Abstract

Image restoration from degraded observations and from properties that the image is supposed to satisfy has been approached by the method of projections onto convex constraint sets. Previous attempts have incorporated only partially the knowledge that we possess about the image to be restored because of difficulties in the implementation of some of the projections. In the parallel-projection algorithm presented here the a priori knowledge can be fully exploited. Moreover, the algorithm operates well even if the constraints are nonconvex and/or if the constraints have an empty intersection, without a limitation on the (finite) number of constraint sets.

© 1995 Optical Society of America

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References

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  1. H. J. Trussel, M. R. Civanlar, IEEE Trans Acoust. Speech Signal Process. ASSP-32, 201 (1984).
    [CrossRef]
  2. M. I. Sezan, A. M. Tekalp, IEEE Trans. Acoust. Speech Signal Process. 38, 181 (1990).
    [CrossRef]
  3. M. I. Sezan, H. J. Trussel, IEEE Trans. Acoust. Speech Signal Process. 39, 2275 (1991).
  4. B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
    [CrossRef]
  5. T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Publ. 900 (Technion—Israel Institute of Technology, Haifa, Israel, November1993).
  6. G. Pierra, Math. Program. 28, 96 (1984).
    [CrossRef]
  7. S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).
  8. A. Levi, H. Stark, J. Opt. Soc. Am. A1, 932 (1984).
    [CrossRef]
  9. P. L. Combettes, H. J. Trussel, J. Optim. Theory Appl. 67, 487 (1990).
    [CrossRef]
  10. Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numer. Algorithms (to be published).
  11. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. (to be published).
  12. J. Rosen, Opt. Lett. 19, 369 (1994).
    [CrossRef] [PubMed]
  13. T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Publ. 920 (Technion—Israel Institute of Technology, Haifa, Israel, August1994).

1994 (2)

S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).

J. Rosen, Opt. Lett. 19, 369 (1994).
[CrossRef] [PubMed]

1991 (1)

M. I. Sezan, H. J. Trussel, IEEE Trans. Acoust. Speech Signal Process. 39, 2275 (1991).

1990 (2)

M. I. Sezan, A. M. Tekalp, IEEE Trans. Acoust. Speech Signal Process. 38, 181 (1990).
[CrossRef]

P. L. Combettes, H. J. Trussel, J. Optim. Theory Appl. 67, 487 (1990).
[CrossRef]

1984 (3)

G. Pierra, Math. Program. 28, 96 (1984).
[CrossRef]

A. Levi, H. Stark, J. Opt. Soc. Am. A1, 932 (1984).
[CrossRef]

H. J. Trussel, M. R. Civanlar, IEEE Trans Acoust. Speech Signal Process. ASSP-32, 201 (1984).
[CrossRef]

1973 (1)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

Atlas, L. E.

S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).

Censor, Y.

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numer. Algorithms (to be published).

Civanlar, M. R.

H. J. Trussel, M. R. Civanlar, IEEE Trans Acoust. Speech Signal Process. ASSP-32, 201 (1984).
[CrossRef]

Cohen, N.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Publ. 900 (Technion—Israel Institute of Technology, Haifa, Israel, November1993).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Publ. 920 (Technion—Israel Institute of Technology, Haifa, Israel, August1994).

Combettes, P. L.

P. L. Combettes, H. J. Trussel, J. Optim. Theory Appl. 67, 487 (1990).
[CrossRef]

Elfving, T.

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numer. Algorithms (to be published).

Hunt, B. R.

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

Kotzer, T.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Publ. 900 (Technion—Israel Institute of Technology, Haifa, Israel, November1993).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. (to be published).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Publ. 920 (Technion—Israel Institute of Technology, Haifa, Israel, August1994).

Levi, A.

A. Levi, H. Stark, J. Opt. Soc. Am. A1, 932 (1984).
[CrossRef]

Marks, R. J.

S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).

Oh, S.

S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).

Pierra, G.

G. Pierra, Math. Program. 28, 96 (1984).
[CrossRef]

Rosen, J.

J. Rosen, Opt. Lett. 19, 369 (1994).
[CrossRef] [PubMed]

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. (to be published).

Sezan, M. I.

M. I. Sezan, H. J. Trussel, IEEE Trans. Acoust. Speech Signal Process. 39, 2275 (1991).

M. I. Sezan, A. M. Tekalp, IEEE Trans. Acoust. Speech Signal Process. 38, 181 (1990).
[CrossRef]

Shamir, J.

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Publ. 900 (Technion—Israel Institute of Technology, Haifa, Israel, November1993).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. (to be published).

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Publ. 920 (Technion—Israel Institute of Technology, Haifa, Israel, August1994).

Stark, H.

A. Levi, H. Stark, J. Opt. Soc. Am. A1, 932 (1984).
[CrossRef]

Tekalp, A. M.

M. I. Sezan, A. M. Tekalp, IEEE Trans. Acoust. Speech Signal Process. 38, 181 (1990).
[CrossRef]

Trussel, H. J.

M. I. Sezan, H. J. Trussel, IEEE Trans. Acoust. Speech Signal Process. 39, 2275 (1991).

P. L. Combettes, H. J. Trussel, J. Optim. Theory Appl. 67, 487 (1990).
[CrossRef]

H. J. Trussel, M. R. Civanlar, IEEE Trans Acoust. Speech Signal Process. ASSP-32, 201 (1984).
[CrossRef]

IEEE Trans Acoust. Speech Signal Process. (1)

H. J. Trussel, M. R. Civanlar, IEEE Trans Acoust. Speech Signal Process. ASSP-32, 201 (1984).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (3)

M. I. Sezan, A. M. Tekalp, IEEE Trans. Acoust. Speech Signal Process. 38, 181 (1990).
[CrossRef]

M. I. Sezan, H. J. Trussel, IEEE Trans. Acoust. Speech Signal Process. 39, 2275 (1991).

S. Oh, R. J. Marks, L. E. Atlas, IEEE Trans. Acoust. Speech Signal Process. 42, 1653 (1994).

IEEE Trans. Comput. (1)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

A. Levi, H. Stark, J. Opt. Soc. Am. A1, 932 (1984).
[CrossRef]

J. Optim. Theory Appl. (1)

P. L. Combettes, H. J. Trussel, J. Optim. Theory Appl. 67, 487 (1990).
[CrossRef]

Math. Program. (1)

G. Pierra, Math. Program. 28, 96 (1984).
[CrossRef]

Opt. Lett. (1)

Other (4)

T. Kotzer, N. Cohen, J. Shamir, “A projection algorithm for consistent and inconsistent constraints,” EE Publ. 920 (Technion—Israel Institute of Technology, Haifa, Israel, August1994).

Y. Censor, T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numer. Algorithms (to be published).

T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. (to be published).

T. Kotzer, N. Cohen, J. Shamir, “Extended and alternative projections onto convex constraint sets: theory and applications,” EE Publ. 900 (Technion—Israel Institute of Technology, Haifa, Israel, November1993).

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Figures (2)

Fig. 1
Fig. 1

Restoration results. Clockwise from upper left: The original image, the degraded blurred and noisy image, the Wiener solution, and the result of our algorithm after 1 iteration.

Fig. 2
Fig. 2

Restoration results. Clockwise from upper left: The result of our algorithm after one iteration, after 10 iterations, and again the original image (for comparison).

Equations (14)

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

g = f * h + n ,
d i ( h 1 , h 2 ) : = H 1 - H 2 W i 2 : = - H 1 ( u ) - H 2 ( u ) 2 W i ( u ) d u ,
J ^ ( h ) 2 : = i = 1 N β i d i [ P C i d i ( h ) , h ] = i - 1 N β i F { P C i d i ( h ) } - F { h } W i 2 ,
P i , λ i ( h ) = P C i d i ( h ) + λ i [ h - P C i d i ( h ) ] ,
v i k + 1 ( x ) : = P i , λ i ( h k )             ( i = 1 , 2 , N ) ,
h k + 1 ( x ) = F - 1 { i = 1 N β i W i ( u ) F { v i k + 1 } i = 1 N β i W i ( u ) } .
C 1 : = { f [ f * h ] C ^ 1 } ,
C ^ 1 : = { ρ ρ ( j ) - g ( j ) δ 0 ,             ρ ( j ) R } ,
C 2 : = { f Re { F ( m ) H ( m ) - G ( m ) } δ 1 , Im { F ( m ) H ( m ) - G ( m ) } δ 1 } ,
C 3 : = { f f ( j ) - f w ( j ) δ 2 } ,
C 4 : = { f f ( j ) = 0             for             j [ - a , a ] ,             a > 0 ,             f R + } ,
d 1 ( F 1 , F 2 ) = m W 1 ( m ) F 1 ( m ) - F 2 ( m ) 2 = j F - 1 { W 1 ( m ) } ( j ) * [ f 1 ( j ) - f 2 ( j ) ] 2 .
F - 1 { V 1 } = v 1 : = P C 1 d 1 ( f ) ,             V 1 ( m ) = [ F { ρ ( j ) } ( m ) ] / H ( m ) , ρ ( j ) = { g ( j ) + δ 0 sng [ Re { ρ ( j ) } - g ( j ) ] if Re { ρ ( j ) } - g ( j ) > δ 0 Re { ρ ( j ) } if Re { ρ ( j ) } - g ( j ) δ 0 ,
ρ ( j ) = ρ ( j ) exp [ + ( - 1 ) φ ρ ( j ) ] = F - 1 { H ( m ) F ( m ) } = f ( j ) * h ( j ) .

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