. Xavier, An incremental clustering algorithm based on hyperbolic smoothing, Comput. Optim. Appl., 61 (2015), 219-241.  doi: 10.1007/s10589-014-9711-7.  Google Scholar

[3]

A. M. BagirovN. SultanovaA. Al Nuaimat and S. Taheri, Solving minimax problems: Local smoothing versus global smoothing, Numerical Analysis and Optimization, Springer Proceedings in Mathematics an Statistic, 235 (2018), 23-43.  doi: 10.1007/978-3-319-90026-1_2.  Google Scholar

[4]

D. P. Bertsekas, Nondifferentiable optimization via approximation, Math. Programming Stud., 3 (1975), 1-25.  doi: 10.1007/BFb0120696.  Google Scholar

[5]

W. Bian and X. Chen, Smoothing neural network for constrained non-Lipschitz optimization with applications, IEEE Trans. Neural Netw. Learn. Syst., 23 (2012), 399-411.  doi: 10.1109/TNNLS.2011.2181867.  Google Scholar

[6]

W. Bian and X. Chen, Linearly constrained non-Lipschitz optimization for image restoration, SIAM J. Imaging Sci., 8 (2015), 2294-2322.  doi: 10.1137/140985639.  Google Scholar

[7]

L. CaccettaB. Qu and G. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45-58.  doi: 10.1007/s10589-009-9242-9.  Google Scholar

[8]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Prog. Ser. B., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[9]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.  doi: 10.1007/BF00249052.  Google Scholar

[10]

X. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-regularization and box constrained model for image restoration, IEEE Trans. Image Process., 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[11]

X. Chen and W. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sciences, 3 (2010), 765-790.  doi: 10.1137/080740167.  Google Scholar

[12]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. doi: 10.1137/1.9781611971309.  Google Scholar

[13]

C. Grossmann, Smoothing techniques for exact penalty methods, Contemporary Mathematics, In book Panaroma of Mathematics: Pure and Applied, 658 (2016), 249-265.   Google Scholar

[14]

Y. HuangH. Liu and W. Cong, A note on the smoothing quadratic regularization method for non-Lipschitz optimization, Numer. Algor., 69 (2015), 863-874.  doi: 10.1007/s11075-014-9929-6.  Google Scholar

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X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, J. Ind. Manag. Optim., 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[16]

M. Kang and M. Jung, Simultaneous image enhancement and restoration with non-convex total variation, J. Sci. Comput., 87 (2021), Paper No. 83, 46 pp. doi: 10.1007/s10915-021-01488-x.  Google Scholar

[17]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0074500.  Google Scholar

[18]

G. Landi, A modified Newton projection method for $\ell_1$-regularized least squares image deblurring, J. Math. Imaging Vis., 51 (2015), 195-208.  doi: 10.1007/s10851-014-0514-3.  Google Scholar

[19]

S.-J. Lian, Smoothing approximation to $l_1$ exact penalty function for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.  doi: 10.1016/j.amc.2012.09.042.  Google Scholar

[20]

M. M. Mäkelä and P. Neitaanmäki, Nonsmooth Optimization, World Scientific, Singapore, 1992. doi: 10.1142/1493.  Google Scholar

[21]

N. Mau NamL. T. H. AnD. Giles and N. Thai An, Smoothing techniques and difference of convex functions algorithms for image reconstructions, Optimization, 69 (2020), 1601-1633.  doi: 10.1080/02331934.2019.1648467.  Google Scholar

[22]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[23]

M. Nikolova, Minimizers of cost functions involving nonsmooth data-fidelity terms. Application to the processing outliers, SIAM J. Numer. Anal., 40 (2002), 965-994.  doi: 10.1137/S0036142901389165.  Google Scholar

[24]

M. NikolovaM. K. NgS. Zheng and W. K. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 1 (2008), 2-25.  doi: 10.1137/070692285.  Google Scholar

[25]

O. N. OnakY. Serinagaoglu-Dogrusoz and G.-W. Weber, Effects of a priori parameter selection in minimum relative entropy method on inverse electrocardiography problem, Inverse Probl. Sci. Eng., 26 (2018), 877-897.  doi: 10.1080/17415977.2017.1369979.  Google Scholar

[26]

C. T. PhamG. GamardA. Kopylov and T. T. T. Tran, An algorithm for image restoration with mixed noise using total variation regularization, Turk. J. Elec. Eng. Comp. Sci., 26 (2018), 2831-2845.  doi: 10.3906/elk-1803-100.  Google Scholar

[27]

C. T. PhamT. T. T. Tran and G. Gamard, An efficient total variation minimization method for image restoration, Informatica, 31 (2020), 539-560.  doi: 10.15388/20-INFOR407.  Google Scholar

[28]

R. A. Polyak, Smooth optimization methods for minimax problems, SIAM J. Control Optim., 26 (1988), 1274-1286.  doi: 10.1137/0326071.  Google Scholar

[29]

L. Qi and D. Sun, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl., 113 (2002), 121-147.  doi: 10.1023/A:1014861331301.  Google Scholar

[30]

R. T. Rockefellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[31]

L. I. RudinS. 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

[32]

B. SaheyaC.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equations, J. Appl. Math. Comput., 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0.  Google Scholar

[33]

A. SahinerG. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Contol Optim., 6 (2016), 161-173.  doi: 10.3934/naco.2016006.  Google Scholar

[34]

M. SouzaA. E. XavierC. Lavor and N. Maculan, Hyperbolic smoothing and penalty techniques applied to molecular structure determination, Oper. Res. Lett., 39 (2011), 461-465.  doi: 10.1016/j.orl.2011.07.007.  Google Scholar

[35]

P. TaylanG.-W. Weber and F. Yerlikaya-Ozkurt, A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization, TOP, 18 (2010), 377-395.  doi: 10.1007/s11750-010-0155-7.  Google Scholar

[36]

A. H. Tor, Hyperbolic smoothing method for sum-max problems, Neural, Parallel Sci. Comput., 24 (2016), 381-391.   Google Scholar

[37]

S. Voronin, G. Ozkaya and D. Yoshida, Convolution based smooth approximations to the absolute value function with application to non-smooth regularization, Preprint, (2015). arXiv: 1408.6795. Google Scholar

[38]

C. Wu, J. Zhan, Y. Lu and J.-S. Chen, Signal reconstruction by conjugate gradient algorithm based on smoothing $l_1$-norm, Calcolo, 56 (2019), Paper No. 42, 26 pp. doi: 10.1007/s10092-019-0340-5.  Google Scholar

[39]

A. E. Xavier, Penalizacao Hiperbolica, I Congresso Latino-Americano de Pesquisa Operacional e Engenharia de Sistemas, 8 a 11 de Novembro, Rio de Janeiro, Brasil, 1982. Google Scholar

[40]

A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 (2010), 731-737.  doi: 10.1016/j.patcog.2009.06.018.  Google Scholar

[41]

A. E. Xavier and A. A. F. de Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Global Optim., 31 (2005), 493-504.  doi: 10.1007/s10898-004-0737-8.  Google Scholar

[42]

A. E. Xavier and V. L. Xavier, Solving the minimum sum-of-squares clustering problem by hyperbolic smoothing and partition into boundary and gravitational regions, Patt. Recog., 44 (2011), 70-77.  doi: 10.1016/j.patcog.2010.07.004.  Google Scholar

[43]

V. L. Xavier and A. E. Xavier, Accelerated hyperbolic smoothing method for solving the multisource Fermat-Weber and k-Median problems, Knowl. Based Syst., 191 (2020), 105226.  doi: 10.1016/j.knosys.2019.105226.  Google Scholar

[44]

N. Yilmaz and A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Contol Optim., 10 (2020), 317-330.  doi: 10.3934/naco.2020004.  Google Scholar

[45]

H. Yin, An adaptive smoothing method for continuous minimax problems, Asia-Pac. J. Oper. Res., 32 (2015), 1540001, 19 pp. doi: 10.1142/S0217595915400011.  Google Scholar

[46]

L. Yuan, C. Fei, Z. Wan, W. Li and W. Wang, A nonmonotone smoothing Newton method for system of nonlinear inequalities based on a new smoothing function, Comput. Appl. Math., 38 (2019), Paper No. 91, 11 pp. doi: 10.1007/s40314-019-0856-y.  Google Scholar

[47]

I. Zang, A smoothing out technique for min-max optimization, Math. Programm., 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

show all references

References:
[1]

A. M. BagirovA. Al Nuaimat and N. Sultanova, Hyperbolic smoothing function method for minimax problems, Optimization, 62 (2013), 759-782.  doi: 10.1080/02331934.2012.675335.  Google Scholar

[2]

A. M. BagirovB. OrdinG. Ozturk and A. E. Xavier, An incremental clustering algorithm based on hyperbolic smoothing, Comput. Optim. Appl., 61 (2015), 219-241.  doi: 10.1007/s10589-014-9711-7.  Google Scholar

[3]

A. M. BagirovN. SultanovaA. Al Nuaimat and S. Taheri, Solving minimax problems: Local smoothing versus global smoothing, Numerical Analysis and Optimization, Springer Proceedings in Mathematics an Statistic, 235 (2018), 23-43.  doi: 10.1007/978-3-319-90026-1_2.  Google Scholar

[4]

D. P. Bertsekas, Nondifferentiable optimization via approximation, Math. Programming Stud., 3 (1975), 1-25.  doi: 10.1007/BFb0120696.  Google Scholar

[5]

W. Bian and X. Chen, Smoothing neural network for constrained non-Lipschitz optimization with applications, IEEE Trans. Neural Netw. Learn. Syst., 23 (2012), 399-411.  doi: 10.1109/TNNLS.2011.2181867.  Google Scholar

[6]

W. Bian and X. Chen, Linearly constrained non-Lipschitz optimization for image restoration, SIAM J. Imaging Sci., 8 (2015), 2294-2322.  doi: 10.1137/140985639.  Google Scholar

[7]

L. CaccettaB. Qu and G. Zhou, A globally and quadratically convergent method for absolute value equations, Comput. Optim. Appl., 48 (2011), 45-58.  doi: 10.1007/s10589-009-9242-9.  Google Scholar

[8]

X. Chen, Smoothing methods for nonsmooth, nonconvex minimization, Math. Prog. Ser. B., 134 (2012), 71-99.  doi: 10.1007/s10107-012-0569-0.  Google Scholar

[9]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.  doi: 10.1007/BF00249052.  Google Scholar

[10]

X. ChenM. K. Ng and C. Zhang, Non-Lipschitz $\ell_{p}$-regularization and box constrained model for image restoration, IEEE Trans. Image Process., 21 (2012), 4709-4721.  doi: 10.1109/TIP.2012.2214051.  Google Scholar

[11]

X. Chen and W. Zhou, Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization, SIAM J. Imaging Sciences, 3 (2010), 765-790.  doi: 10.1137/080740167.  Google Scholar

[12]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. doi: 10.1137/1.9781611971309.  Google Scholar

[13]

C. Grossmann, Smoothing techniques for exact penalty methods, Contemporary Mathematics, In book Panaroma of Mathematics: Pure and Applied, 658 (2016), 249-265.   Google Scholar

[14]

Y. HuangH. Liu and W. Cong, A note on the smoothing quadratic regularization method for non-Lipschitz optimization, Numer. Algor., 69 (2015), 863-874.  doi: 10.1007/s11075-014-9929-6.  Google Scholar

[15]

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, J. Ind. Manag. Optim., 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[16]

M. Kang and M. Jung, Simultaneous image enhancement and restoration with non-convex total variation, J. Sci. Comput., 87 (2021), Paper No. 83, 46 pp. doi: 10.1007/s10915-021-01488-x.  Google Scholar

[17]

K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1985. doi: 10.1007/BFb0074500.  Google Scholar

[18]

G. Landi, A modified Newton projection method for $\ell_1$-regularized least squares image deblurring, J. Math. Imaging Vis., 51 (2015), 195-208.  doi: 10.1007/s10851-014-0514-3.  Google Scholar

[19]

S.-J. Lian, Smoothing approximation to $l_1$ exact penalty function for inequality constrained optimization, Appl. Math. Comput., 219 (2012), 3113-3121.  doi: 10.1016/j.amc.2012.09.042.  Google Scholar

[20]

M. M. Mäkelä and P. Neitaanmäki, Nonsmooth Optimization, World Scientific, Singapore, 1992. doi: 10.1142/1493.  Google Scholar

[21]

N. Mau NamL. T. H. AnD. Giles and N. Thai An, Smoothing techniques and difference of convex functions algorithms for image reconstructions, Optimization, 69 (2020), 1601-1633.  doi: 10.1080/02331934.2019.1648467.  Google Scholar

[22]

Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program., 103 (2005), 127-152.  doi: 10.1007/s10107-004-0552-5.  Google Scholar

[23]

M. Nikolova, Minimizers of cost functions involving nonsmooth data-fidelity terms. Application to the processing outliers, SIAM J. Numer. Anal., 40 (2002), 965-994.  doi: 10.1137/S0036142901389165.  Google Scholar

[24]

M. NikolovaM. K. NgS. Zheng and W. K. Ching, Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization, SIAM J. Imaging Sci., 1 (2008), 2-25.  doi: 10.1137/070692285.  Google Scholar

[25]

O. N. OnakY. Serinagaoglu-Dogrusoz and G.-W. Weber, Effects of a priori parameter selection in minimum relative entropy method on inverse electrocardiography problem, Inverse Probl. Sci. Eng., 26 (2018), 877-897.  doi: 10.1080/17415977.2017.1369979.  Google Scholar

[26]

C. T. PhamG. GamardA. Kopylov and T. T. T. Tran, An algorithm for image restoration with mixed noise using total variation regularization, Turk. J. Elec. Eng. Comp. Sci., 26 (2018), 2831-2845.  doi: 10.3906/elk-1803-100.  Google Scholar

[27]

C. T. PhamT. T. T. Tran and G. Gamard, An efficient total variation minimization method for image restoration, Informatica, 31 (2020), 539-560.  doi: 10.15388/20-INFOR407.  Google Scholar

[28]

R. A. Polyak, Smooth optimization methods for minimax problems, SIAM J. Control Optim., 26 (1988), 1274-1286.  doi: 10.1137/0326071.  Google Scholar

[29]

L. Qi and D. Sun, Smoothing functions and smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl., 113 (2002), 121-147.  doi: 10.1023/A:1014861331301.  Google Scholar

[30]

R. T. Rockefellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[31]

L. I. RudinS. 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

[32]

B. SaheyaC.-H. Yu and J.-S. Chen, Numerical comparisons based on four smoothing functions for absolute value equations, J. Appl. Math. Comput., 56 (2018), 131-149.  doi: 10.1007/s12190-016-1065-0.  Google Scholar

[33]

A. SahinerG. Kapusuz and N. Yilmaz, A new smoothing approach to exact penalty functions for inequality constrained optimization problems, Numer. Algebra Contol Optim., 6 (2016), 161-173.  doi: 10.3934/naco.2016006.  Google Scholar

[34]

M. SouzaA. E. XavierC. Lavor and N. Maculan, Hyperbolic smoothing and penalty techniques applied to molecular structure determination, Oper. Res. Lett., 39 (2011), 461-465.  doi: 10.1016/j.orl.2011.07.007.  Google Scholar

[35]

P. TaylanG.-W. Weber and F. Yerlikaya-Ozkurt, A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization, TOP, 18 (2010), 377-395.  doi: 10.1007/s11750-010-0155-7.  Google Scholar

[36]

A. H. Tor, Hyperbolic smoothing method for sum-max problems, Neural, Parallel Sci. Comput., 24 (2016), 381-391.   Google Scholar

[37]

S. Voronin, G. Ozkaya and D. Yoshida, Convolution based smooth approximations to the absolute value function with application to non-smooth regularization, Preprint, (2015). arXiv: 1408.6795. Google Scholar

[38]

C. Wu, J. Zhan, Y. Lu and J.-S. Chen, Signal reconstruction by conjugate gradient algorithm based on smoothing $l_1$-norm, Calcolo, 56 (2019), Paper No. 42, 26 pp. doi: 10.1007/s10092-019-0340-5.  Google Scholar

[39]

A. E. Xavier, Penalizacao Hiperbolica, I Congresso Latino-Americano de Pesquisa Operacional e Engenharia de Sistemas, 8 a 11 de Novembro, Rio de Janeiro, Brasil, 1982. Google Scholar

[40]

A. E. Xavier, The hyperbolic smoothing clustering method, Patt. Recog., 43 (2010), 731-737.  doi: 10.1016/j.patcog.2009.06.018.  Google Scholar

[41]

A. E. Xavier and A. A. F. de Oliveira, Optimal covering of plane domains by circles via hyperbolic smoothing, J. Global Optim., 31 (2005), 493-504.  doi: 10.1007/s10898-004-0737-8.  Google Scholar

[42]

A. E. Xavier and V. L. Xavier, Solving the minimum sum-of-squares clustering problem by hyperbolic smoothing and partition into boundary and gravitational regions, Patt. Recog., 44 (2011), 70-77.  doi: 10.1016/j.patcog.2010.07.004.  Google Scholar

[43]

V. L. Xavier and A. E. Xavier, Accelerated hyperbolic smoothing method for solving the multisource Fermat-Weber and k-Median problems, Knowl. Based Syst., 191 (2020), 105226.  doi: 10.1016/j.knosys.2019.105226.  Google Scholar

[44]

N. Yilmaz and A. Sahiner, On a new smoothing technique for non-smooth, non-convex optimization, Numer. Algebra Contol Optim., 10 (2020), 317-330.  doi: 10.3934/naco.2020004.  Google Scholar

[45]

H. Yin, An adaptive smoothing method for continuous minimax problems, Asia-Pac. J. Oper. Res., 32 (2015), 1540001, 19 pp. doi: 10.1142/S0217595915400011.  Google Scholar

[46]

L. Yuan, C. Fei, Z. Wan, W. Li and W. Wang, A nonmonotone smoothing Newton method for system of nonlinear inequalities based on a new smoothing function, Comput. Appl. Math., 38 (2019), Paper No. 91, 11 pp. doi: 10.1007/s40314-019-0856-y.  Google Scholar

[47]

I. Zang, A smoothing out technique for min-max optimization, Math. Programm., 19 (1980), 61-77.  doi: 10.1007/BF01581628.  Google Scholar

Figure 1.  The graphs of smoothing functions $ \phi_1(x, \varepsilon) $ and $ \phi_2(x, \varepsilon) $ with different $ \varepsilon $ values
Figure 2.  The graphs of smoothing functions $ \phi_1(x, \varepsilon) $ and $ \phi_2(x, \varepsilon) $ with different $ q $ values
Figure 3.  The graphs of smoothing function $ \phi_3(t, \varepsilon) $ with different $ \varepsilon $ and $ r $ values
Figure 4.  The graphs of smoothing functions $ \phi_1(x, \varepsilon) $, $ \phi_2(x, \varepsilon) $ and $ \phi_3(x, \varepsilon) $
Figure 5.  The original version of images (a) Barbara, (b) Cameraman, (c) House and (d) Peppers
Figure 6.  The noisy image and denoised images
Figure 7.  The noisy image and denoised images
Figure 8.  The noisy image and denoised images
Figure 9.  The noisy image and denoised images
Table 1.  The computational results
Noisy Image Denoised by $ \Phi_1 $ Denoised by $ \Phi_2 $ Denoised by $ \Phi_3 $ Denoised by TV
Image Name PSNR Iter PSNR Time Iter PSNR Time Iter PSNR Time Iter PSNR Time
Barbara $ 20.156 $ $ 98 $ $ 23.272 $ $ 10.545 $ $ 116 $ $ 23.378 $ $ 13.194 $ $ 112 $ $ \textbf{24.684} $ $ 13.757 $ $ 124 $ $ 24.661 $ $ 14.278 $
$ 18.161 $ $ 96 $ $ 23.155 $ $ 11.121 $ $ 120 $ $ 23.152 $ $ 13.440 $ $ 89 $ $ \textbf{23.340} $ $ 9.0128 $ $ 99 $ $ 23.147 $ $ 11.554 $
Cameraman $ 20.217 $ $ 94 $ $ 26.597 $ $ 8.8602 $ $ 111 $ $ 26.510 $ $ 12.142 $ $ 88 $ $ \textbf{26.643} $ $ 12.9414 $ $ 129 $ $ 26.497 $ $ 15.26 $
$ 18.149 $ $ 96 $ $ 25.126 $ $ 12.283 $ $ 119 $ $ 24.976 $ $ 13.705 $ $ 93 $ $ \textbf{25.201} $ $ 11.0368 $ $ 102 $ $ 24.943 $ $ 10.336 $
House $ 20.143 $ $ 80 $ $ 25.877 $ $ 1.4441 $ $ 94 $ $ 26.332 $ $ 1.8774 $ $ 45 $ $ \textbf{26.806} $ $ 0.8385 $ $ 111 $ $ 26.186 $ $ 2.2779 $
$ 18.299 $ $ 82 $ $ 24.775 $ $ 1.3958 $ $ 100 $ $ 24.701 $ $ 1.9154 $ $ 52 $ $ \textbf{25.590} $ $ 0.7164 $ $ 88 $ $ 25.368 $ $ 1.7774 $
Peppers $ 20.075 $ $ 98 $ $ 26.591 $ $ 8.9505 $ $ 117 $ $ 26.430 $ $ 11.302 $ $ 86 $ $ \textbf{27.496} $ $ 10.9760 $ $ 140 $ $ 27.111 $ $ 14.11 $
$ 18.307 $ $ 99 $ $ 25.099 $ $ 10.472 $ $ 123 $ $ 25.013 $ $ 12.721 $ $ 111 $ $ \textbf{26.980} $ $ 11.7570 $ $ 108 $ $ 25.687 $ $ 11.339 $
Noisy Image Denoised by $ \Phi_1 $ Denoised by $ \Phi_2 $ Denoised by $ \Phi_3 $ Denoised by TV
Image Name PSNR Iter PSNR Time Iter PSNR Time Iter PSNR Time Iter PSNR Time
Barbara $ 20.156 $ $ 98 $ $ 23.272 $ $ 10.545 $ $ 116 $ $ 23.378 $ $ 13.194 $ $ 112 $ $ \textbf{24.684} $ $ 13.757 $ $ 124 $ $ 24.661 $ $ 14.278 $
$ 18.161 $ $ 96 $ $ 23.155 $ $ 11.121 $ $ 120 $ $ 23.152 $ $ 13.440 $ $ 89 $ $ \textbf{23.340} $ $ 9.0128 $ $ 99 $ $ 23.147 $ $ 11.554 $
Cameraman $ 20.217 $ $ 94 $ $ 26.597 $ $ 8.8602 $ $ 111 $ $ 26.510 $ $ 12.142 $ $ 88 $ $ \textbf{26.643} $ $ 12.9414 $ $ 129 $ $ 26.497 $ $ 15.26 $
$ 18.149 $ $ 96 $ $ 25.126 $ $ 12.283 $ $ 119 $ $ 24.976 $ $ 13.705 $ $ 93 $ $ \textbf{25.201} $ $ 11.0368 $ $ 102 $ $ 24.943 $ $ 10.336 $
House $ 20.143 $ $ 80 $ $ 25.877 $ $ 1.4441 $ $ 94 $ $ 26.332 $ $ 1.8774 $ $ 45 $ $ \textbf{26.806} $ $ 0.8385 $ $ 111 $ $ 26.186 $ $ 2.2779 $
$ 18.299 $ $ 82 $ $ 24.775 $ $ 1.3958 $ $ 100 $ $ 24.701 $ $ 1.9154 $ $ 52 $ $ \textbf{25.590} $ $ 0.7164 $ $ 88 $ $ 25.368 $ $ 1.7774 $
Peppers $ 20.075 $ $ 98 $ $ 26.591 $ $ 8.9505 $ $ 117 $ $ 26.430 $ $ 11.302 $ $ 86 $ $ \textbf{27.496} $ $ 10.9760 $ $ 140 $ $ 27.111 $ $ 14.11 $
$ 18.307 $ $ 99 $ $ 25.099 $ $ 10.472 $ $ 123 $ $ 25.013 $ $ 12.721 $ $ 111 $ $ \textbf{26.980} $ $ 11.7570 $ $ 108 $ $ 25.687 $ $ 11.339 $
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