Abstract

A relationship is derived between the irradiance moments of an object’s and the irradiance moments of the object’s image. Accurate knowledge of the system’s point-spread-function irradiance moments permits inversion of this relationship. It is shown that the inversion is well conditioned, and Monte-Carlo simulation results of a two-point object that demonstrate super resolution are presented.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. R. Teague, J. Opt. Soc. Am. 70, 920 (1980).
    [CrossRef]
  2. M. K. Hu, IRE Trans. Inf. Theory IT-8, 179 (1962).
  3. S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
    [CrossRef]
  4. Y. S. Abu-Mostafa, D. Psaltis, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
    [CrossRef]
  5. S. Maitra, Proc. IEEE 67, 697 (1979).
    [CrossRef]
  6. Y. Li, Jp. J. Appl Phys. 30, (1991).
  7. Y. Li, Pattern Recogn. 25, 723 (1992).
    [CrossRef]
  8. T. W. Körner, Fourier Analysis (Cambridge U. Press, Cambridge, 1989), pp. 21–23.
  9. R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991), p. 41.

1992 (1)

Y. Li, Pattern Recogn. 25, 723 (1992).
[CrossRef]

1991 (1)

Y. Li, Jp. J. Appl Phys. 30, (1991).

1985 (1)

Y. S. Abu-Mostafa, D. Psaltis, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

1980 (1)

1979 (1)

S. Maitra, Proc. IEEE 67, 697 (1979).
[CrossRef]

1977 (1)

S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

1962 (1)

M. K. Hu, IRE Trans. Inf. Theory IT-8, 179 (1962).

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

Breeding, K. J.

S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Dudari, S. A.

S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Hu, M. K.

M. K. Hu, IRE Trans. Inf. Theory IT-8, 179 (1962).

Körner, T. W.

T. W. Körner, Fourier Analysis (Cambridge U. Press, Cambridge, 1989), pp. 21–23.

Li, Y.

Y. Li, Pattern Recogn. 25, 723 (1992).
[CrossRef]

Y. Li, Jp. J. Appl Phys. 30, (1991).

Maitra, S.

S. Maitra, Proc. IEEE 67, 697 (1979).
[CrossRef]

Marks, R. J.

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991), p. 41.

McGhee, R. B.

S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Psaltis, D.

Y. S. Abu-Mostafa, D. Psaltis, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

Teague, M. R.

IEEE Trans. Comput. (1)

S. A. Dudari, K. J. Breeding, R. B. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. S. Abu-Mostafa, D. Psaltis, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, IRE Trans. Inf. Theory IT-8, 179 (1962).

J. Opt. Soc. Am. (1)

Jp. J. Appl Phys. (1)

Y. Li, Jp. J. Appl Phys. 30, (1991).

Pattern Recogn. (1)

Y. Li, Pattern Recogn. 25, 723 (1992).
[CrossRef]

Proc. IEEE (1)

S. Maitra, Proc. IEEE 67, 697 (1979).
[CrossRef]

Other (2)

T. W. Körner, Fourier Analysis (Cambridge U. Press, Cambridge, 1989), pp. 21–23.

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991), p. 41.

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 (4)

Fig. 1
Fig. 1

Condition numbers of the moment imaging matrix versus moment order for the PSF used in the simulations.

Fig. 2
Fig. 2

Simulation results for I = 100 photons and an image size of 11 × 11 pixels.

Fig. 3
Fig. 3

Simulation results for I = 1000 photons and an image size of 11 × 11 pixels. (Note: error bars are very small at larger separations.)

Fig. 4
Fig. 4

Simulation results for I = 1000 photons and an image size of 21 × 21 pixels.

Equations (12)

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

m f j , k - - x j y k f ( x , y ) d x d y .
G ( w x , w y ) = F ( w x , w y ) H ( w x , w y ) ,
G ( k 1 , k 2 ) ( w x , w y ) = j 2 = 0 k 2 ( k 2 j 2 ) j 1 = 0 k 1 ( k 1 j 1 ) H ( j 1 , j 2 ) ( w x , w y ) × F ( k 1 - j 1 , k 2 - j 2 ) ( w x , w y ) ,
G ( k 1 , k 2 ) ( w x , w y ) k 1 w x k 1 k 2 w y k 2 G ( w x , w y ) .
G ( k 1 , k 2 ) ( 0 , 0 ) = ( - j ) k 1 + k 2 - - x k 1 y k 2 g ( x , y ) d x d y .
m g k 1 , k 2 = j 2 = 0 k 2 ( k 2 j 2 ) j 1 = 0 k 1 ( k 1 j 1 ) m h j 1 , j 2 m f k 1 - j 1 , k 2 - j 2 .
[ m g 00 m g 01 m g 10 m g 11 m g 02 m g 20 ] = [ m h 00 0 0 0 0 0 m h 01 m h 00 0 0 0 0 m h 10 0 m h 00 0 0 0 m h 11 m h 10 m h 01 m h 00 0 0 m h 02 2 m h 01 0 0 m h 00 0 m h 20 0 2 m h 10 0 0 m h 00 ] [ m f 00 m f 01 m f 10 m f 11 m f 02 m f 20 ] .
M g k l = m = - N N n = - N N ( m / N ) k ( n / N ) l g ( m , n ) .
f ( x , y ) = I [ δ ( x + d / 2 , y ) + δ ( x - d / 2 , y ) ] .
h ( x , y ) = J 1 2 ( π x 2 + y 2 ) π ( x 2 + y 2 ) ,
g ( x , y ) = I [ h ( x + d / 2 , y ) + h ( x - d / 2 , y ) ] .
d ^ = 2 m ^ f 20 / m ^ f 00 .

Metrics