Abstract

For incoherent imaging systems, wave-front aberrations can have the effect of severely reducing the contrast of high spatial frequencies. Although postdetection image processing can improve image detail, the success of such an approach is limited by image noise. If a simple spatial filter consisting of a nonredundant array of clear openings in an opaque mask is placed in the pupil plane of the system, postdetection image processing yields a restored image superior to that obtainable from a system without the pupil-plane filter. Experimental results confirm the predictions of the theory.

© 1971 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 486.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 108.
  3. Reference 2,p. 123.
  4. J. Tsujiuchi, Progress in Optics, II, edited by E. Wolf (North- Holland, Amsterdam, 1963), p. 131.
    [Crossref]
  5. A. Maréchal and P. Croce, Compt. Rend. 237, 607 (1953).
  6. J. L. Horner, J. Opt. Soc. Am. 59, 553 (1969).
    [Crossref]
  7. Reference 1,p. 211.
  8. Reference 1,p. 210.
  9. R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
    [Crossref]
  10. H. H. Hopkins, Proc. Roy. Soc. (London) A231, 98 (1955).
  11. M. De, Proc. Roy. Soc. (London) A233, 91 (1955).
  12. P. A. Stokseth, J. Opt. Soc. Am. 59, 1314 (1969).
    [Crossref]
  13. P. Jaquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 29.
    [Crossref]
  14. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.
  15. F. D. Russell, thesis, Stanford University, Stanford, Calif. (Available from University Microfilms, Ann Arbor, Mich.)
  16. A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
    [Crossref]
  17. J. Singer, Trans. Am. Math. Soc. 43, 377 (1938).
    [Crossref]
  18. M. Golay, private communication.

1969 (2)

1968 (1)

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

1961 (1)

1955 (2)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 98 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

1953 (1)

A. Maréchal and P. Croce, Compt. Rend. 237, 607 (1953).

1938 (1)

J. Singer, Trans. Am. Math. Soc. 43, 377 (1938).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 486.

Croce, P.

A. Maréchal and P. Croce, Compt. Rend. 237, 607 (1953).

De, M.

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Golay, M.

M. Golay, private communication.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 108.

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 98 (1955).

Horner, J. L.

Jaquinot, P.

P. Jaquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 29.
[Crossref]

Jones, R. C.

Maréchal, A.

A. Maréchal and P. Croce, Compt. Rend. 237, 607 (1953).

Moffet, A. T.

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.

Roizen-Dossier, B.

P. Jaquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 29.
[Crossref]

Russell, F. D.

F. D. Russell, thesis, Stanford University, Stanford, Calif. (Available from University Microfilms, Ann Arbor, Mich.)

Singer, J.

J. Singer, Trans. Am. Math. Soc. 43, 377 (1938).
[Crossref]

Stokseth, P. A.

Tsujiuchi, J.

J. Tsujiuchi, Progress in Optics, II, edited by E. Wolf (North- Holland, Amsterdam, 1963), p. 131.
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 486.

Compt. Rend. (1)

A. Maréchal and P. Croce, Compt. Rend. 237, 607 (1953).

IEEE Trans. (1)

A. T. Moffet, IEEE Trans. AP-16, 172 (1968).
[Crossref]

J. Opt. Soc. Am. (3)

Proc. Roy. Soc. (London) (2)

H. H. Hopkins, Proc. Roy. Soc. (London) A231, 98 (1955).

M. De, Proc. Roy. Soc. (London) A233, 91 (1955).

Trans. Am. Math. Soc. (1)

J. Singer, Trans. Am. Math. Soc. 43, 377 (1938).
[Crossref]

Other (10)

M. Golay, private communication.

P. Jaquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 29.
[Crossref]

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley, Reading, Mass., 1963), p. 114.

F. D. Russell, thesis, Stanford University, Stanford, Calif. (Available from University Microfilms, Ann Arbor, Mich.)

Reference 1,p. 211.

Reference 1,p. 210.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, London, 1965), p. 486.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1968), p. 108.

Reference 2,p. 123.

J. Tsujiuchi, Progress in Optics, II, edited by E. Wolf (North- Holland, Amsterdam, 1963), p. 131.
[Crossref]

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Figures (14)

F. 1
F. 1

Notation for the pupil-plane filter.

F. 2
F. 2

Modulation transfer functions, aberration-free case: (a) full pupil, (b) 2Δf < δf, and (c) Δf = δf.

F. 3
F. 3

Modulation transfer functions with aberrations: (a) full pupil, (b) 2Δf < δf, and (c) Δf = δf.

F. 4
F. 4

Attenuation of the peaks of the spectral segments when Δf = (Δx)−1.

F. 5
F. 5

Geometrical-optics spread function (——) and rectangular bound (- - - - - -).

F. 6
F. 6

MTF’s, one-wavelength focusing error. (——) full pupil, (- - - - - -) five-element array.

F. 7
F. 7

Renormalized MTF’s for various focusing errors. (——) full pupil, (- - - -) array: (a) one-wavelength error, five elements, (b) two-wavelengths error, six elements, and (c) four-wavelengths error, eight elements.

F. 8
F. 8

Geometry of the experiment. The focal length of the len is 0.5 m.

F. 9
F. 9

Pupil-plane filters: (a) full pupil and (b) six-element array (negative prints).

F. 10
F. 10

The target (negative print).

F. 11
F. 11

Spectrum of the target. The straight-line diffraction-limited MTF is also shown.

F. 12
F. 12

Microphotometer scans of the recorded image for (a) full pupil in focus, (b) full pupil out of focus, and (c) array out of focus. The focusing error is two wavelengths.

F. 13
F. 13

Restored images: (a) full pupil and (b) array.

F. 14
F. 14

Restored images: (a) full pupil, 10% added noise and (b) array, 17% added noise.

Tables (2)

Tables Icon

Table I Some perfect-difference sets.

Tables Icon

Table II Parameters calculated and chosen for three focusing errors.

Equations (43)

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P ( x ) = a ( x ) exp [ j w ( x ) ] ,
a ( f ) = m = 1 K rect ( f f m Δ f ) ,
w ( f ) = w ( f m ) + w ( f m ) ( f f m ) + ,
w ( f ) w ( f m ) + w ( f m ) ( f f m ) = a m + b m ( f f m ) | f f m | ( Δ f / 2 ) .
H ( f ) = 1 K m = 1 K n = 1 K exp [ j ( a m a n ) ] × exp [ j ( b m + b n 2 ) ( f f m + f n ) ] Λ ( f f m + f n Δ f ) × sinc [ ( b m b n ) 2 π Δ f Λ ( f f m + f n Δ f ) ] ,
f m f n = p m n δ f ( p m n an integer ) ,
δ f = min m n | f m f n | .
H ( f ) = 1 K m = 1 K n = 1 K exp [ j ( a m a n ) ] × exp [ j ( b m + b n 2 ) ( f p m n δ f ) ] Λ ( f p m n δ f Δ f ) × sinc [ ( b m b n ) Δ f 2 π Λ ( f p m n δ f Δ f ) ] .
H ( f ) = Λ ( f Δ f ) + 1 K m = 1 K n = 1 K m n exp [ j ( a m a n ) ] × Λ ( f p m n δ f Δ f ) .
H ( f ) = Λ ( f Δ f ) + 1 K p = ( K 2 K ) / 2 p 0 ( K 2 K ) / 2 exp ( j α p ) Λ ( f p δ f Δ f ) ,
2 F 0 = ( K 2 K ) Δ f K 2 Δ f .
max [ | b m b n | ( Δ f / 2 π ) ] 1 .
Δ f [ 2 π / ( 2 | w max | ) ] ,
w ( f ) = c f r r = 1 , 2 , 3 , or 4 ,
w ( f ) = c r f r 1 .
Δ f = 2 π / ( 2 | w max | ) ,
Δ x = ( 2 | w max | ) / 2 π .
Δ f = 1 / Δ x .
R ( f ) = 1 L | L / 2 L / 2 r ( x ) e j 2 π f x d x | 2 ,
I ( f ) = S ( f ) | H ( f ) | 2 ,
s = [ f 1 F 0 I ( f ) d f ] 1 2 ,
n = [ f 1 F 0 N ( f ) d f ] 1 2 .
d = ( s 1 / n 1 ) / ( s 2 / n 2 ) .
S ( f ) = S 0 N ( f ) = N 0 } f 1 f F 0 .
s 1 = [ ( M 1 ) / M ] 1 2 [ ( F 0 S 0 ) 1 2 / K ] .
s 1 0.76 [ ( M 1 ) / M ] 1 2 [ ( F 0 S 0 ) 1 2 / K ] .
| 1 Δ x rect ( x Δ x ) e j 2 π f x d x | = | sinc Δ x f | .
| sinc Δ x f | ( 1 / π Δ x f ) .
s 2 [ F 0 / M F 0 S 0 ( 1 π Δ x f ) 2 d f ] 1 2 = ( M 1 ) 1 2 π Δ x ( S 0 F 0 ) 1 2 .
A 2 / A 1 = F 0 / K Δ f ,
n 2 / n 1 = ( K Δ f / F 0 ) 1 2 .
d 0.76 π ( K / 2 M ) 1 2 .
d [ ( 0.33 F 0 π 3 | w max | ) / 2 M 2 ] 1 4 ;
d ( 0.59 π 2 / 2 M ) K .
D = K ( K 1 ) / 2 a K
D > K ( K 1 ) / 2 ( K 2 K + 1 ) 1 2 .
w ( f ) = A ( 2 f / F 0 ) 2 ,
Δ f max = π F 0 / 4 A .
{ c i } = ( i , i 2 ) ( modulo K )
1 8
1 12
1 16
1 32