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# On the GSOR iteration method for image restoration

• * Corresponding author: Davod Khojasteh Salkuyeh
• 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.

Mathematics Subject Classification: Primary: 65F10; Secondary: 65D18.

 Citation: • • 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 $(0.27,0.36)$ $(0.31,0.22)$ $(0.02,0.34)$ $(0.01,0.28)$

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 $21.23$ $22.25$ $24.23$ $24.46$

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 $0.1767$ $0.1571$ $0.1252$ $0.1218$

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 $0.48$ $0.52$ $0.49$ $0.51$

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 $(0.01,0.09)$ $(0.001,0.15)$ $(0.005,0.22)$ $(0.01,0.25)$

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 $27.61$ $27.83$ $29.43$ $29.16$

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 $0.2740$ $0.2671$ $0.2222$ $0.2292$

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 $0.49$ $0.51$ $0.53$ $0.50$

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 $(0.003,0.19)$ $(0.008,0.13)$ $(0.006,0.13)$ $(0.05,0.09)$

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 $22.75$ $28.43$ $26.87$ $27.93$

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 $0.1276$ $0.0664$ $0.0795$ $0.0704$

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 $2.26$ $2.28$ $2.29$ $2.33$
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