Abstract

Given only a portion of an arbitrary finite-energy [L2(−∞, ∞)] image and a portion of that image's spectrum, we present a closed-form method by which the entire image can be reconstructed. The algorithm is a basic augmentation of a recently proposed iterative restoration algorithm presented by Stark et al. [J. Opt. Soc. Am. 71, 635 (1981); Opt. Lett. 6, 259 (1981)]. Experimental results are presented.

© 1981 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Youla, “Generalized image restoration by method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  2. H. Stark, D. Cahana, H. Webb, “Restoration of arbitrary finite-energy optical objects from limited spatial and spectral information,” J. Opt. Soc. Am. 71, 635–642 (1981).
    [CrossRef]
  3. H. Stark, D. Cahana, G. J. Habetler, “Is it possible to restore an optical object from its low-pass spectrum and its truncated image?” Opt. Lett. 6, 259–260 (1981).
    [CrossRef] [PubMed]
  4. M. S. Sabri, W. Steenaart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
    [CrossRef]
  5. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  6. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  7. M. S. Sabri, W. Steenaart, “Discrete Hilbert transform filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 452–454 (1977).
    [CrossRef]
  8. T. K. Sarkar, D. D. Weiner, V. K. Jain, “Some mathematical considerations in dealing with the inverse problem,” IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
    [CrossRef]
  9. D. K. Smith, R. J. Marks, “Closed form bandlimited image extrapolation,” Appl. Opt. 20, 2476–2483 (1981).
    [CrossRef] [PubMed]

1981 (4)

1978 (2)

D. C. Youla, “Generalized image restoration by method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

M. S. Sabri, W. Steenaart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

1977 (1)

M. S. Sabri, W. Steenaart, “Discrete Hilbert transform filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 452–454 (1977).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Cahana, D.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Habetler, G. J.

Jain, V. K.

T. K. Sarkar, D. D. Weiner, V. K. Jain, “Some mathematical considerations in dealing with the inverse problem,” IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

Marks, R. J.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Sabri, M. S.

M. S. Sabri, W. Steenaart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

M. S. Sabri, W. Steenaart, “Discrete Hilbert transform filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 452–454 (1977).
[CrossRef]

Sarkar, T. K.

T. K. Sarkar, D. D. Weiner, V. K. Jain, “Some mathematical considerations in dealing with the inverse problem,” IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

Smith, D. K.

Stark, H.

Steenaart, W.

M. S. Sabri, W. Steenaart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

M. S. Sabri, W. Steenaart, “Discrete Hilbert transform filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 452–454 (1977).
[CrossRef]

Webb, H.

Weiner, D. D.

T. K. Sarkar, D. D. Weiner, V. K. Jain, “Some mathematical considerations in dealing with the inverse problem,” IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

Youla, D. C.

D. C. Youla, “Generalized image restoration by method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

M. S. Sabri, W. Steenaart, “Discrete Hilbert transform filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 452–454 (1977).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

T. K. Sarkar, D. D. Weiner, V. K. Jain, “Some mathematical considerations in dealing with the inverse problem,” IEEE Trans. Antennas Propag. AP-29, 373–379 (1981).
[CrossRef]

IEEE Trans. Circuits Syst. (3)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

D. C. Youla, “Generalized image restoration by method of alternating projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

M. S. Sabri, W. Steenaart, “An approach to band-limited signal extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Lett. (1)

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 (6)

Fig. 1
Fig. 1

Restoration of a pulse from Pb f and Qaf.

Fig. 2
Fig. 2

Restoration of a single-sided exponential from Pbf and Qaf.

Fig. 3
Fig. 3

Restoration of a double-sided exponential from= Pbf and Qaf.

Fig. 4
Fig. 4

Restoration of a pulse from Pbf and Paf. The vectors f and Pb f are shown in Fig. 1.

Fig. 5
Fig. 5

Restoration of a single-sided exponential from Pbf and Paf. The vectors f and Pbf are shown in Fig. 2.

Fig. 6
Fig. 6

Restoration of a double-sided exponential from Pbf and Paf. The vectors f and Pbf are shown in Fig. 3.

Equations (15)

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

f k ( x ) = r = 0 k 1 ( Q b P a ) r m 1 ( x ) ,
m 1 ( x ) = P b f ( x ) + Q b [ Q a f ( x ) ] .
P a = diag { 0 , 0 , , 0 , 1 , , 1 , 1 , 1 , , 1 , 0 , , 0 , 0 } ,
P b = 1 Q b
P b = D 1 B D ,
B = diag { 0 , 0 , , 0 , 1 , , 1 , 1 , 1 , , 1 , 0 , , 0 , 0 }
Q b = I P b ,
f k = r = 0 k 1 ( Q b P a ) r m 1 = R k m 1 ,
R k = r = 0 k 1 ( Q b P a ) r
R R = r = 0 ( Q b P a ) r = ( I Q b P a ) 1 ,
f = R m 1 .
m 1 = P b f + Q b Q a f .
R = ( I Q a Q b ) 1 .
r = P a f + Q a P b f
f = R r .

Metrics