Abstract

We investigate the problem of obtaining minimum-bias high-resolution spectral estimators with a coherent optical system. Circular symmetry and isotropic statistics are assumed. We consider the class of spectral smoothing windows W(ρ) > 0 and derive a window function that ensures both minimum-bias and high-resolution spectral estimation. The effect of aberrations, as summarized by the optical transfer function of the system, on the spectral estimate is also considered. In the general case it is shown that minimum-bias, direct frequency-plane smoothing is not physically realizable.

© 1975 Optical Society of America

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Corrections

H. Stark and B. Dimitriadis, "Errata: Minimum-bias spectral estimation with a coherent optical spectrum analyzer," J. Opt. Soc. Am. 65, 973-973 (1975)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-65-8-973

References

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  1. E. N. Leith, Photogr. Sci. Eng. 6, 75 (1962).
  2. H. Thiry, J. Photogr. Sci. 2, 2 (1963).
  3. H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
    [CrossRef]
  4. L. B. Lambert, A. Aimette, and M. Arm, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett, D. A. Berkovitz, L. C. Clapp, C. J. Koester, and A. Vandenburgh (MIT Press, Cambridge, Mass., 1965), p. 720.
  5. J. O. Palgen, Patt. Recog. 2, 4 (1970).
    [CrossRef]
  6. H. Stark and F. B. Tuteur, J. Opt. Soc. Am. 63, 6 (1973).
    [CrossRef]
  7. A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
    [CrossRef]
  8. H. Stark and F. B. Tuteur, in Proceedings of the Second International Joint Conference on Pattern Recognition, Lyngby—Copenhagen, Denmark, 13–15 August 1974. Proceedings available from IEEE under Cat. No. 74CH0885-4C.
  9. A. Papoulis, IEEE Trans. IT-19, 1 (1973).
  10. G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden–Day, San Francisco, 1969), p. 209.
  11. By stability, we mean that the irradiance distribution in the back-focal plane, which serves as the spectral estimate, does not vary much from sample to sample. Bias is a measure of fidelity: a large bias implies that the spectrum has been blurred or distorted. High stability by itself is not a suitable criterion for a spectral estimator. We could always conjure up a stable estimator that, unfortunately, would convey very little information about the process.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967), p. 83.
  13. Reference 12, p. 119.
  14. Reference 10, p. 209.
  15. A. Papoulis, J. Opt. Soc. Am. 62, 1423 (1972).
    [CrossRef]
  16. N. I. Achieser, Theory of Approximation (Ungar, New York, 1956), p. 152.
  17. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, Vol. II, edited by E. Wolf (North-Holland, Amsterdam, 1966), Ch. 1, p. 31.
  18. E. W. Marchand, J. Opt. Soc. Am. 54, 915 (1964).
    [CrossRef]

1974 (1)

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[CrossRef]

1973 (2)

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

H. Stark and F. B. Tuteur, J. Opt. Soc. Am. 63, 6 (1973).
[CrossRef]

1972 (1)

1970 (1)

J. O. Palgen, Patt. Recog. 2, 4 (1970).
[CrossRef]

1969 (1)

H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
[CrossRef]

1964 (1)

1963 (1)

H. Thiry, J. Photogr. Sci. 2, 2 (1963).

1962 (1)

E. N. Leith, Photogr. Sci. Eng. 6, 75 (1962).

Achieser, N. I.

N. I. Achieser, Theory of Approximation (Ungar, New York, 1956), p. 152.

Aimette, A.

L. B. Lambert, A. Aimette, and M. Arm, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett, D. A. Berkovitz, L. C. Clapp, C. J. Koester, and A. Vandenburgh (MIT Press, Cambridge, Mass., 1965), p. 720.

Arm, M.

H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
[CrossRef]

L. B. Lambert, A. Aimette, and M. Arm, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett, D. A. Berkovitz, L. C. Clapp, C. J. Koester, and A. Vandenburgh (MIT Press, Cambridge, Mass., 1965), p. 720.

Bennett, W. R.

H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967), p. 83.

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, Vol. II, edited by E. Wolf (North-Holland, Amsterdam, 1966), Ch. 1, p. 31.

Jenkins, G. M.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden–Day, San Francisco, 1969), p. 209.

Lambert, L. B.

L. B. Lambert, A. Aimette, and M. Arm, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett, D. A. Berkovitz, L. C. Clapp, C. J. Koester, and A. Vandenburgh (MIT Press, Cambridge, Mass., 1965), p. 720.

Leith, E. N.

E. N. Leith, Photogr. Sci. Eng. 6, 75 (1962).

Marchand, E. W.

Palgen, J. O.

J. O. Palgen, Patt. Recog. 2, 4 (1970).
[CrossRef]

Papoulis, A.

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

A. Papoulis, J. Opt. Soc. Am. 62, 1423 (1972).
[CrossRef]

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, Vol. II, edited by E. Wolf (North-Holland, Amsterdam, 1966), Ch. 1, p. 31.

Stark, H.

H. Stark and F. B. Tuteur, J. Opt. Soc. Am. 63, 6 (1973).
[CrossRef]

H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
[CrossRef]

H. Stark and F. B. Tuteur, in Proceedings of the Second International Joint Conference on Pattern Recognition, Lyngby—Copenhagen, Denmark, 13–15 August 1974. Proceedings available from IEEE under Cat. No. 74CH0885-4C.

Thiry, H.

H. Thiry, J. Photogr. Sci. 2, 2 (1963).

Tuteur, F. B.

H. Stark and F. B. Tuteur, J. Opt. Soc. Am. 63, 6 (1973).
[CrossRef]

H. Stark and F. B. Tuteur, in Proceedings of the Second International Joint Conference on Pattern Recognition, Lyngby—Copenhagen, Denmark, 13–15 August 1974. Proceedings available from IEEE under Cat. No. 74CH0885-4C.

Vander Lugt, A.

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[CrossRef]

Watts, D. G.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden–Day, San Francisco, 1969), p. 209.

Appl. Opt. (1)

H. Stark, W. R. Bennett, and M. Arm, Appl. Opt. 8, 11 (1969).
[CrossRef]

IEEE Proc. (1)

A. Vander Lugt, IEEE Proc. 62, 1300 (1974).
[CrossRef]

IEEE Trans. (1)

A. Papoulis, IEEE Trans. IT-19, 1 (1973).

J. Opt. Soc. Am. (3)

J. Photogr. Sci. (1)

H. Thiry, J. Photogr. Sci. 2, 2 (1963).

Patt. Recog. (1)

J. O. Palgen, Patt. Recog. 2, 4 (1970).
[CrossRef]

Photogr. Sci. Eng. (1)

E. N. Leith, Photogr. Sci. Eng. 6, 75 (1962).

Other (9)

L. B. Lambert, A. Aimette, and M. Arm, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett, D. A. Berkovitz, L. C. Clapp, C. J. Koester, and A. Vandenburgh (MIT Press, Cambridge, Mass., 1965), p. 720.

H. Stark and F. B. Tuteur, in Proceedings of the Second International Joint Conference on Pattern Recognition, Lyngby—Copenhagen, Denmark, 13–15 August 1974. Proceedings available from IEEE under Cat. No. 74CH0885-4C.

N. I. Achieser, Theory of Approximation (Ungar, New York, 1956), p. 152.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, Vol. II, edited by E. Wolf (North-Holland, Amsterdam, 1966), Ch. 1, p. 31.

G. M. Jenkins and D. G. Watts, Spectral Analysis and Its Applications (Holden–Day, San Francisco, 1969), p. 209.

By stability, we mean that the irradiance distribution in the back-focal plane, which serves as the spectral estimate, does not vary much from sample to sample. Bias is a measure of fidelity: a large bias implies that the spectrum has been blurred or distorted. High stability by itself is not a suitable criterion for a spectral estimator. We could always conjure up a stable estimator that, unfortunately, would convey very little information about the process.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, New York, 1967), p. 83.

Reference 12, p. 119.

Reference 10, p. 209.

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Figures (3)

FIG. 1
FIG. 1

A coherent optical spectrum analyzer. Coherent light is filtered and expanded by a spatial filter (a) and collimated by the lens (b). The sample is held in a phase-matching liquid (c) and placed against the Fourier-transforming lens (d). The spectrum is observed in the back-focal plane (xf, yf).

FIG. 2
FIG. 2

The minimum-bias spectral-smoothing window, W0(ρ).

FIG. 3
FIG. 3

The minimum-bias data-smoothing window, w0(r).

Equations (75)

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S W ( ω ) = I T S 2 ( ω ) ,
P D ( x , y ) = { 1 , ( x 2 + y 2 ) 1 / 2 < b 0 , otherwise ,
F ( u , υ ) = 1 j λ f exp [ j π λ f ( x f 2 + y f 2 ) ] × t ( x , y ) P D ( x , y ) exp [ j 2 π ( u x + υ y ) ] d x d y ,
u = x f λ f = hozontal spatial frequency υ = y f λ f = vertical spatial frequency .
| F ( u , υ ) | 2 = A λ 2 f 2 R ( α , β ) H D ( α , β ) × exp [ j 2 π ( u α + υ β ) ] d α d β ,
R ( α , β ) = t * ( x , y ) t ( x + α , y + β ) ,
H D ( α , β ) = H D ( r ) = { π 2 { cos 1 ( r 2 b ) ( r 2 b ) [ 1 ( r 2 b ) 2 ] 1 l 8 } , r 2 b 0 , otherwise .
| F ( u , υ ) | 2 = A λ 2 f 2 S ( u , υ ) * 4 A ( J 1 ( 2 π b ( u 2 + υ 2 ) 1 / 2 ) 2 π b ( u 2 + υ 2 ) 1 / 2 ) 2 ,
4 A [ J 1 ( 2 π b ( u 2 + υ 2 ) 1 / 2 ) 2 π b ( u 2 + υ 2 ) 1 / 2 ] 2
S ( u , υ ) = λ 2 f 2 A | F ( u , υ ) | 2
S W ( u , υ ) = S ( u ξ , υ η ) W ( ξ , η ) d ξ d η
S W ( u , υ ) S ( u ξ , υ η ) W ( ξ , η ) d ξ d η S W ( u , υ ) .
S W ( u , υ ) = R ( α , β ) w ( α , β ) × exp [ j 2 π ( u α + υ β ) ] d α d β ,
w ( r ) = 0 , r > 2 c 2 b ,
w ( 0 ) = 2 π 0 ρ W ( ρ ) d ρ = 1 ,
B ( u , υ ) = S W ( u , υ ) S ( u , υ ) ,
B ( u , υ ) B ( ω ) = π 2 [ 1 ω S ( ω ) ω ] 0 ρ 3 W ( ρ ) d ρ ,
ω = ( u 2 + υ 2 ) 1 / 2 , ρ = ( ξ 2 + η 2 ) 1 / 2 , S ( u , υ ) = S ( ( u 2 + υ 2 ) 1 / 2 ) S ( ω ) ,   W ( ξ , η ) = W ( ( ξ 2 + η 2 ) 1 / 2 ) W ( ρ ) .
0 ρ 3 W ( ρ ) d ρ
2 π 0 ρ W ( ρ ) d ρ = 1 .
W 0 ( ρ ) = 4 π c 2 ζ 1 2 J 0 2 ( 2 π ρ c ) [ ζ 1 2 ( 2 π ρ c ) 2 ] 2 ,
W 0 ( r ) = 2 ζ 1 2 0 x J 0 2 ( x ) J 0 ( x ( r / c ) ) [ ζ 1 2 x 2 ] 2 d x
W 0 ( 0 ) = 2 ζ 1 2 0 x J 0 2 ( x ) [ ζ 1 2 x 2 ] 2 d x = 1 .
d I = | g ( x f x f , y f y f ) F ( x f , y f ) | 2 d x f d y f .
I ( x f , y f ) = | F ( x f , y f ) | 2 | g ( x f x f , y f y f ) | 2 d x f d y f .
I ( x f , y f ) = A R ( α , β ) Q ( α , β ) × exp [ j 2 π ( α x f λ f + β y f λ f ) ] d α d β ,
Q ( α , β ) | g ( ξ λ f , η λ f ) | 2 exp [ j 2 π ( ξ α + η β ) ] d ξ d η .
S W ( u , υ ) = κ R ( α , β ) Q ( α , β ) × exp [ j 2 π ( u α + υ β ) ] d α d β ,
κ Q 0 ( r ) = w 0 ( r ) .
| g 0 ( λ f ρ ) | 2 = 1 κ W 0 ( ρ )
| g 0 ( λ f ρ ) | = 1 κ | Y 0 ( ρ ) | ,
P ( x , y ) = e j θ ( x , y ) P D ( x , y ) ,
H ( α , β ) = 1 A exp [ j θ ( ξ + α / 2 , η + β / 2 ) ] × exp [ j θ ( ξ α / 2 , η β / 2 ) ] d ξ d η ,
| F ( x f , y f ) | 2 = A λ 8 f 2 R ( α , β ) H ( α , β ) × exp [ j 2 π ( u α + υ β ) ] d α d β .
| I ( x f , y f ) | = A R ( α , β ) H ( α , β ) Q ( α , β ) × exp [ j 2 π ( α x f λ f + β y f λ f ) ] d α d β .
S W ( u , υ ) = κ R ( α , β ) H ( α , β ) Q ( α , β ) × exp [ j 2 π ( α u + β υ ) ] d α d β .
κ H ( r ) Q 0 ( r ) = w 0 ( r )
Q 0 ( r ) = { w 0 ( r ) κ H ( r ) , r < 2 c 0 , otherwise .
I = 2 π 0 W 2 ( ρ ) d ρ ,
D = 2 π 0 ρ 3 W ( ρ ) d ρ ,
D 0 = 0.144 / c 2 , I 0 = 1.148 c 2 .
y ( r ) = 1 c π circ ( r c ) , where circ ( x ) = { 1 , | x | < 1 0 , otherwise W ( ρ ) = J 1 2 ( 2 π ρ c ) / π ρ 2 , D = 1 2 π 2 c 2 0 x J 1 2 ( x ) d x = , I = 1.44 c 2 .
w ( r ) = circ ( r c ) , W ( ρ ) = 2 c ρ J 1 ( 4 π ρ c ) ,   D = 1 16 π 2 c 2 0 x 2 J 1 ( x ) d x = , I = 4 π c 2 .
w ( r ) = 2 π cos 1 ( r 2 c ) circ ( r 2 c ) , W ( ρ ) = 2 c J 0 ( 2 π ρ c ) J 1 ( 2 π ρ c ) ρ , D = 1 2 π 2 c 2 0 x 2 J 0 ( x ) J 1 ( x ) d x = , I = 3.74 c 2 .
y ( r ) = 4 c ( 3 π ( 3 π 2 16 ) ) 1 / 2 × { cos 1 ( r c ) r c [ 1 ( r c ) 2 ] 1 / 2 } circ ( r c ) , W ( ρ ) = 12 c 2 π ( 3 π 2 16 ) J 1 4 ( π ρ c ) ρ 4 , D = 0.179 / c 2 , I = 0.891 c 2 ( by numerical integration ) .
W 0 ( ρ ) = 4 π c 2 ζ 1 2 J 0 2 ( 2 π ρ c ) [ ζ 1 2 ( 2 π ρ c ) 2 ] 2 .
w ( ξ ) = 0 , | ξ | > 2 c .
w ( u ) 0 for all u ,
W ( u ) = | Y ( u ) | 2
y ( ξ ) = 0 , | ξ | > c .
w ( r ) = 0 , r > 2 c .
2 π 0 2 c r | w ( r ) | d r < .
W ˜ ( ρ ) = | Y ˜ ( ρ ) | 2
q ( ξ ) = w ( ξ , η ) d η ,
W ( ρ , 0 ) = Q ( ρ ) = W ˜ ( ρ ) ,
W ˜ ( ρ ) = 2 π 0 r w ( r ) J 0 ( 2 π r ρ ) d r .
q ( ξ ) = 0 , | ξ | > 2 c
Q ( ρ ) 0 for all ρ .
| P ( ρ ) | 2 = Q ( ρ ) ,
p ( ξ ) = 0 , | ξ | > c .
y ( r ) = 2 π d d r 0 c p ( ξ 2 + r ) d ξ ,
P ( ρ ) = Y ˜ ( ρ ) ,
0 c r [ y r ] 2 d r ,
0 c r [ y ( r ) ] 2 d r = 1 2 π
r y ( r ) + y ( r ) Λ r y = 0 ,
y ( r ) = n C n J 0 ( ζ n r / c ) ,
2 π 0 c r [ y ( r ) ] 2 d r = 1
n C n 2 β n = 1 ,
β n = π c 2 J 1 2 ( ζ n ) .
β 1 > β i for all i 1 ,
C 1 = [ c π J 1 ( ζ 1 ) ] 1 K , C 2 = C 3 = = 0 .
0 c r [ y r ] 2 d r = n C n 2 α n D ,
α n 0 ζ n x J 1 2 ( x ) d x .
y ( r ) = K J 0 ( ζ r / c ) ,
D min = α 1 K .