Abstract

This paper presents a formal solution to the problem of object restoration in a one-dimensional, diffraction-limited imaging system. It is found that if the illumination in the object space is confined to a finite region, then the imaging equation can be solved for the object in terms of the image. The solution can be expressed as a series expansion on the eigenfunctions of the imaging operator.

© 1966 Optical Society of America

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References

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  1. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
    [Crossref]
  2. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  3. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [Crossref]
  4. H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, Chap. V, Sec. 4.6.
  5. The point response given by Eq. (2) describes an imaging system with unity magnification. This involves no loss of generality since it is always possible to normalize the image-plane coordinate so that the magnification is unity.
  6. D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
    [Crossref]
  7. Carson Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964; Dover Publications, Inc., New York, 1965).
  9. The writer would like to thank the anonymous reader for suggesting this form.
  10. M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), Chap. 7.

1964 (1)

1961 (1)

D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
[Crossref]

1955 (1)

1952 (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964; Dover Publications, Inc., New York, 1965).

Beran, M. J.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), Chap. 7.

Flammer, Carson

Carson Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).

Harris, J. L.

Landau, H. S.

D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
[Crossref]

Parrent, G. B.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), Chap. 7.

Pollak, H. O.

D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
[Crossref]

Slepian, D.

D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964; Dover Publications, Inc., New York, 1965).

Toraldo di Francia, G.

G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
[Crossref]

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Wolter, H.

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, Chap. V, Sec. 4.6.

Bell System Tech. J. (1)

D. Slepian, H. O. Pollak, and H. S. Landau, Bell System Tech. J. 40, 43 (1961); Bell System Tech. J. 40, 65 (1961); Bell System Tech. J. 41, 1295 (1962).
[Crossref]

J. Opt. Soc. Am. (2)

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

Other (6)

H. Wolter, in Progress in Optics, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1961), Vol. I, Chap. V, Sec. 4.6.

The point response given by Eq. (2) describes an imaging system with unity magnification. This involves no loss of generality since it is always possible to normalize the image-plane coordinate so that the magnification is unity.

Carson Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, Calif., 1957).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Applied Mathematics Series, Vol. 55, 1964; Dover Publications, Inc., New York, 1965).

The writer would like to thank the anonymous reader for suggesting this form.

M. J. Beran and G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964), Chap. 7.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of composite object restoration system.

Fig. 2
Fig. 2

Point response of composite system compared with the point response of the imaging system alone for the case where N = 2 and α = 4d/π.

Fig. 3
Fig. 3

Point response of composite system compared with the point response of the imaging system alone for the case where N = 4 and α = 4d/π.

Fig. 4
Fig. 4

Point response of composite system compared with the point response of the imaging system alone for the case where N = 6 and α = 4d/π.

Fig. 5
Fig. 5

Point response of composite system compared with the point response of the imaging system alone for the case where N = 8 and α = 4d/π.

Equations (18)

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b ( ξ ) = g ( ξ , x ) a ( x ) d x ,
g ( ξ , x ) = d - 1 sinc [ ( ξ - x ) / d ] ,
sinc ( x ) sin ( π x ) / π x
b ( ξ ) = - α α d - 1 sinc [ ( ξ - x ) / d ] a ( x ) d x .
d - 1 - α α sinc [ ( ξ - x ) / d ] σ n ( x ) d x = λ n σ n ( ξ ) ,
σ n ( x ) S 0 n ( c , x / α ) { - α α [ S 0 n ( c , x / α ) ] 2 d x } - 1 2 ,
λ n = ( 2 α / d ) R 0 n ( 1 ) ( c , 1 ) 2 ,
- α α σ n ( x ) σ m ( x ) d x = δ n m ,
0 σ n ( x ) σ n ( ξ ) = δ ( x - ξ ) ,
a ( x ) = n = 0 - α α σ n ( x ) σ n ( ξ ) λ n b ( ξ ) d ξ .
a ( x ) = lim N a N ( x ) ,
a N ( x ) = - α α k N ( x , ξ ) b ( ξ ) d ξ
k N ( x , ξ ) = n = 0 N σ n ( x ) σ n ( ξ ) λ n .
a N ( ζ ) = - α α h N ( ζ , x ) a ( x ) d x ,
h N ( ζ , x ) = d - 1 - α α k N ( ζ , ξ ) sinc [ ( ξ - x ) / d ] d ξ = n = 0 N σ n ( ζ ) σ n ( x ) .
Γ ˆ b ( ξ 1 , ξ 2 ) = d - 2 - α α d x 1 - α α d x 2 sinc [ ( ξ 1 - x 1 ) / d ] × sinc [ ( ξ 2 - x 2 ) / d ] Γ ˆ a ( x 1 , x 2 ) .
Γ ˆ a ( x 1 , x 2 ) = lim N - α α d ξ 1 - α α d ξ 2 k N ( x 1 , ξ 1 ) × k N ( x 2 , ξ 2 ) Γ ˆ b ( ξ 1 , ξ 2 ) ,
g ( x , y ; ξ , η ) = ( d 1 d 2 ) - 1 sinc [ ( x - ξ ) / d 1 ] × sinc [ ( y - η ) / d 2 ] .