Abstract

A partially coherent imaging system distorts images nonlinearly. We investigate the restoration of such images by use of a second-degree nonlinear filter which is optimum for low-contrast images. The performance of this filter is shown to be better than that of a linear inverse filter, especially as the imaging becomes more coherent.

© 1981 Optical Society of America

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References

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  1. B. E. A. Saleh, Opt. Acta 26, 777 (1979).
    [CrossRef]
  2. R. K. Raney, J. Opt. Soc. Am. 59, 1149 (1969).
    [CrossRef]
  3. B. E. A. Saleh, Opt. Commun. 10, 247 (1974).
    [CrossRef]
  4. B. E. A. Saleh, Opt. Commun. 27, 13 (1978).
    [CrossRef]
  5. B. E. A. Saleh, W. C. Goeke, J. Opt. Soc. Am. 70, 506 (1980).
    [CrossRef]
  6. B. E. A. Saleh, Appl. Opt. 19, 3646 (1980).
    [CrossRef] [PubMed]
  7. B. E. A. Saleh, M. Rabbani, “Bilinear Transformation in Optics. IV: Bayesian Restoration,” U. Wisconsin, Technical Report ECE-80-40 (1980).
  8. M. Schetzen, IEEE Trans. Circuits Syst. CS-23, 285 (1976).
    [CrossRef]
  9. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).
  10. G. W. Stroke, M. Halioua, Optik 35, 489 (1972).
  11. B. E. A. Saleh, Appl. Opt. 17, 3408 (1978).
    [CrossRef] [PubMed]
  12. B. E. A. Saleh, “Bilinear Transformations in Optical Signal Processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. Fienup, B. E. A. Saleh, Eds., SPIE Advanced Institute Series, (SPIE, in press 1982).

1980

1979

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

1978

1976

M. Schetzen, IEEE Trans. Circuits Syst. CS-23, 285 (1976).
[CrossRef]

1974

B. E. A. Saleh, Opt. Commun. 10, 247 (1974).
[CrossRef]

1972

G. W. Stroke, M. Halioua, Optik 35, 489 (1972).

1969

Goeke, W. C.

Halioua, M.

G. W. Stroke, M. Halioua, Optik 35, 489 (1972).

Rabbani, M.

B. E. A. Saleh, M. Rabbani, “Bilinear Transformation in Optics. IV: Bayesian Restoration,” U. Wisconsin, Technical Report ECE-80-40 (1980).

Raney, R. K.

Saleh, B. E. A.

B. E. A. Saleh, W. C. Goeke, J. Opt. Soc. Am. 70, 506 (1980).
[CrossRef]

B. E. A. Saleh, Appl. Opt. 19, 3646 (1980).
[CrossRef] [PubMed]

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

B. E. A. Saleh, Opt. Commun. 27, 13 (1978).
[CrossRef]

B. E. A. Saleh, Appl. Opt. 17, 3408 (1978).
[CrossRef] [PubMed]

B. E. A. Saleh, Opt. Commun. 10, 247 (1974).
[CrossRef]

B. E. A. Saleh, M. Rabbani, “Bilinear Transformation in Optics. IV: Bayesian Restoration,” U. Wisconsin, Technical Report ECE-80-40 (1980).

B. E. A. Saleh, “Bilinear Transformations in Optical Signal Processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. Fienup, B. E. A. Saleh, Eds., SPIE Advanced Institute Series, (SPIE, in press 1982).

Schetzen, M.

M. Schetzen, IEEE Trans. Circuits Syst. CS-23, 285 (1976).
[CrossRef]

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

Stroke, G. W.

G. W. Stroke, M. Halioua, Optik 35, 489 (1972).

Appl. Opt.

IEEE Trans. Circuits Syst.

M. Schetzen, IEEE Trans. Circuits Syst. CS-23, 285 (1976).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

B. E. A. Saleh, Opt. Acta 26, 777 (1979).
[CrossRef]

Opt. Commun.

B. E. A. Saleh, Opt. Commun. 10, 247 (1974).
[CrossRef]

B. E. A. Saleh, Opt. Commun. 27, 13 (1978).
[CrossRef]

Optik

G. W. Stroke, M. Halioua, Optik 35, 489 (1972).

Other

B. E. A. Saleh, “Bilinear Transformations in Optical Signal Processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, J. Fienup, B. E. A. Saleh, Eds., SPIE Advanced Institute Series, (SPIE, in press 1982).

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

B. E. A. Saleh, M. Rabbani, “Bilinear Transformation in Optics. IV: Bayesian Restoration,” U. Wisconsin, Technical Report ECE-80-40 (1980).

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Figures (5)

Fig. 1
Fig. 1

Models of distortion and restoration systems. 1 is a linear system, and 2 is a bilinear system (as denoted by double box).

Fig. 2
Fig. 2

True signal f(x) and restored signal f ^ (x) using (a) a linear restoration system and (b) a second-order nonlinear system. Distortion system is assumed memoryless.

Fig. 3
Fig. 3

Models for distortion and restoration systems when the distortion system is (a) completely coherent [parameter a = 2 f ¯ H1(0)] and (b) completely incoherent (parameter b = 2 f ¯).

Fig. 4
Fig. 4

Restoration of two images (one of a single pulse and another of two pulses) distorted by a partially coherent imaging system (xx = xc = 8 pixels): (a) original images |f(x)|2; (b) distorted images g(x); (c) restored images | f ^ (x)|2 using linear restoration; and (d) restored images | f ^ (x)|2 using a second-degree nonlinear restoration system.

Fig. 5
Fig. 5

As in Fig. 4 but for a completely coherent system (xs = 8, xc = 32 pixels).

Equations (25)

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g ( x ) = H 2 [ f ( x ) ] = - f ( x 1 ) f ( x 2 ) h 2 ( x - x 1 , x - x 2 ) d x 1 d x 2 ,
h 2 ( x 1 , x 2 ) = h * ( x 1 ) h ( x 2 ) γ ( x 1 - x 2 ) ,
f ( x ) = f ¯ + f 1 ( x ) ,
g ( x ) = g ¯ + g 1 ( x ) ,
g ¯ = H 2 { f ¯ } = f ¯ 2 - h 2 ( x 1 , x 2 ) d x 1 d x 2
H 2 ( ω 1 , ω 2 ) = - h 2 ( x 1 , x 2 ) exp [ - j ( ω 1 · x 1 + ω 2 · x 2 ) ] d x 1 d x 2 ,
g ¯ = f ¯ 2 H 2 ( 0 , 0 ) .
g 1 ( x ) = H 1 { f 1 ( x ) } + H 2 { f 1 ( x ) } ,
H 1 { f 1 ( x ) } = - f 1 ( x 1 ) h 1 ( x - x 1 ) d x 1 ,
h 1 ( x ) = 2 f ¯ - h 2 ( x , x 1 ) d x 1 ,
H 1 ( ω ) = 2 f ¯ H 2 ( ω , 0 ) .
f ^ 1 ( x ) = H ^ 1 { g 1 ( x ) } + H ^ 2 { g 1 ( x ) } .
f ^ 1 ( x ) = H ˜ 1 { f 1 } + H ˜ 2 { f 1 } + H ˜ 3 { f 1 } + H ˜ 4 { f 1 } ,
H ˜ 1 = H ^ 1 H 1
H ˜ 2 = H ^ 1 H 2 + H ^ 2 H 1
f ^ ( x ) f ( x ) .
H ^ 1 = H 1 - 1
H ^ 2 = - H 1 - 1 H 2 H 1 - 1 = - H ^ 1 H 2 H ^ 1 .
H ^ 1 ( ω ) = 1 / H 1 ( ω ) = [ 2 f ¯ H 2 ( ω , 0 ) ] - 1 .
H 2 ( ω 1 , ω 2 ) = 1 ,
H 1 ( ω ) = 2 f ¯ .
g 1 ( x ) = 2 f ¯ f 1 ( x ) + f 1 2 ( x ) ,
f 1 ( x ) = ( 1 / 2 f ¯ ) g 1 ( x ) - ( 1 / 2 f ¯ ) 3 g 1 2 ( x ) .
f ^ 1 ( x ) = { f 1 ( x ) + f 1 2 ( x ) / 2 f ¯ } - { f 1 2 ( x ) / 2 f ¯ + f 1 3 ( x ) / 2 f ¯ 2 + f 1 4 ( x ) / 8 f ¯ 3 }
= f 1 ( x ) { 1 - 1 / 2 [ f 1 ( x ) / f ¯ ] 2 - 1 / 8 [ f 1 ( x ) / f ¯ ] 3 } ,

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