Abstract

Under coherent illumination, the interference between signal and noise produces an amplification of the noise. That is why image processing by optical methods may be done better with incoherent light than with coherent light. A method is described where image deblurring is performed in spatially incoherent illumination. The two main features are an additional step of illuminance subtraction to simulate negative intensities, and the use of computer-generated holograms of a low number of cells.

© 1976 Optical Society of America

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References

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  1. T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
    [Crossref]
  2. J. L. Harris, J. Opt. Soc. Am. 56, 569 (1966).
    [Crossref]
  3. Y. Biraud, Astron. Astrophys. 1, 121 (1969).
  4. B. Roy Frieden, J. Opt. Soc. Am. 62, 511 (1972).
    [Crossref]
  5. B. Roy Frieden and J. J. Burke, J. Opt. Soc. Am. 62, 1202 (1972).
    [Crossref]
  6. C. Helstrom, J. Opt. Soc. Am. 57, 297 (1967).
    [Crossref]
  7. D. A. Tichenor and J. W. Goodman, Digest of papers, International Optical Conference, Washington D. C. (1975), IEEE Catalog no 75, CHO 941-5 C, p. 82.
  8. J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
    [Crossref] [PubMed]
  9. S. Lowenthal and A. Werts, C. R. Acad. Sci. Paris 266B, 542 (1968).
  10. A. W. Lohmann, Appl. Opt. 7, 561 (1968).
    [Crossref]
  11. W. Swindell, Appl. Opt. 9, 2459 (1970).
    [Crossref] [PubMed]
  12. W. T. Maloney, Appl. Opt. 10, 2127 (1971).
    [Crossref] [PubMed]
  13. M. Lasserre and R. W. Smith, Opt. Commun. 12, 260 (1974).
    [Crossref]
  14. K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
    [Crossref]
  15. S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
    [Crossref]
  16. Y. Belvaux, S. Lowenthal, and T. Saimi, Opt. Commun. 5, 143 (1972).
    [Crossref]
  17. S. R. Dashiell and A. W. Lohmann, Opt. Commun. 8, 100 (1973).
    [Crossref]
  18. S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
    [Crossref]
  19. W. H. Lee and M. O. Greer, Appl. Opt. 13, 925 (1974).
    [Crossref] [PubMed]
  20. S. Lowenthal and P. Chavel, Appl. Opt. 13, 718 (1974).
    [Crossref] [PubMed]
  21. M. Inuiya and Y. Ichioka, Opt. Commun. 8, 382 (1973).
    [Crossref]
  22. J. Braat and S. Lowenthal, J. Opt. Soc. Am. 63, 388 (1973).
    [Crossref]
  23. J. W. Goodman and A. M. Silvestri, IBM J. Res. Develop. 14, 478 (1969).
    [Crossref]
  24. W. J. Dallas, Appl. Opt. 10, 674 (1971).
    [Crossref] [PubMed]
  25. A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [Crossref] [PubMed]
  26. M. Born and E. Wolf, Priniciples of Optics, 4th ed. (PergamonNew York, 1970), Ch. X.

1975 (1)

D. A. Tichenor and J. W. Goodman, Digest of papers, International Optical Conference, Washington D. C. (1975), IEEE Catalog no 75, CHO 941-5 C, p. 82.

1974 (3)

1973 (4)

M. Inuiya and Y. Ichioka, Opt. Commun. 8, 382 (1973).
[Crossref]

J. Braat and S. Lowenthal, J. Opt. Soc. Am. 63, 388 (1973).
[Crossref]

S. R. Dashiell and A. W. Lohmann, Opt. Commun. 8, 100 (1973).
[Crossref]

S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

1972 (3)

1971 (4)

W. J. Dallas, Appl. Opt. 10, 674 (1971).
[Crossref] [PubMed]

T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[Crossref]

W. T. Maloney, Appl. Opt. 10, 2127 (1971).
[Crossref] [PubMed]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

1970 (2)

K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
[Crossref]

W. Swindell, Appl. Opt. 9, 2459 (1970).
[Crossref] [PubMed]

1969 (2)

Y. Biraud, Astron. Astrophys. 1, 121 (1969).

J. W. Goodman and A. M. Silvestri, IBM J. Res. Develop. 14, 478 (1969).
[Crossref]

1968 (2)

S. Lowenthal and A. Werts, C. R. Acad. Sci. Paris 266B, 542 (1968).

A. W. Lohmann, Appl. Opt. 7, 561 (1968).
[Crossref]

1967 (3)

1966 (1)

Belvaux, Y.

Y. Belvaux, S. Lowenthal, and T. Saimi, Opt. Commun. 5, 143 (1972).
[Crossref]

Biraud, Y.

Y. Biraud, Astron. Astrophys. 1, 121 (1969).

Born, M.

M. Born and E. Wolf, Priniciples of Optics, 4th ed. (PergamonNew York, 1970), Ch. X.

Braat, J.

Burke, J. J.

Chavel, P.

Dallas, W. J.

Dashiell, S. R.

S. R. Dashiell and A. W. Lohmann, Opt. Commun. 8, 100 (1973).
[Crossref]

S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

Debrus, S.

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Françon, M.

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Goodman, J. W.

D. A. Tichenor and J. W. Goodman, Digest of papers, International Optical Conference, Washington D. C. (1975), IEEE Catalog no 75, CHO 941-5 C, p. 82.

J. W. Goodman and A. M. Silvestri, IBM J. Res. Develop. 14, 478 (1969).
[Crossref]

J. W. Goodman, J. Opt. Soc. Am. 57, 493 (1967).
[Crossref] [PubMed]

Greer, M. O.

Grover, C. P.

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Harris, J. L.

Helstrom, C.

Huang, T. S.

T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[Crossref]

Ichioka, Y.

M. Inuiya and Y. Ichioka, Opt. Commun. 8, 382 (1973).
[Crossref]

Inuiya, M.

M. Inuiya and Y. Ichioka, Opt. Commun. 8, 382 (1973).
[Crossref]

Lasserre, M.

M. Lasserre and R. W. Smith, Opt. Commun. 12, 260 (1974).
[Crossref]

Lee, W. H.

Lohmann, A. W.

S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

S. R. Dashiell and A. W. Lohmann, Opt. Commun. 8, 100 (1973).
[Crossref]

A. W. Lohmann, Appl. Opt. 7, 561 (1968).
[Crossref]

A. W. Lohmann and D. P. Paris, Appl. Opt. 6, 1739 (1967).
[Crossref] [PubMed]

Lowenthal, S.

S. Lowenthal and P. Chavel, Appl. Opt. 13, 718 (1974).
[Crossref] [PubMed]

J. Braat and S. Lowenthal, J. Opt. Soc. Am. 63, 388 (1973).
[Crossref]

Y. Belvaux, S. Lowenthal, and T. Saimi, Opt. Commun. 5, 143 (1972).
[Crossref]

S. Lowenthal and A. Werts, C. R. Acad. Sci. Paris 266B, 542 (1968).

Maloney, W. T.

Michaelson, J. D.

S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

Paris, D. P.

Pennington, K. S.

K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
[Crossref]

Roy Frieden, B.

Saimi, T.

Y. Belvaux, S. Lowenthal, and T. Saimi, Opt. Commun. 5, 143 (1972).
[Crossref]

Schrieber, W. F.

T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[Crossref]

Shelton, G. L.

K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
[Crossref]

Silvestri, A. M.

J. W. Goodman and A. M. Silvestri, IBM J. Res. Develop. 14, 478 (1969).
[Crossref]

Smith, R. W.

M. Lasserre and R. W. Smith, Opt. Commun. 12, 260 (1974).
[Crossref]

Swindell, W.

Tichenor, D. A.

D. A. Tichenor and J. W. Goodman, Digest of papers, International Optical Conference, Washington D. C. (1975), IEEE Catalog no 75, CHO 941-5 C, p. 82.

Tretiak, O. J.

T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[Crossref]

Werts, A.

S. Lowenthal and A. Werts, C. R. Acad. Sci. Paris 266B, 542 (1968).

Will, P. M.

K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Priniciples of Optics, 4th ed. (PergamonNew York, 1970), Ch. X.

Appl. Opt. (7)

Astron. Astrophys. (1)

Y. Biraud, Astron. Astrophys. 1, 121 (1969).

C. R. Acad. Sci. Paris (1)

S. Lowenthal and A. Werts, C. R. Acad. Sci. Paris 266B, 542 (1968).

Digest of papers, International Optical Conference, Washington D. C. (1)

D. A. Tichenor and J. W. Goodman, Digest of papers, International Optical Conference, Washington D. C. (1975), IEEE Catalog no 75, CHO 941-5 C, p. 82.

IBM J. Res. Develop. (1)

J. W. Goodman and A. M. Silvestri, IBM J. Res. Develop. 14, 478 (1969).
[Crossref]

J. Opt. Soc. Am. (6)

Opt. Commun. (7)

M. Lasserre and R. W. Smith, Opt. Commun. 12, 260 (1974).
[Crossref]

K. S. Pennington, P. M. Will, and G. L. Shelton, Opt. Commun. 2, 113 (1970).
[Crossref]

S. Debrus, M. Françon, and C. P. Grover, Opt. Commun. 4, 172 (1971).
[Crossref]

Y. Belvaux, S. Lowenthal, and T. Saimi, Opt. Commun. 5, 143 (1972).
[Crossref]

S. R. Dashiell and A. W. Lohmann, Opt. Commun. 8, 100 (1973).
[Crossref]

S. R. Dashiell, A. W. Lohmann, and J. D. Michaelson, Opt. Commun. 8, 105 (1973).
[Crossref]

M. Inuiya and Y. Ichioka, Opt. Commun. 8, 382 (1973).
[Crossref]

Proc. IEEE (1)

T. S. Huang, W. F. Schrieber, and O. J. Tretiak, Proc. IEEE 59, 1586 (1971).
[Crossref]

Other (1)

M. Born and E. Wolf, Priniciples of Optics, 4th ed. (PergamonNew York, 1970), Ch. X.

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Figures (14)

FIG. 1
FIG. 1

Block diagram of an image processing experiment; r denotes the point with coordinates x, y.

FIG. 2
FIG. 2

Block diagram of the method.

FIG. 3
FIG. 3

Image processing using CGH; So = area of the object; SH = area of the hologram; D = distance between object and pupil; the number of samples in the hologram must be at least SoSH2D2.

FIG. 4
FIG. 4

Two-step holographic filter: recording of the natural hologram NH from the computer hologram CH; S = reference wave, m = hole selecting one order.

FIG. 5
FIG. 5

Synthesis of incoherence by scanning the pupil plane; NH = natural hologram, A = object, L and L′ = lenses, Def = xy deflector.

FIG. 6
FIG. 6

Amplitude quantization: Δx×Δx = size of one cell; Z = amplitude to be coded; Z0 = maximum value of Z; cΔx = maximum rectangle width; pΔx = pen width, (a) if Z/Z0 > p; (b) if Z/Z0 < p.

FIG. 7
FIG. 7

Two-step holographic filter: recording of the natural hologram NH from a synthetic image-hologram CH; S = reference wave, m = hole; CH′ = image of CH given by lens L.

FIG. 8
FIG. 8

The two image-holograms of the positive and negative parts of the impulse response.

FIG. 9
FIG. 9

Positive and negative parts of the processing impulse response (filtered images of the holograms of Fig. 8).

FIG. 10
FIG. 10

Photometric scan of the cross sections of the two images to be subtracted.

FIG. 11
FIG. 11

Result of the processing: (a) blur; (b) corrected impulse response; (c) computer simulation of (b).

FIG. 12
FIG. 12

Synthesis of incoherence; M = point in the object plane; p(t) = point scanning the pupil; M′ = point in the image plane.

FIG. 13
FIG. 13

Characterization of the incoherence created by a line-by-line scanning; distance between peaks along y: NλD/L2. N = number of lines; L2 = length of scanning along y.

FIG. 14
FIG. 14

Validity of the approximation of computer-generated image holograms; (a) the approximated function Y2(x); (b) Y1(x) for α = 0, 4; (c) Y1(x) for α = 1.

Equations (41)

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h ˜ ( μ , ν ) = d ˜ * ( μ , ν ) / [ d ˜ ( μ , ν ) 2 + ( μ , ν ) ] ,
ρ = E i / σ E ,
ρ c = R / 2 R + 1 ,
R = E i / E n .
A ( r ) = O ( r ) * d ( r ) .
B ( r ) = h ( r ) * A ( r ) = O ( r ) * [ d ( r ) * h ( r ) ] .
{ h ( r ) = h + ( r ) - h - ( r ) , h + ( r ) 0 , h - ( r ) 0.
h + ( r ) = { h ( r ) , if h ( r ) 0 , 0 , elsewhere ; h - ( r ) = { 0 , if h ( r ) 0 , - h ( r ) , elsewhere .
B ( r ) = A ( r ) * h + ( r ) - A ( r ) * h - ( r ) = A ( r ) * h ( r ) .
N = S o S H / λ 2 D 2 ,
H = ( Z / Z 0 ) Δ x w = c Δ x }             if Z / Z 0 > p , H = p Δ x w = ( Z / p Z 0 ) c Δ x }             if Z / Z 0 < p
A ( r ) = d ( r ) .
B ( r ) = d ( r ) * h ( r ) ,
B + ( r ) = d ( r ) * h + ( r )             and             B - ( r ) = d ( r ) * h - ( r )
B + ( r ) = k + A ( r ) * h + ( r ) , B - ( r ) = k - A ( r ) * h - ( r ) .
B ( r ) = B + ( r ) - α B - ( r ) = k + A ( r ) * [ h + ( r ) - α ( k - / k + ) h - ( r ) ] ,
α = k + / k - .
I + = h + ( r ) d r , I - = h - ( r ) d r
e + e - = k + h + ( r ) d r k - h - ( r ) d r ,
α = I - e + I + e - .
E ( r , t ) = 1 λ 4 d 4 | a ( r ) τ ( r - r ) exp ( 2 i π r · p ( t ) λ d ) d r | 2 ,
l 1 ( r ) = t t + T 1 E ( r , θ ) d θ .
l ( r ) = t t + T E ( r , θ ) d θ = 1 λ 4 d 4 - + a ( r 1 ) τ ( r - r 1 ) a * ( r 2 ) × τ * ( r - r 2 ) D ( r 1 - r 2 ) d r 1 d r 2 ,
D ( r ) = t t + T exp ( 2 i π r · p ( θ ) λ d ) d θ .
l ( r ) = a ( r ) 2 τ ( r - r ) 2 d r ,
p ( t ) = { ( t N T - k - 1 2 ) L 1 ( k N - 1 2 - 1 2 N ) L 2 for { k T N < t < ( k + 1 ) T N and 0 < k < N - 1
D ( x , y ) = k = 0 N - 1 ( k T / N ) ( k + 1 ) T / N exp ( 2 i π × x ( t N / T - k - 1 2 ) L 1 + y ( k / N - 1 2 - 1 2 N ) L 2 λ d ) d t = T sinc x L 1 d sin ( π y L 2 / λ d ) sin ( π y L 2 / N λ d ) ,
D { x < Δ x / 2 , y < Δ x / 2 ,
D { μ < Δ μ / 2 , ν < Δ μ / 2.
f i j = f ( i δ x , j δ x ) ,
f ( x , y ) = ( i δ x , i δ x ) D f i j sinc x - i δ x δ x sinc y - j δ x x .
τ 1 ( x , y ) = ( i δ x , j δ x ) D δ ( i δ x , j δ x ) * rect c δ x ( x ) * rect K f i j δ x ( y ) ,
rect a ( u ) = { 1 if u < a / 2 , 0 if u > a / 2.
f 1 ( x , y ) = A 0 C K δ x 2 τ 1 ( x , y ) * ( sinc x δ x sinc y δ x ) = A 0 C K δ x 2 ( i δ x , j δ x ) D ( sinc x - i δ x δ x sinc y - j δ x δ x ) * rect c δ x ( x ) * rect K f i j δ x ( y ) .
{ sinc x δ x * rect c δ x ( x ) c δ x sinc x δ x sinc y δ x * rect K f i j δ x ( y ) K f i j δ x sinc y δ x .
Y 1 ( x ) = α sinc x δ x * rect α δ x ( x ) = δ x π α [ Si ( π α 2 - π x δ x ) - Si ( - π α 2 - π x δ x ) ] , Y 2 ( x ) = δ x sinc ( x / δ x )
Si ( x ) = 0 x sin u u d u .
f ( x , y ) = F ( x , y )
D { ( m - 1 2 ) Δ μ < μ < ( m + 1 2 ) Δ μ , ( n - 1 2 ) Δ ν < v < ( n + 1 2 ) Δ ν , with m or n > 1 2 ,
f 1 ( x , y ) = A 0 C K δ x 2 ( i δ x , j δ x ) D [ sinc x - i δ x δ x sinc y - j δ x δ x exp ( 2 i π ( x - i δ x ) m + ( y - j δ x ) n x ) ] * rect c δ x ( x ) * rect K f i j δ x ( y ) ,
{ sinc x x exp ( 2 i π m x x ) * rect c δ x ( x ) c δ x sinc x δ x exp ( 2 i π m x δ x ) , sinc y x exp ( 2 i π n y x ) * rect K f i j δ x ( y ) K f i j δ x sinc y δ x exp ( 2 i π n y δ x ) .