Abstract

A constrained least-squares solution based on a second difference smoothing measure is obtained for images which are distorted by a bilinear (quadratic and with nonzero memory) system in the presence of additive signal-independent noise. Results are applied to a partially coherent diffraction-limited imaging system. It is found that the optimum weight given to the smoothness factor is larger for a coherent (nonlinear) system than for an incoherent (linear) system. However, the restoration quality improves as the imaging system approaches incoherence.

© 1983 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. M. Schetzen, The Volterra and Wiener Theorems of Nonlinear Systems (Wiley, New York, 1980).
  3. B. E. A. Saleh, Opt. Commun. 109, 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, S. I. Sayegh, Appl. Opt. 20, 4089 (1981).
    [CrossRef] [PubMed]
  8. B. E. A. Saleh, M. Rabbani, J. Opt. Soc. Am. 73, 66 (1983).
    [CrossRef]
  9. B. E. A. Saleh, M. Rabbani, J. Opt. Soc. Am. 73, 71 (1983).
    [CrossRef]
  10. M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta30, in press (1983).
  11. D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
    [CrossRef]
  12. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).
  13. S. Twomey, J. Franklin Inst. 279, 95 (1965).
    [CrossRef]
  14. S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
    [CrossRef]
  15. G. Wahba, “Smoothing and III-Posed Problems,” in Solution Methods for Integral Equations, M. Golberg, Ed. (Plenum, New York, 1979), p. 183.
  16. G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
    [CrossRef]
  17. B. R. Hunt, IEEE Trans. Autom. Control AC-17, 703 (1972).
    [CrossRef]
  18. A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).
  19. B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
    [CrossRef]
  20. S. S. Reddi, Appl. Opt. 17, 2340 (1978).
    [CrossRef] [PubMed]
  21. N. N. Abdelmalek, T. Kasvand, J. Olmstead, M. M. Tremblay, Appl. Opt. 20, 4227 (1981).
    [CrossRef] [PubMed]
  22. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1974).
  23. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  24. L. C. W. Dixon, Nonlinear OptimizationCrane, Russak, New York, 1972).
  25. B. R. Hunt, IEEE Trans. Comput. C-26, 57 (1979).
  26. E. Isaacson, H. B. Keller, Analysis of Numerical Methods (New York, Wiley, 1966).
  27. B. E. A. Saleh, M. Rabbani, Appl. Opt. 21, 2770 (1982).
    [CrossRef] [PubMed]

1983 (2)

1982 (1)

1981 (2)

1980 (2)

1979 (3)

B. R. Hunt, IEEE Trans. Comput. C-26, 57 (1979).

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

G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
[CrossRef]

1978 (2)

1974 (1)

B. E. A. Saleh, Opt. Commun. 109, 247 (1974).
[CrossRef]

1973 (1)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

1972 (1)

B. R. Hunt, IEEE Trans. Autom. Control AC-17, 703 (1972).
[CrossRef]

1965 (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

1963 (1)

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

1962 (1)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Abdelmalek, N. N.

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1974).

Dixon, L. C. W.

L. C. W. Dixon, Nonlinear OptimizationCrane, Russak, New York, 1972).

Goeke, W. C.

Golub, G.

G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
[CrossRef]

Heath, M.

G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
[CrossRef]

Hunt, B. R.

B. R. Hunt, IEEE Trans. Comput. C-26, 57 (1979).

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

B. R. Hunt, IEEE Trans. Autom. Control AC-17, 703 (1972).
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Isaacson, E.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (New York, Wiley, 1966).

Kasvand, T.

Keller, H. B.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (New York, Wiley, 1966).

Olmstead, J.

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1974).

Phillips, D. L.

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

Rabbani, M.

B. E. A. Saleh, M. Rabbani, J. Opt. Soc. Am. 73, 66 (1983).
[CrossRef]

B. E. A. Saleh, M. Rabbani, J. Opt. Soc. Am. 73, 71 (1983).
[CrossRef]

B. E. A. Saleh, M. Rabbani, Appl. Opt. 21, 2770 (1982).
[CrossRef] [PubMed]

M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta30, in press (1983).

Reddi, S. S.

Saleh, B. E. A.

Sayegh, S. I.

Schetzen, M.

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

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

Tremblay, M. M.

Twomey, S.

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

Wahba, G.

G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
[CrossRef]

G. Wahba, “Smoothing and III-Posed Problems,” in Solution Methods for Integral Equations, M. Golberg, Ed. (Plenum, New York, 1979), p. 183.

Appl. Opt. (5)

IEEE Trans. Autom. Control (1)

B. R. Hunt, IEEE Trans. Autom. Control AC-17, 703 (1972).
[CrossRef]

IEEE Trans. Comput. (2)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

B. R. Hunt, IEEE Trans. Comput. C-26, 57 (1979).

J. Assoc. Comput. Mach. (2)

D. L. Phillips, J. Assoc. Comput. Mach. 9, 84 (1962).
[CrossRef]

S. Twomey, J. Assoc. Comput. Mach. 10, 97 (1963).
[CrossRef]

J. Franklin Inst. (1)

S. Twomey, J. Franklin Inst. 279, 95 (1965).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Acta (1)

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

Opt. Commun. (2)

B. E. A. Saleh, Opt. Commun. 109, 247 (1974).
[CrossRef]

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

Technometrics (1)

G. Golub, M. Heath, G. Wahba, Technometrics 21, 215 (1979).
[CrossRef]

Other (9)

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Posed Problems (Wiley, New York, 1977).

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1974).

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

L. C. W. Dixon, Nonlinear OptimizationCrane, Russak, New York, 1972).

M. Rabbani, B. E. A. Saleh, “Bayesian restoration of partially coherent imagery in conventional and scanning microscopy,” Opt. Acta30, in press (1983).

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

G. Wahba, “Smoothing and III-Posed Problems,” in Solution Methods for Integral Equations, M. Golberg, Ed. (Plenum, New York, 1979), p. 183.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, New York, 1977).

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (New York, Wiley, 1966).

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Figures (5)

Fig. 1
Fig. 1

Partially coherent optical system.

Fig. 2
Fig. 2

Original image used in Sec. IV.

Fig. 3
Fig. 3

Dotted lines represent images that correspond to the object in Fig. 2 after distortion by (a) coherent imaging system, (b) partially coherent imaging system (three modes), (c) partially coherent imaging system (five modes), and (d) incoherent imaging system. Solid lines are the same as above but in the presence of white Gaussian noise (SNR = 27 dB).

Fig. 4
Fig. 4

(a), (b), (c), and (d) are the unconstrained least-squares restoration of the images in Figs. 3(a), (b), (c), and (d), respectively.

Fig. 5
Fig. 5

Solid lines in (a), (b), (c), and (d) are the smoothed least-squares restoration of the images in Figs. 3(a), (b), (c), and (d), respectively. Dotted lines show the original object for comparison.

Equations (26)

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g ( x ) = f ( x 1 ) f ( x 2 ) q ( x ; x 1 , x 2 ) d x 1 d x 2 + n ( x ) .
q ( x ; x 1 x 2 ) = γ ( x 1 , x 2 ) h * ( x ; x 1 ) h ( x ; x 2 ) ,
g i = i + n i , i = 1 , 2 , , N ,
i = j = 1 N k = 1 N f k f j q j k i , i = 1 , 2 , , N ,
g = + n ,
i = f T q i f , i = 1 , 2 , , N ,
g s ( f ̂ ) 2 = n 2 = n T n
s i ( f ̂ ) = j = 1 N k = 1 N q j k i f ̂ j f ̂ k , i = 1 , 2 , , N .
ϕ ( f ̂ ) [ g s ( f ̂ ) ] = 0 ,
ϕ l i ( f ̂ ) = 4 k = 1 N Re ( q l k i ) f ̂ k , l , i = 1 , , N ,
f ̂ ( m ) 2 = i [ f ̂ i ( m ) ] 2 ,
f i ( 2 ) = f i + 1 2 f i + f i 1 .
Minimize f W ( f ) = g s ( f ) 2 + λ f ( m ) 2 .
ϕ ( f ̂ ) [ g s ( f ̂ ) ] + λ A f ̂ = 0 ,
A l l = 12 , A l , l ± 1 = 8 , l = 3 , 4 , , N 3 , A l , l ± 2 = 2 .
g s ( f ̂ ) 2 = n 2 ,
ϕ ( f ̂ MAP ) [ g s ( f ̂ MAP ) ] + λ R f 1 ( f ̂ MAP f ̂ ) = 0 ,
y ( m + 1 ) = y ( m ) α ( m ) ψ [ y ( m ) ] , m = 0 , 1 , ,
f ̂ ( m + 1 ) = f ̂ ( m ) α ( m ) ( ϕ [ f ̂ ( m ) ] { g s [ f ̂ ( m ) ] } + λ A f ̂ ( m ) ) ,
ϕ l i ( m ) = 4 k = 1 N Re ( q l k i ) f ̂ k ( m ) , s i ( m ) = j = 1 N k = 1 N q j k i f ̂ j ( m ) f ̂ k ( m ) .
q ( x ; x 1 , x 2 ) = h * ( x x 1 ) h ( x x 2 ) γ ( x 1 x 2 ) .
q j k i = h i j * h i k γ j k ,
γ ( x ) = sinc ( ω Γ x / π ) , Γ ( ω ) = 1 2 ω Γ rect ( ω / 2 ω Γ ) ,
H ( ω ) = rect ( ω / 2 ω H ) ,
H i = rect ( i 2 ω s ) i = 7 , , 0 , , + 7 .
Γ i = 1 N Γ , | i | ( N Γ 1 ) / 2 , i = 15 , , 0 , , + 15 0 , otherwise ,

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