Abstract

We report an optimization approach to restore degraded binary images by using positive semidefinite programming when the point spread function (PSF) is known. The approach takes advantage of the combinatorial nature of the problem, considering not only local similarity and spatial context but also the relationship between individual pixel values and the PSF. Numerical experiments confirm the superiority of the approach.

© 2006 Optical Society of America

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References

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  1. D. Hitchcock and C. A. Glasbey, Biometrics 53, 1010 (1997).
    [CrossRef]
  2. M. Neifeld, R. Xuan, and M. Marcellin, Appl. Opt. 39, 269 (2000).
    [CrossRef]
  3. J. Meloche and R. H. Zamar, Can. J. Stat. 22, 335 (1994).
    [CrossRef]
  4. X. D. Gu, H. M. Wang, and D. H. Yu, in 8th International Conference on Neural Information Processing (Fudan University Press, 2001), pp. 922-927.
  5. T. F. Chan, S. Esedoglu, and M. Nikolova, in IEEE International Conference on Image Processing (IEEE, 2005), pp. 121-124.
  6. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).
  7. J. Myrheim and H. Rue, Comput. Vis. Graph. Image Process. 54, 223 (1992).
  8. M. R. Banham and A. K. Katsaggelos, IEEE Signal Process. Mag. 14(2), 24 (1997).
    [CrossRef]
  9. J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
    [CrossRef]
  10. Y. Shen, E. Y. Lam, and N. Wong, 2006 IAENG International Workshop on Imaging Engineering (IWIE'06) (International Association of Engineers, 2006), pp. 537-542.
  11. J. F. Sturm, Optim. Methods Software 11, 625 (1999).
    [CrossRef]
  12. M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
    [CrossRef]

2003

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

2000

1999

J. F. Sturm, Optim. Methods Software 11, 625 (1999).
[CrossRef]

1997

M. R. Banham and A. K. Katsaggelos, IEEE Signal Process. Mag. 14(2), 24 (1997).
[CrossRef]

D. Hitchcock and C. A. Glasbey, Biometrics 53, 1010 (1997).
[CrossRef]

1995

M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
[CrossRef]

1994

J. Meloche and R. H. Zamar, Can. J. Stat. 22, 335 (1994).
[CrossRef]

1992

J. Myrheim and H. Rue, Comput. Vis. Graph. Image Process. 54, 223 (1992).

Banham, M. R.

M. R. Banham and A. K. Katsaggelos, IEEE Signal Process. Mag. 14(2), 24 (1997).
[CrossRef]

Chan, T. F.

T. F. Chan, S. Esedoglu, and M. Nikolova, in IEEE International Conference on Image Processing (IEEE, 2005), pp. 121-124.

Cremers, D.

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

Esedoglu, S.

T. F. Chan, S. Esedoglu, and M. Nikolova, in IEEE International Conference on Image Processing (IEEE, 2005), pp. 121-124.

Glasbey, C. A.

D. Hitchcock and C. A. Glasbey, Biometrics 53, 1010 (1997).
[CrossRef]

Goemans, M. X.

M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

Gu, X. D.

X. D. Gu, H. M. Wang, and D. H. Yu, in 8th International Conference on Neural Information Processing (Fudan University Press, 2001), pp. 922-927.

Hitchcock, D.

D. Hitchcock and C. A. Glasbey, Biometrics 53, 1010 (1997).
[CrossRef]

Katsaggelos, A. K.

M. R. Banham and A. K. Katsaggelos, IEEE Signal Process. Mag. 14(2), 24 (1997).
[CrossRef]

Keuchel, J.

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

Lam, E. Y.

Y. Shen, E. Y. Lam, and N. Wong, 2006 IAENG International Workshop on Imaging Engineering (IWIE'06) (International Association of Engineers, 2006), pp. 537-542.

Marcellin, M.

Meloche, J.

J. Meloche and R. H. Zamar, Can. J. Stat. 22, 335 (1994).
[CrossRef]

Myrheim, J.

J. Myrheim and H. Rue, Comput. Vis. Graph. Image Process. 54, 223 (1992).

Neifeld, M.

Nikolova, M.

T. F. Chan, S. Esedoglu, and M. Nikolova, in IEEE International Conference on Image Processing (IEEE, 2005), pp. 121-124.

Rue, H.

J. Myrheim and H. Rue, Comput. Vis. Graph. Image Process. 54, 223 (1992).

Schellewald, C.

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

Schnörr, C.

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

Shen, Y.

Y. Shen, E. Y. Lam, and N. Wong, 2006 IAENG International Workshop on Imaging Engineering (IWIE'06) (International Association of Engineers, 2006), pp. 537-542.

Sturm, J. F.

J. F. Sturm, Optim. Methods Software 11, 625 (1999).
[CrossRef]

Wang, H. M.

X. D. Gu, H. M. Wang, and D. H. Yu, in 8th International Conference on Neural Information Processing (Fudan University Press, 2001), pp. 922-927.

Williamson, D. P.

M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
[CrossRef]

Wong, N.

Y. Shen, E. Y. Lam, and N. Wong, 2006 IAENG International Workshop on Imaging Engineering (IWIE'06) (International Association of Engineers, 2006), pp. 537-542.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

Xuan, R.

Yu, D. H.

X. D. Gu, H. M. Wang, and D. H. Yu, in 8th International Conference on Neural Information Processing (Fudan University Press, 2001), pp. 922-927.

Zamar, R. H.

J. Meloche and R. H. Zamar, Can. J. Stat. 22, 335 (1994).
[CrossRef]

Appl. Opt.

Biometrics

D. Hitchcock and C. A. Glasbey, Biometrics 53, 1010 (1997).
[CrossRef]

Can. J. Stat.

J. Meloche and R. H. Zamar, Can. J. Stat. 22, 335 (1994).
[CrossRef]

Comput. Vis. Graph. Image Process.

J. Myrheim and H. Rue, Comput. Vis. Graph. Image Process. 54, 223 (1992).

IEEE Signal Process. Mag.

M. R. Banham and A. K. Katsaggelos, IEEE Signal Process. Mag. 14(2), 24 (1997).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

J. Keuchel, C. Schellewald, D. Cremers, and C. Schnörr, IEEE Trans. Pattern Anal. Mach. Intell. 25, 1364 (2003).
[CrossRef]

J. ACM

M. X. Goemans and D. P. Williamson, J. ACM 42, 1115 (1995).
[CrossRef]

Optim. Methods Software

J. F. Sturm, Optim. Methods Software 11, 625 (1999).
[CrossRef]

Other

Y. Shen, E. Y. Lam, and N. Wong, 2006 IAENG International Workshop on Imaging Engineering (IWIE'06) (International Association of Engineers, 2006), pp. 537-542.

X. D. Gu, H. M. Wang, and D. H. Yu, in 8th International Conference on Neural Information Processing (Fudan University Press, 2001), pp. 922-927.

T. F. Chan, S. Esedoglu, and M. Nikolova, in IEEE International Conference on Image Processing (IEEE, 2005), pp. 121-124.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Prentice-Hall, 2002).

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Figures (4)

Fig. 1
Fig. 1

(a) True image. Image (b) blurred by h 3 , (c) blurred by h 3 and with noise, (d) blurred by h 5 , (e) blurred by h 5 and with noise.

Fig. 2
Fig. 2

Results of restoration (a)–(d) of Fig. 1b, (e)–(h) of Fig. 1c without overlapping, with overlapping of 1–3 rows and 1–3 columns.

Fig. 3
Fig. 3

Results of restoration (a)–(f) of Fig. 1d, (g)–(l) of Fig. 1e without overlapping, with overlapping of 1–5 rows and 1–5 columns.

Fig. 4
Fig. 4

Results of restoration of Figs. 1b, 1c, 1d, 1e by using the Wiener filter.

Tables (3)

Tables Icon

Table 1 Accuracy of Restored Images in Fig. 2

Tables Icon

Table 2 Accuracy of Restored Images in Fig. 2

Tables Icon

Table 3 Accuracy of Restored Images in Fig. 4

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

g ( x 1 , x 2 ) = f ( x 1 , x 2 ) h ( x 1 , x 2 ) + n ( x 1 , x 2 ) ,
z ( x ) = i ( x i g i ) 2 + λ 2 i , j ( x i x j ) 2 ,
x i b = j = 1 m k = 1 n x i + ( j m ̃ ) + ( k n ̃ ) u h r ( j , k ) ,
z ( x ) = i ( x i b g i ) 2 + λ 2 i , j ( x i x j ) 2 .
2 g i i x i b + i ( x i b ) 2 λ i , j x i x j .
b i = 4 j = 1 m k = 1 n g ( j m ̃ , k n ̃ ) i h r ( j , k ) ,
inf x ( 1 4 x T Q x + 1 2 b T x ) , x { 1 , + 1 } n .
x T Q x + 2 b T x = ( x 1 ) T L ( x 1 ) , L = [ Q b b T 0 ] .
z d = sup y e T y , L D ( y ) S + n ,
z p = inf x S + n L X , D ( X ) = I ,

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