March  2021, 11(1): 27-43. doi: 10.3934/naco.2020013

On the GSOR iteration method for image restoration

1. 

Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

2. 

Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

* Corresponding author: Davod Khojasteh Salkuyeh

Received  June 2019 Revised  August 2019 Published  February 2020

In this study, we present a generalization of the successive overrelaxation (GSOR) iteration method to find the solution of the image restoration problem. Moreover, an improved version of the GSOR (IGSOR) method is also given to solve the proposed problem. Convergence of the GSOR and IGSOR methods are investigated. Three numerical examples are given to illustrate the effectiveness and accuracy of the methods.

Citation: Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
References:
[1] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511624100.  Google Scholar
[2]

L. R. BerrielJ. Bescos and A. Santistebao, Image restoration for a defocused optical system, Appl. Opt., 22 (1983), 2772-2780.   Google Scholar

[3]

A. Bouhamidi and K. Jbilou, Sylvester Tikhonov-regularization methods in image restoration, J. Comput. Appl. Math., 206 (2007), 86-98.  doi: 10.1016/j.cam.2006.05.028.  Google Scholar

[4]

S. Serra-Capizzano, A note on antireflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25 (2003), 1307-1325.  doi: 10.1137/S1064827502410244.  Google Scholar

[5]

T. ChanA. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar

[6]

L.-J. DengT.-Z. Huang and X.-L. Zhao, Wavelet-based two-level methods for image restoration, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 5079-5087.  doi: 10.1016/j.cnsns.2012.04.001.  Google Scholar

[7]

B. Fisher, Digital restoration of snow white: 120,000 famous frames are back, , Advanced Imaging, (1993), 32–36. Google Scholar

[8]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.2307/1268518.  Google Scholar

[9]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2$^nd$ edition, Prentice Hall, New Jersey, 2002. Google Scholar

[10]

Y.-S. Han, D. M. Herrington and W. E. Snyder, Quantitative angiography using mean field annealing, Proc. of Computers in Cardiology, (1992), 119–122. Google Scholar

[11]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[12]

P. C. Hansen, J. G. Nagy and D. P. O'leary, Deblurring Images: Matrices Spectra and Filtering, SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718874.  Google Scholar

[13]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[14]

G. Landi, A fast truncated Lagrange method for large-scale image restoration problems, Appl. Math. Comput., 186 (2007), 1075-1082.  doi: 10.1016/j.amc.2006.08.039.  Google Scholar

[15]

X.-G. LvT.-Z. HuangZ.-B. Xu and X.-L. Zhao, A special Hermitian and skew-Hermitian splitting method for image restoration, Appl. Math. Model., 37 (2013), 1069-1082.  doi: 10.1016/j.apm.2012.03.019.  Google Scholar

[16]

V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), 414-417.   Google Scholar

[17]

J. G. NagyM. K. Ng and L. Perrone, Kronecker product approximations for image restoration with reflexive boundary conditions, SIAM J. Matrix Anal. Appl., 25 (2003), 829-841.  doi: 10.1137/S0895479802419580.  Google Scholar

[18]

M. K. NgR. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866.  doi: 10.1137/S1064827598341384.  Google Scholar

[19]

L. Perrone, Kronecker product approximations for image restoration with anti-reflective boundary conditions, Numer. Linear Algebra Appl., 13 (2006), 1-22.  doi: 10.1002/nla.458.  Google Scholar

[20]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2$^{nd}$ edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718003.  Google Scholar

[21]

D. K. SalkuyehD. Hezari and V. Edalatpour, Generalized SOR iterative method for a class of complex symmetric linear system of equations, Int. J. Comput. Math., 92 (2015), 802-815.  doi: 10.1080/00207160.2014.912753.  Google Scholar

[22]

J. L. Starck and F. Murtagh, Astronomical Image and Data Analysis, 2$^nd$ edition, Springer, Berlin, 2006. Google Scholar

[23]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

show all references

References:
[1] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1996.  doi: 10.1017/CBO9780511624100.  Google Scholar
[2]

L. R. BerrielJ. Bescos and A. Santistebao, Image restoration for a defocused optical system, Appl. Opt., 22 (1983), 2772-2780.   Google Scholar

[3]

A. Bouhamidi and K. Jbilou, Sylvester Tikhonov-regularization methods in image restoration, J. Comput. Appl. Math., 206 (2007), 86-98.  doi: 10.1016/j.cam.2006.05.028.  Google Scholar

[4]

S. Serra-Capizzano, A note on antireflective boundary conditions and fast deblurring models, SIAM J. Sci. Comput., 25 (2003), 1307-1325.  doi: 10.1137/S1064827502410244.  Google Scholar

[5]

T. ChanA. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar

[6]

L.-J. DengT.-Z. Huang and X.-L. Zhao, Wavelet-based two-level methods for image restoration, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 5079-5087.  doi: 10.1016/j.cnsns.2012.04.001.  Google Scholar

[7]

B. Fisher, Digital restoration of snow white: 120,000 famous frames are back, , Advanced Imaging, (1993), 32–36. Google Scholar

[8]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.2307/1268518.  Google Scholar

[9]

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2$^nd$ edition, Prentice Hall, New Jersey, 2002. Google Scholar

[10]

Y.-S. Han, D. M. Herrington and W. E. Snyder, Quantitative angiography using mean field annealing, Proc. of Computers in Cardiology, (1992), 119–122. Google Scholar

[11]

P. C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev., 34 (1992), 561-580.  doi: 10.1137/1034115.  Google Scholar

[12]

P. C. Hansen, J. G. Nagy and D. P. O'leary, Deblurring Images: Matrices Spectra and Filtering, SIAM, Philadelphia, 2006. doi: 10.1137/1.9780898718874.  Google Scholar

[13]

A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1989. Google Scholar

[14]

G. Landi, A fast truncated Lagrange method for large-scale image restoration problems, Appl. Math. Comput., 186 (2007), 1075-1082.  doi: 10.1016/j.amc.2006.08.039.  Google Scholar

[15]

X.-G. LvT.-Z. HuangZ.-B. Xu and X.-L. Zhao, A special Hermitian and skew-Hermitian splitting method for image restoration, Appl. Math. Model., 37 (2013), 1069-1082.  doi: 10.1016/j.apm.2012.03.019.  Google Scholar

[16]

V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), 414-417.   Google Scholar

[17]

J. G. NagyM. K. Ng and L. Perrone, Kronecker product approximations for image restoration with reflexive boundary conditions, SIAM J. Matrix Anal. Appl., 25 (2003), 829-841.  doi: 10.1137/S0895479802419580.  Google Scholar

[18]

M. K. NgR. H. Chan and W.-C. Tang, A fast algorithm for deblurring models with Neumann boundary conditions, SIAM J. Sci. Comput., 21 (1999), 851-866.  doi: 10.1137/S1064827598341384.  Google Scholar

[19]

L. Perrone, Kronecker product approximations for image restoration with anti-reflective boundary conditions, Numer. Linear Algebra Appl., 13 (2006), 1-22.  doi: 10.1002/nla.458.  Google Scholar

[20]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2$^{nd}$ edition, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898718003.  Google Scholar

[21]

D. K. SalkuyehD. Hezari and V. Edalatpour, Generalized SOR iterative method for a class of complex symmetric linear system of equations, Int. J. Comput. Math., 92 (2015), 802-815.  doi: 10.1080/00207160.2014.912753.  Google Scholar

[22]

J. L. Starck and F. Murtagh, Astronomical Image and Data Analysis, 2$^nd$ edition, Springer, Berlin, 2006. Google Scholar

[23]

C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002. doi: 10.1137/1.9780898717570.  Google Scholar

Figure 1.  True image, PSF and degraded image in Example 1
Figure 2.  Restored images with GSOR method for various BCs in Example 1
Figure 3.  Restored images with IGSOR method for various BCs in Example 1
Figure 4.  True image and degraded image in Example 2
Figure 5.  Restored images with GSOR method for various BCs in Example 2
Figure 6.  Restored images with IGSOR method for various BCs in Example 2
Figure 7.  True image, PSF and degraded image in Example 3
Figure 8.  Restored images with GSOR method for various BCs in Example 3
Figure 9.  Restored images with IGSOR method for various BCs in Example 3
Table 1.  Values of $ (\alpha,\omega) $ in Example 1
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3283,-) $ $ (0.3333,-) $ $ (0.3290,-) $ $ (0.4650,-) $
GSOR $ (-,0.22) $ $ (-,0.20) $ $ (-,0.14) $ $ (-,0.19) $
IGSOR $ (0.27,0.36) $ $ (0.31,0.22) $ $ (0.02,0.34) $ $ (0.01,0.28) $
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3283,-) $ $ (0.3333,-) $ $ (0.3290,-) $ $ (0.4650,-) $
GSOR $ (-,0.22) $ $ (-,0.20) $ $ (-,0.14) $ $ (-,0.19) $
IGSOR $ (0.27,0.36) $ $ (0.31,0.22) $ $ (0.02,0.34) $ $ (0.01,0.28) $
Table 2.  PSNR values of various methods in Example 1
Method Zero Periodic Reflexive Antireflective
SHSS $ 21.08 $ $ 22.09 $ $ 23.98 $ $ 24.17 $
GSOR $ 21.12 $ $ 22.18 $ $ 24.13 $ $ 24.40 $
IGSOR $ 21.23 $ $ 22.25 $ $ 24.23 $ $ 24.46 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 21.08 $ $ 22.09 $ $ 23.98 $ $ 24.17 $
GSOR $ 21.12 $ $ 22.18 $ $ 24.13 $ $ 24.40 $
IGSOR $ 21.23 $ $ 22.25 $ $ 24.23 $ $ 24.46 $
Table 3.  Relative error of various methods in Example 1
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.1796 $ $ 0.1592 $ $ 0.1287 $ $ 0.1259 $
GSOR $ 0.1790 $ $ 0.1582 $ $ 0.1265 $ $ 0.1224 $
IGSOR $ 0.1767 $ $ 0.1571 $ $ 0.1252 $ $ 0.1218 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.1796 $ $ 0.1592 $ $ 0.1287 $ $ 0.1259 $
GSOR $ 0.1790 $ $ 0.1582 $ $ 0.1265 $ $ 0.1224 $
IGSOR $ 0.1767 $ $ 0.1571 $ $ 0.1252 $ $ 0.1218 $
Table 4.  CPU times of various methods in Example 1
Method Zero Periodic Reflexive Antireflective
SHSS $ 5.26 $ $ 5.05 $ $ 5.74 $ $ 5.31 $
GSOR $ 0.49 $ $ 0.52 $ $ 0.49 $ $ 0.51 $
IGSOR $ 0.48 $ $ 0.52 $ $ 0.49 $ $ 0.51 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 5.26 $ $ 5.05 $ $ 5.74 $ $ 5.31 $
GSOR $ 0.49 $ $ 0.52 $ $ 0.49 $ $ 0.51 $
IGSOR $ 0.48 $ $ 0.52 $ $ 0.49 $ $ 0.51 $
Table 5.  Values of $ (\alpha,\omega) $ in Example 2
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3277,-) $ $ (0.3333,-) $ $ (0.3339,-) $ $ (0.6039,-) $
GSOR $ (-,0.32) $ $ (-,0.30) $ $ (-,0.17) $ $ (-,0.18) $
IGSOR $ (0.01,0.09) $ $ (0.001,0.15) $ $ (0.005,0.22) $ $ (0.01,0.25) $
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3277,-) $ $ (0.3333,-) $ $ (0.3339,-) $ $ (0.6039,-) $
GSOR $ (-,0.32) $ $ (-,0.30) $ $ (-,0.17) $ $ (-,0.18) $
IGSOR $ (0.01,0.09) $ $ (0.001,0.15) $ $ (0.005,0.22) $ $ (0.01,0.25) $
Table 6.  PSNR values of various methods in Example 2
Method Zero Periodic Reflexive Antireflective
SHSS $ 27.50 $ $ 27.71 $ $ 28.81 $ $ 28.39 $
GSOR $ 27.53 $ $ 27.76 $ $ 29.41 $ $ 29.14 $
IGSOR $ 27.61 $ $ 27.83 $ $ 29.43 $ $ 29.16 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 27.50 $ $ 27.71 $ $ 28.81 $ $ 28.39 $
GSOR $ 27.53 $ $ 27.76 $ $ 29.41 $ $ 29.14 $
IGSOR $ 27.61 $ $ 27.83 $ $ 29.43 $ $ 29.16 $
Table 7.  Relative error of various methods in Example 2
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.2771 $ $ 0.2704 $ $ 0.2384 $ $ 0.2502 $
GSOR $ 0.2764 $ $ 0.2690 $ $ 0.2224 $ $ 0.2296 $
IGSOR $ 0.2740 $ $ 0.2671 $ $ 0.2222 $ $ 0.2292 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.2771 $ $ 0.2704 $ $ 0.2384 $ $ 0.2502 $
GSOR $ 0.2764 $ $ 0.2690 $ $ 0.2224 $ $ 0.2296 $
IGSOR $ 0.2740 $ $ 0.2671 $ $ 0.2222 $ $ 0.2292 $
Table 8.  CPU times of various methods in Example 2
Method Zero Periodic Reflexive Antireflective
SHSS $ 4.98 $ $ 5.08 $ $ 5.21 $ $ 5.40 $
GSOR $ 0.49 $ $ 0.52 $ $ 0.51 $ $ 0.52 $
IGSOR $ 0.49 $ $ 0.51 $ $ 0.53 $ $ 0.50 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 4.98 $ $ 5.08 $ $ 5.21 $ $ 5.40 $
GSOR $ 0.49 $ $ 0.52 $ $ 0.51 $ $ 0.52 $
IGSOR $ 0.49 $ $ 0.51 $ $ 0.53 $ $ 0.50 $
Table 9.  Values of $ (\alpha,\omega) $ in Example 3
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3330,-) $ $ (0.3333,-) $ $ (0.3333,-) $ $ (0.5894,-) $
GSOR $ (-,0.26) $ $ (-,0.25) $ $ (-,0.25) $ $ (-,0.29) $
IGSOR $ (0.003,0.19) $ $ (0.008,0.13) $ $ (0.006,0.13) $ $ (0.05,0.09) $
Method Zero Periodic Reflexive Antireflective
SHSS $ (0.3330,-) $ $ (0.3333,-) $ $ (0.3333,-) $ $ (0.5894,-) $
GSOR $ (-,0.26) $ $ (-,0.25) $ $ (-,0.25) $ $ (-,0.29) $
IGSOR $ (0.003,0.19) $ $ (0.008,0.13) $ $ (0.006,0.13) $ $ (0.05,0.09) $
Table 10.  PSNR values of various methods in Example 3
Method Zero Periodic Reflexive Antireflective
SHSS $ 22.68 $ $ 27.65 $ $ 26.22 $ $ 26.69 $
GSOR $ 22.72 $ $ 27.90 $ $ 26.54 $ $ 26.80 $
IGSOR $ 22.75 $ $ 28.43 $ $ 26.87 $ $ 27.93 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 22.68 $ $ 27.65 $ $ 26.22 $ $ 26.69 $
GSOR $ 22.72 $ $ 27.90 $ $ 26.54 $ $ 26.80 $
IGSOR $ 22.75 $ $ 28.43 $ $ 26.87 $ $ 27.93 $
Table 11.  Relative error of various methods in Example 3
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.1287 $ $ 0.0726 $ $ 0.0857 $ $ 0.0811 $
GSOR $ 0.1281 $ $ 0.0706 $ $ 0.0825 $ $ 0.0801 $
IGSOR $ 0.1276 $ $ 0.0664 $ $ 0.0795 $ $ 0.0704 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 0.1287 $ $ 0.0726 $ $ 0.0857 $ $ 0.0811 $
GSOR $ 0.1281 $ $ 0.0706 $ $ 0.0825 $ $ 0.0801 $
IGSOR $ 0.1276 $ $ 0.0664 $ $ 0.0795 $ $ 0.0704 $
Table 12.  CPU times of various methods in Example 3
Method Zero Periodic Reflexive Antireflective
SHSS $ 24.62 $ $ 22.81 $ $ 22.95 $ $ 24.65 $
GSOR $ 2.27 $ $ 2.30 $ $ 2.41 $ $ 2.35 $
IGSOR $ 2.26 $ $ 2.28 $ $ 2.29 $ $ 2.33 $
Method Zero Periodic Reflexive Antireflective
SHSS $ 24.62 $ $ 22.81 $ $ 22.95 $ $ 24.65 $
GSOR $ 2.27 $ $ 2.30 $ $ 2.41 $ $ 2.35 $
IGSOR $ 2.26 $ $ 2.28 $ $ 2.29 $ $ 2.33 $
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