Abstract

Analog spatial-filtering techniques are applied to the restoration of images degraded by propagation through random media. The image-forming system is assumed to be linear and stationary so that optically dividing the degraded image spectrum by the degrading transfer function suppresses the degradation. Images are stored and filters fabricated by controlled photographic methods. Preliminary experimental results have successfully demonstrated the feasibility of this filtering technique. In particular, degradations produced by cosine fringes and hot turbulent air were suppressed and enhanced images obtained.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. L. Harris, J. Opt. Soc. Am. 54, 606 (1964).
    [Crossref]
  2. J. L. Harris, J. Opt. Soc. Am. 54, 931 (1964).
    [Crossref]
  3. J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
    [Crossref]
  4. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
    [Crossref]
  5. G. O. Reynolds and T. J. Skinner, J. Opt. Soc. Am. 54, 1302 (1964).
    [Crossref]
  6. G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
    [Crossref]
  7. M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
    [Crossref]
  8. M. J. Beran and G. B. Parrent, The Theory of Partial Coherence (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., Ch. 10.
  10. L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
    [Crossref]
  11. B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957); J. Opt. Soc. Am. 48, 95 (1958).
    [Crossref]
  12. E. L. Bouche and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).
  13. E. L. O’Neill, IRE Trans. Information Theory IT-2, 56 (1956).
    [Crossref]
  14. A. Marshal and P. Croce, Compt. Rend. 237, 607 (1953).
  15. J. Tsujiuchi, in Progress in Optics II, E. Wolf, Ed. (North-Holland Publishing Co., Amsterdam, 1963).
  16. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill Book Co., New York, 1962), p. 37.
  17. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
    [Crossref]
  18. A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
    [Crossref]
  19. The illumination need be spatially coherent only over intervals larger than the impulse response.
  20. The delta functions used here are not rigorously justified because, strictly speaking, the integration could not proceed over infinite limits since the cosine field is limited by the lens aperture. Thus, the lens spread function should actually be convolved about each delta function. Omitting this step keeps the notation simple without sacrificing any vital information.
  21. D. G. Falconer, Phot. Sci. Eng. 10, 133 (1966).
  22. R. F. van Ligten, J. Opt. Soc. Am. 56, 1 (1966).
    [Crossref]

1966 (4)

1964 (5)

1963 (1)

E. L. Bouche and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

1962 (1)

G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
[Crossref]

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

1957 (1)

1956 (1)

E. L. O’Neill, IRE Trans. Information Theory IT-2, 56 (1956).
[Crossref]

1953 (2)

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

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
[Crossref]

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
[Crossref]

Beran, M. J.

M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
[Crossref]

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., Ch. 10.

Bouche, E. L.

E. L. Bouche and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

Croce, P.

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

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

Falconer, D. G.

D. G. Falconer, Phot. Sci. Eng. 10, 133 (1966).

Harris, J. L.

Hufnagel, R. E.

Kellen, P. F.

E. L. Bouche and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

Marshal, A.

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

O’Neill, E. L.

E. L. O’Neill, IRE Trans. Information Theory IT-2, 56 (1956).
[Crossref]

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill Book Co., New York, 1962), p. 37.

Parrent, G. B.

G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
[Crossref]

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

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

Reynolds, G. O.

Shore, R. A.

G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
[Crossref]

Skinner, T. J.

G. O. Reynolds and T. J. Skinner, J. Opt. Soc. Am. 54, 1302 (1964).
[Crossref]

G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
[Crossref]

Stanley, N. R.

Thompson, B. J.

Tsujiuchi, J.

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

van Ligten, R. F.

Vander Lugt, A.

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[Crossref]

Wolf, E.

B. J. Thompson and E. Wolf, J. Opt. Soc. Am. 47, 895 (1957); J. Opt. Soc. Am. 48, 95 (1958).
[Crossref]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., Ch. 10.

Compt. Rend. (1)

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

IEEE Trans. Inform. Theory (1)

A. Vander Lugt, IEEE Trans. Inform. Theory IT-10, 139 (1964).
[Crossref]

IRE Trans. Inform. Theory (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, IRE Trans. Inform. Theory IT-6, 386 (1960).
[Crossref]

IRE Trans. Information Theory (1)

E. L. O’Neill, IRE Trans. Information Theory IT-2, 56 (1956).
[Crossref]

J. Math. Phys. (1)

G. B. Parrent, R. A. Shore, and T. J. Skinner, J. Math. Phys. 34, 678 (1962).
[Crossref]

J. Opt. Soc. Am. (9)

Phot. Sci. Eng. (1)

D. G. Falconer, Phot. Sci. Eng. 10, 133 (1966).

Proc. Phys. Soc. (London) (1)

L. R. Baker, Proc. Phys. Soc. (London) B66, 975 (1953).
[Crossref]

Other (6)

The illumination need be spatially coherent only over intervals larger than the impulse response.

The delta functions used here are not rigorously justified because, strictly speaking, the integration could not proceed over infinite limits since the cosine field is limited by the lens aperture. Thus, the lens spread function should actually be convolved about each delta function. Omitting this step keeps the notation simple without sacrificing any vital information.

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

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill Book Co., New York, 1962), p. 37.

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

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964), 2nd ed., Ch. 10.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Coherent optical system.

Fig. 2
Fig. 2

Incoherent recording system.

Fig. 3
Fig. 3

Degraded bar-target image [ I ob ( y ) * I ( y ) ].

Fig. 4
Fig. 4

(a) Inverse-transfer-function filter [Cav(μ)], and (b) spectrum of degraded bar target [ I ¯ ob ( μ ) I ( μ ) ].

Fig. 5
Fig. 5

Retrieved bar target.

Fig. 6
Fig. 6

(a) Bar-target image through severe turbulence, and (b) binary-target image through moderate turbulence (corresponding transfer-function filters for each target are shown at right).

Fig. 7
Fig. 7

(a) Enhanced bar-target image, and (b) enhanced binary-target image.

Equations (29)

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

I ( x ) = T ( ξ 1 ) T * ( ξ 2 ) Γ ( ξ 1 , ξ 2 ) e ( i k x / f ) · ( ξ 1 ξ 2 ) d ξ 1 d ξ 2 ,
Γ ( ξ 1 , ξ 2 ) = Γ ( ξ 1 , ξ 2 ) .
τ syst ( u ) = γ ( u ) av τ 0 ( u ) ,
I im ( x ) av = I ob ( x ) I ( x x ) av d x ,
I ˜ im ( μ ) av = I ˜ ob ( μ ) I ˜ ( μ ) ,
I ˜ ( μ ) = τ syst ( μ ) .
I ˜ im ( μ ) av = I ˜ ob ( μ ) γ ( μ ) av τ 0 ( μ ) .
I ˜ im ( μ ) av = I ˜ ob ( μ ) γ ( μ ) av .
I ˜ im ( μ ) av / γ ( μ ) av = I ˜ ob ( μ ) .
T A ( x ) = K | I im ( x ) av + I 0 | ,
t A ( μ ) = C / γ ( μ ) av ,
T A ( μ ) t A ( μ ) = K C [ I ˜ ob ( μ ) γ ( μ ) av γ ( μ ) av + I 0 δ ( μ ) γ ( μ ) av ] .
A ( y ) A * ( y ) = ( K C ) 2 | I ob ( y ) + I 0 | 2 .
P A ( μ ) = a + b cos 2 π ω 0 μ with b < a
P ˜ A ( y ) = a δ ( y / λ f 2 ) + b / 2 [ δ ( y / λ f 2 ω 0 ) + δ ( y / λ f 2 + ω 0 ) ] ,
I ( y ) = a 2 δ ( y / λ f 2 ) + b 2 / 4 [ δ ( y / λ f 2 ω 0 ) + δ ( y / λ f 2 + ω 0 ) ] .
I ˜ ( μ ) = a 2 + b 2 / 2 cos 2 π ω 0 μ ,
T I = ( E / E b ) γ ,
| T A | = ( T I ) 1 2 = ( E / E b ) γ / 2 .
T A = ( E / E b ) γ / 2 e i K D
T A = ( E / E b ) γ / 2 ( E / E b ) i K γ ,
T A ( x ) = [ I ( x ) ( t / E b ) ] γ / 2 ,
T I ( x ) = [ I ( x ) ( t / E b ) ] γ .
T A ( x ) = [ T I ( x ) ( t / E b ) ] γ / 2
T A ( x ) = { [ I ( x ) ( t / E b ) ] γ ( t / E b ) } γ / 2 = [ I ( x ) ] γ γ / 2 ( t / E b ) γ γ / 2 ( t / E b ) γ / 2 ,
A ( μ ) = | A ( μ ) | ;
T A ( μ ) = [ | A ( μ ) | 2 t / E b ] γ / 2 .
T A ( μ ) = ( t / E b ) 1 2 [ | A ( μ ) | ] 1 .
| A ( μ ) Max | 2 | A ( μ ) min | 2 E M E m and E m | A ( μ ) min | 2 < t < E M | A ( μ ) Max | 2 ,