Abstract

Given a truncated portion of a band-limited image with known x and y bandwidths, the extrapolation problem is to determine the signal over all x and y. An iterative extrapolation algorithm, recently proposed by Gerchberg, requires only the repeated operations of Fourier transformation and truncation. A coherent optical processor is presented that implements Gerchberg's iterative extrapolation algorithm in two dimensions. Iteration is performed by simple passive feedback.

© 1980 Optical Society of America

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References

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  1. D. Slepian, H. Pollak, Bell Syst. Tech. J. 40, 43 (1961).
  2. G. T. Di Francia, J. Opt. Soc. Am. 59, 799 (1969).
    [Crossref]
  3. W. K. Pratt, Digital Image Processing (Wiley, New York, 1968, pp. 437–440.
  4. R. W. Gerchberg, Opt. Acta 21, 709 (1974).
    [Crossref]
  5. M. S. Sabri, W. Steenaart, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
    [Crossref]
  6. A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [Crossref]
  7. D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
    [Crossref]
  8. J. R. Fienup, Opt. Lett. 3, 27 (1978).
    [Crossref] [PubMed]
  9. Spatial frequency is measured on the (fx,fy) plane by dividing the vertical and horizontal displacements from the origin by λF, respectively. λ is the wavelength of the spatially coherent illumination, and F is the focal length of the transforming lens.
  10. R. J. Marks, D. K. Smith, J. Opt. Soc. Am. 69, 1467A (1979).

1979 (1)

R. J. Marks, D. K. Smith, J. Opt. Soc. Am. 69, 1467A (1979).

1978 (3)

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[Crossref]

J. R. Fienup, Opt. Lett. 3, 27 (1978).
[Crossref] [PubMed]

M. S. Sabri, W. Steenaart, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[Crossref]

1975 (1)

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

1974 (1)

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[Crossref]

1969 (1)

1961 (1)

D. Slepian, H. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Di Francia, G. T.

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[Crossref]

Marks, R. J.

R. J. Marks, D. K. Smith, J. Opt. Soc. Am. 69, 1467A (1979).

Papoulis, A.

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

Pollak, H.

D. Slepian, H. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1968, pp. 437–440.

Sabri, M. S.

M. S. Sabri, W. Steenaart, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[Crossref]

Slepian, D.

D. Slepian, H. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

Smith, D. K.

R. J. Marks, D. K. Smith, J. Opt. Soc. Am. 69, 1467A (1979).

Steenaart, W.

M. S. Sabri, W. Steenaart, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[Crossref]

Youla, D. C.

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[Crossref]

Bell Syst. Tech. J. (1)

D. Slepian, H. Pollak, Bell Syst. Tech. J. 40, 43 (1961).

IEEE Trans. Circuits Syst. (3)

M. S. Sabri, W. Steenaart, IEEE Trans. Circuits Syst. CAS-25, 74 (1978).
[Crossref]

A. Papoulis, IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[Crossref]

D. C. Youla, IEEE Trans. Circuits Syst. CAS-25, 694 (1978).
[Crossref]

J. Opt. Soc. Am. (2)

G. T. Di Francia, J. Opt. Soc. Am. 59, 799 (1969).
[Crossref]

R. J. Marks, D. K. Smith, J. Opt. Soc. Am. 69, 1467A (1979).

Opt. Acta (1)

R. W. Gerchberg, Opt. Acta 21, 709 (1974).
[Crossref]

Opt. Lett. (1)

Other (2)

Spatial frequency is measured on the (fx,fy) plane by dividing the vertical and horizontal displacements from the origin by λF, respectively. λ is the wavelength of the spatially coherent illumination, and F is the focal length of the transforming lens.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1968, pp. 437–440.

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Figures (2)

Fig. 1
Fig. 1

Illustration of Gerchberg's algorithm for extrapolation of band-limited signals.

Fig. 2
Fig. 2

Coherent optical processor for implementing Gerchberg's algorithm in two dimensions.

Equations (13)

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U ( f ) = F [ u ( x ) ] = u ( x ) exp ( j 2 π f x ) dx .
u ( x ) = W W U ( f ) exp ( j 2 π f x ) df .
u T ( x ) = u ( x ) G ( x 2 a ) ,
G ( x ) = { 1 ; | x | ½ , 0 ; | x | > ½ .
F 1 [ S ( f ) ] = S ( f ) exp ( j 2 π f x ) df .
u ( t ) = n = 0 H n u T ( t ) .
H s ( t ) = 2 W [ s ( t ) * sinc 2 Wt ] [ 1 G ( t 2 a ) ] ,
H o s ( t ) = s ( t ) .
u ( x , y ) = W x W x W y W y U ( f x , f y ) exp [ j 2 π ( f x x + f y y ) ] df x df y ,
U ( f x , f y ) = u ( x , y ) exp [ j 2 π ( f x x + f y y ) ] dxdy ,
u ( x , y ) = n = 0 H n u T ( x , y ) ,
H s ( x , y ) = 4 W x W y [ s ( x , y ) * ( sinc 2 W x x sinc 2 W y y ) ] × [ 1 G ( x 2 a ) G ( y 2 a ) ] .
n = 0 ( p H ) n u T ( x , y ) ,

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