Abstract

The storage capacity of holographic associative memories is estimated. An argument based on the available degrees of freedom shows that the number of patterns that can be stored is limited by the space–bandwidth product of the hologram divided by the number of pixels in each pattern. A statistical calculation shows that if we attempt to store associations by multiply exposing the hologram, the cross talk among the stored items severely degrades the output fidelity. This confirms the storage capacity predicted by the degrees-of-freedom argument.

© 1986 Optical Society of America

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References

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  1. P. J. van Heerden, Appl. Opt. 2, 387 (1963).
    [CrossRef]
  2. D. Gabor, IBM J. Res. Devel. 13, 156 (1969).
    [CrossRef]
  3. H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
    [CrossRef]
  4. D. Psaltis, N. Farhat, Opt. Lett. 10, 98 (1985).
    [CrossRef] [PubMed]
  5. H. Mada, Appl. Opt. 24, 2063 (1985).
    [CrossRef] [PubMed]
  6. H. J. Caulfield, Opt. Commun. 55, 80 (1985).
    [CrossRef]
  7. B. Soffer, G. J. Dunning, Y. Owechko, E. Marom, Opt. Lett. 11, 118 (1986).
    [CrossRef] [PubMed]
  8. D. Anderson, Opt. Lett. 11, 56 (1986).
    [CrossRef] [PubMed]
  9. M. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 625, 30 (1986).
  10. A. Yariv et al., Proc. Soc. Photo-Opt. Instrum. Eng. 613, 1 (1986).
  11. Y. S. Abu-Mostafa, D. Psaltis, presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Miami Beach, Fla., November 1985.

1986

B. Soffer, G. J. Dunning, Y. Owechko, E. Marom, Opt. Lett. 11, 118 (1986).
[CrossRef] [PubMed]

D. Anderson, Opt. Lett. 11, 56 (1986).
[CrossRef] [PubMed]

M. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 625, 30 (1986).

A. Yariv et al., Proc. Soc. Photo-Opt. Instrum. Eng. 613, 1 (1986).

1985

1973

H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
[CrossRef]

1969

D. Gabor, IBM J. Res. Devel. 13, 156 (1969).
[CrossRef]

1963

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Miami Beach, Fla., November 1985.

Anderson, D.

Caulfield, H. J.

H. J. Caulfield, Opt. Commun. 55, 80 (1985).
[CrossRef]

Cohen, M.

M. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 625, 30 (1986).

Dunning, G. J.

Farhat, N.

Gabor, D.

D. Gabor, IBM J. Res. Devel. 13, 156 (1969).
[CrossRef]

Mada, H.

Mager, H. J.

H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
[CrossRef]

Marom, E.

Owechko, Y.

Psaltis, D.

D. Psaltis, N. Farhat, Opt. Lett. 10, 98 (1985).
[CrossRef] [PubMed]

Y. S. Abu-Mostafa, D. Psaltis, presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Miami Beach, Fla., November 1985.

Soffer, B.

van Heerden, P. J.

Waidelich, W.

H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
[CrossRef]

Wess, O.

H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
[CrossRef]

Yariv, A.

A. Yariv et al., Proc. Soc. Photo-Opt. Instrum. Eng. 613, 1 (1986).

Appl. Opt.

IBM J. Res. Devel.

D. Gabor, IBM J. Res. Devel. 13, 156 (1969).
[CrossRef]

Opt. Commun.

H. J. Mager, O. Wess, W. Waidelich, Opt. Commun. 9, 156 (1973).
[CrossRef]

H. J. Caulfield, Opt. Commun. 55, 80 (1985).
[CrossRef]

Opt. Lett.

Proc. Soc. Photo-Opt. Instrum. Eng.

M. Cohen, Proc. Soc. Photo-Opt. Instrum. Eng. 625, 30 (1986).

A. Yariv et al., Proc. Soc. Photo-Opt. Instrum. Eng. 613, 1 (1986).

Other

Y. S. Abu-Mostafa, D. Psaltis, presented at the IEEE Computer Society Workshop on Computer Architecture for Pattern Analysis and Image Database Management, Miami Beach, Fla., November 1985.

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Figures (2)

Fig. 1
Fig. 1

General holographic model.

Fig. 2
Fig. 2

Holographic recording.

Equations (15)

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g m ( k ) = j = 1 N 2 i = 1 N 1 R ( k , j ) t H ( j ) Q ( j , i ) f m ( i )             for k = 1 , 2 , , N 2 ,             m = 1 , 2 , , M ,
t H ( j ) = m = 1 M [ l = 1 N 1 Q * ( j , l ) f m * ( l ) ] [ n = 1 N 3 S ( j , n ) g m ( n ) ] ,             j = 1 , 2 , , N 2 ,
g ( k ) = j = 1 N 2 R ( k , j ) t H ( j ) i = 1 N 1 Q ( j , i ) f m 0 ( i ) ,             k = 1 , 2 , , N 3 ,
g ( k ) = i = 1 N 1 l = 1 N 1 j = l N 2 n = 1 N 3 R ( k , j ) S ( j , n ) Q * ( j , l ) Q ( j , i ) × f m 0 * ( l ) f m 0 ( i ) g m 0 ( n ) + α ( k ) , α ( k ) = m m 0 M i = 1 N 1 l = 1 N 1 j = l N 2 n = 1 N 3 R ( k , j ) S ( j , n ) Q * ( j , l ) Q ( j , i ) × f m * ( l ) f m 0 ( i ) g m ( n ) ,
E [ g ( k ) ] = n = 1 N 3 j = 1 N 2 R ( k , j ) S ( j , n ) i = 1 N 1 Q ( j , i ) 2 g m 0 ( n ) ,
E [ g ( k ) ] = n = 1 N 3 j = 1 N 2 R ( k , j ) S ( j , n ) g m 0 ( n ) .
j = 1 N 2 R ( k , j ) S ( j , n ) = c δ ( k , n ) ,
j = 1 N 2 S * ( j , k ) S ( j , n ) δ ( k , n ) .
S ( j , n ) = R * ( n , j ) ,             j = 1 N 2 R ( k , j ) R * ( n , j ) = c δ ( k , n ) .
σ 2 ( k ) = E [ α ( k ) 2 ] = ( M - 1 ) i = 1 N 1 l = 1 N 1 j 1 = 1 N 2 j 2 = 1 N 2 n = 1 N 3 R ( k , j 1 ) R * ( k , j 2 ) × R * ( n , j 1 ) R ( n , j 2 ) × Q * ( j 1 , l ) Q ( j 1 , i ) Q ( j 2 , l ) Q * ( j 2 , i ) .
σ 2 = ( 1 / N 3 ) k = 1 N 3 σ 2 ( k ) = ( M - 1 ) / N 3 j 1 = 1 N 2 j 2 = 1 N 2 | k = 1 N 3 R ( k , j 1 ) R * ( k , j 2 ) | 2 × | i = 1 N 1 Q ( j 1 , i ) Q * ( j 2 , i ) | 2 .
σ 2 [ ( M - 1 ) / N 3 ] × j = 1 N 2 [ | k = 1 N 3 R ( k , j ) 2 | 2 | i = 1 N 1 Q ( j , i ) 2 | 2 ] .
σ 2 [ ( M - 1 ) / N 3 ] j = 1 N 2 | k = 1 N 3 R ( k , j ) 2 | 2 = ( M - 1 ) c 2 N 3 / N 2 ,
SNR ( signal amplitude ) / σ [ N 2 / ( M - 1 ) N 3 ] 1 / 2 .
( M - 1 ) N 2 / N 3 ,

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