Abstract

Experimental studies dealing with the restoration of photographic images by optical spatial-filtering techniques are reported. The spatial filters are of a class incorporating predictions of optimum-filtering theory in the presence of random additive noise. This theory was derived using the criterion of least-mean-square error. Measured-modulation-transfer-function curves of both filtered and unfiltered scenes are presented, as well as restorations of continuous tone and binary (Sayce) images distorted by linear motion during exposure. Results show that this type of filtering is superior to that of the infinite signal-to-noise-ratio spatial filter.

© 1969 Optical Society of America

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Corrections

Joseph L. Horner, "Errata: Optical Spatial Filtering with the Least Mean-Square-Error Filter," J. Opt. Soc. Am. 59, 1008_1-1008 (1969)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-59-8-1008_1

References

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  1. A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).
  2. J. Tsujiuchi, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963).
  3. P. F. Mueller and G. O. Reynolds, J. Opt. Soc. Am. 57, 1338 (1967).
    [Crossref]
  4. J. Minkoff, S. Hilal, W. Konig, M. Arm, and L. Lambert, Appl. Opt. 7, 633 (1968).
    [Crossref] [PubMed]
  5. J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
    [Crossref]
  6. L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).
  7. Lower-case letters will denote the Fourier transform of the corresponding spatial function that appears in upper-case letters.
  8. C. W. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
    [Crossref]
  9. E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963), Ch. 2.
  10. The irradiance pattern of the point-spread function on the recording film (aerial image) is further degraded by nonlinear effects of the photographic emulsion. This is discussed further in Sec. IV.
  11. A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).
  12. Although Kodak gives the same designation (649F) to both the plate and film material, the film speed of these two differs by a factor of about 5. Therefore, it is not so surprising that the adjacency effect we measured also differs.
  13. C. E. Thomas, Appl. Opt. 7, 517 (1968).
    [Crossref] [PubMed]
  14. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]

1968 (2)

1967 (2)

1966 (1)

1964 (1)

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

1963 (1)

1960 (1)

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

1953 (1)

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

Arm, M.

Croce, P.

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

Cutrona, L.

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

Harris, J. L.

Helstrom, C. W.

Hilal, S.

Konig, W.

Lambert, L.

Leith, E.

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

Leith, E. N.

Maréchal, A.

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

Minkoff, J.

Mueller, P. F.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963), Ch. 2.

Palermo, C.

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

Porcello, L.

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

Reynolds, G. O.

Thomas, C. E.

Tsujiuchi, J.

J. Tsujiuchi, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963).

Upatnieks, J.

Vander Lugt, A.

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

Appl. Opt. (2)

Compt. Rend. (1)

A. Maréchal and P. Croce, Compt. Rend. 237, 706 (1953).

J. Opt. Soc. Am. (4)

Trans. IEEE (1)

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

Trans. IRE (1)

L. Cutrona, E. Leith, C. Palermo, and L. Porcello, Trans. IRE IT-6, 388 (1960).

Other (5)

Lower-case letters will denote the Fourier transform of the corresponding spatial function that appears in upper-case letters.

E. L. O’Neill, Introduction to Statistical Optics (Addison–Wesley Publ. Co., Reading, Mass., 1963), Ch. 2.

The irradiance pattern of the point-spread function on the recording film (aerial image) is further degraded by nonlinear effects of the photographic emulsion. This is discussed further in Sec. IV.

J. Tsujiuchi, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1963).

Although Kodak gives the same designation (649F) to both the plate and film material, the film speed of these two differs by a factor of about 5. Therefore, it is not so surprising that the adjacency effect we measured also differs.

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Figures (10)

Fig. 1
Fig. 1

Optical system with variable transform scale for spatial filtering. (a) Lens–pinhole assembly, (b) coherent optical noise suppressor (CONS).

Fig. 2
Fig. 2

Proper inverse spatial filter in absence of noise.

Fig. 3
Fig. 3

Required inverse spatial filter for assumed signal-to-noise ratio of 1000.

Fig. 4
Fig. 4

Plot of the optimum spatial filter for assumed signal-to-noise ϕ0/ϕn=50 exp(−ν/30 cycles/mm).

Fig. 5
Fig. 5

Smeared image, unrestored.

Fig. 6
Fig. 6

Smeared image, restored with filter ϕ0/ϕn=∞.

Fig. 7
Fig. 7

Smeared image, restored with filter ϕ0/ϕn=100.

Fig. 8
Fig. 8

Smeared image, restored with filter ϕ0/ϕn=50 exp(−ν/30 cycles/mm).

Fig. 9
Fig. 9

Continuous-tone image restoration. Top, unrestored; bottom, restored. Filter ϕ0/ϕn=50 exp(−ν/30 cycles/mm).

Fig. 10
Fig. 10

Modulation-transfer function of smeared unrestored (dashed), and smeared restored Sayce pattern (solid).

Equations (13)

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I ( x , y ) = - S ( x , x ; y , y ) I ( x , y ) d x d y ,
I ( x , y ) = - A A S ( x - x ; y - y ) I ( x , y ) d x d y .
A ( x , y ) = k · S I ,
a ( ν x , ν y ) = c exp [ j π λ f ( u 2 + ν 2 ) ] . A ( x , y ) exp { - j 2 π ( ν x x + ν y y ) } d x d y ,
a ( ν x , ν y ) = k s ( ν x , ν y ) i ( ν x , ν y ) .
t a f = k / s ( ν x , ν y ) .
F - 1 [ k s ( ν x , ν y ) i ( ν x , ν y ) k s ( ν x , ν y ) ] = k k I ( x , y ) .
t a f = 1 s ( ν x , ν y ) [ ϕ 0 ( ν x , ν y ) / ϕ n ( ν x , ν y ) ] [ ϕ 0 ( ν x , ν y ) / ϕ n ( ν x , ν y ) ] + [ s ( ν x , ν y ) ] - 2 ,
E m i n = ϕ 0 ( ν x , ν y ) ϕ n ( ν x , ν y ) s ( ν x , ν y ) 2 ϕ 0 ( ν x , ν y ) + ϕ n ( ν x , ν y ) d ν x d ν y .
S ( x , y ) = ( 1 / L ) rect ( x / L ) P ( y ) ,
ϕ 0 / ϕ n = k exp ( - ν x / ν 0 ) .
T a I .
T a = 1.0 - k I .