Abstract

We describe a shift-invariant all-optical holographic associative memory implemented using phase conjugate mirrors and Fourier transform holograms. A key feature of our system is the large storage capacity obtained through the use of nonlinearities in the correlation domain. The use of angularly multiplexed plane wave reference beams allows access to the correlation domain where nonlinearities in the phase conjugate mirrors can be used to reduce greatly crosstalk and correlation noise.

© 1987 Optical Society of America

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  1. T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).
  2. For a review of this field, see Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.
  3. For a review of the associative properties of holograms, see R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  4. H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
    [CrossRef]
  5. R. J. Collier, K. S. Pennington, “Ghost Imaging by Holograms Formed in The Near Field,” Appl. Phys. Lett. 8, 44 (1966).
    [CrossRef]
  6. E. Marom, J. W. Goodman, “Ghost Image Interferometry,” Appl. Opt. 17, 172 (1978).
    [CrossRef] [PubMed]
  7. B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory with Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118 (1986).
    [CrossRef] [PubMed]
  8. G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “Optical Holographic Associative Memory Using a Phase Conjugate Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 205 (1986).
  9. D. Z. Anderson, “Coherent Optical Eigenstate Memory,” Opt. Lett. 11, 45 (1986).
    [CrossRef]
  10. A. Yariv, S. K. Kwong, “Associative Memories Based on Message-Bearing Optical Modes in Phase-Conjugate Resonators,” Opt. Lett. 11, 186 (1986).
    [CrossRef] [PubMed]
  11. G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
    [CrossRef]
  12. R. A. Athale, H. H. Szu, C. B. Friedlander, “Optical Implementation of Associative Memory with Controlled Nonlinearity in the Correlation Domain,” Opt. Lett. 11, 482 (1986).
    [CrossRef] [PubMed]
  13. D. Psaltis, C. H. Park, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.
  14. H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.
  15. W. S. McCulloch, W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bull. Math. Biophys. 5, 115 (1943).
    [CrossRef]
  16. S. Grossberg, “On Learning and Energy-Entropy Dependence in Recurrent and Nonrecurrent Signed Networks,” J. Stat. Phys. 1, 319 (1969).
    [CrossRef]
  17. J. A. Anderson, “A Simple Neural Network Generating an Interactive Memory,” Math. Biosci. 14, 197 (1972).
    [CrossRef]
  18. T. Kohonen, “Correlation Matrix Memories,” IEEE Trans. Comput. C-21, 353 (1972).
    [CrossRef]
  19. J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
    [CrossRef] [PubMed]
  20. Y. S. Abu-Mostafa, J. N. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
    [CrossRef]
  21. D. Gabor, “Associative Holographic Memories,” IBM J. Res. Dev. 156 (1969).
  22. D. Psaltis, J. Hong, S. Venkatest, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189 (1986).
  23. E.-L. Kral, J. F. Walkup, M. O. Hagler, “Optical Frequency Shifter for Heterodyne Interferometry Using Counterrotating Wave Plates,” Appl. Opt. 21, 1281 (1982).
    [CrossRef] [PubMed]
  24. N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical Implementation of the Hopfield Method,” Appl. Opt. 24, 1469 (1985).
    [CrossRef] [PubMed]
  25. K. H. Brenner, A. Huang, N. Streibl, “Digital Optical Computing with Symbolic Substitution,” Appl. Opt. 25, 3054 (1986).
    [CrossRef] [PubMed]

1986 (7)

1985 (2)

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical Implementation of the Hopfield Method,” Appl. Opt. 24, 1469 (1985).
[CrossRef] [PubMed]

Y. S. Abu-Mostafa, J. N. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

1982 (2)

E.-L. Kral, J. F. Walkup, M. O. Hagler, “Optical Frequency Shifter for Heterodyne Interferometry Using Counterrotating Wave Plates,” Appl. Opt. 21, 1281 (1982).
[CrossRef] [PubMed]

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

1978 (1)

1972 (2)

J. A. Anderson, “A Simple Neural Network Generating an Interactive Memory,” Math. Biosci. 14, 197 (1972).
[CrossRef]

T. Kohonen, “Correlation Matrix Memories,” IEEE Trans. Comput. C-21, 353 (1972).
[CrossRef]

1969 (2)

S. Grossberg, “On Learning and Energy-Entropy Dependence in Recurrent and Nonrecurrent Signed Networks,” J. Stat. Phys. 1, 319 (1969).
[CrossRef]

D. Gabor, “Associative Holographic Memories,” IBM J. Res. Dev. 156 (1969).

1968 (1)

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

1966 (1)

R. J. Collier, K. S. Pennington, “Ghost Imaging by Holograms Formed in The Near Field,” Appl. Phys. Lett. 8, 44 (1966).
[CrossRef]

1965 (1)

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

1943 (1)

W. S. McCulloch, W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bull. Math. Biophys. 5, 115 (1943).
[CrossRef]

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, J. N. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

Anderson, D. Z.

Anderson, J. A.

J. A. Anderson, “A Simple Neural Network Generating an Interactive Memory,” Math. Biosci. 14, 197 (1972).
[CrossRef]

Athale, R. A.

Brenner, K. H.

Brumm, D.

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

Burckhardt, C. B.

For a review of the associative properties of holograms, see R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Chen, H. H.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

Collier, R. J.

R. J. Collier, K. S. Pennington, “Ghost Imaging by Holograms Formed in The Near Field,” Appl. Phys. Lett. 8, 44 (1966).
[CrossRef]

For a review of the associative properties of holograms, see R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Dunning, G. J.

B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory with Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118 (1986).
[CrossRef] [PubMed]

G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “Optical Holographic Associative Memory Using a Phase Conjugate Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 205 (1986).

Farhat, N. H.

Friedlander, C. B.

Funkhouser, A.

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

Gabor, D.

D. Gabor, “Associative Holographic Memories,” IBM J. Res. Dev. 156 (1969).

Giles, C. L.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

Goodman, J. W.

Grossberg, S.

S. Grossberg, “On Learning and Energy-Entropy Dependence in Recurrent and Nonrecurrent Signed Networks,” J. Stat. Phys. 1, 319 (1969).
[CrossRef]

Hagler, M. O.

Hong, J.

D. Psaltis, J. Hong, S. Venkatest, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189 (1986).

Hopfield, J. J.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Huang, A.

Jacques, J. N. St.

Y. S. Abu-Mostafa, J. N. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

Kohonen, T.

T. Kohonen, “Correlation Matrix Memories,” IEEE Trans. Comput. C-21, 353 (1972).
[CrossRef]

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).

Kral, E.-L.

Kwong, S. K.

Lee, H. Y.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

Lee, Y. C.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

Lin, L. H.

For a review of the associative properties of holograms, see R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

Marom, E.

Maxwell, T.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

McCulloch, W. S.

W. S. McCulloch, W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bull. Math. Biophys. 5, 115 (1943).
[CrossRef]

Owechko, Y.

B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory with Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118 (1986).
[CrossRef] [PubMed]

G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “Optical Holographic Associative Memory Using a Phase Conjugate Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 205 (1986).

Paek, E.

Park, C. H.

D. Psaltis, C. H. Park, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.

Pennington, K. S.

R. J. Collier, K. S. Pennington, “Ghost Imaging by Holograms Formed in The Near Field,” Appl. Phys. Lett. 8, 44 (1966).
[CrossRef]

Pitts, W.

W. S. McCulloch, W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bull. Math. Biophys. 5, 115 (1943).
[CrossRef]

Prata, A.

Psaltis, D.

D. Psaltis, J. Hong, S. Venkatest, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189 (1986).

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical Implementation of the Hopfield Method,” Appl. Opt. 24, 1469 (1985).
[CrossRef] [PubMed]

D. Psaltis, C. H. Park, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.

Restrick, R.

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

Soffer, B. H.

B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory with Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118 (1986).
[CrossRef] [PubMed]

G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “Optical Holographic Associative Memory Using a Phase Conjugate Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 205 (1986).

Streibl, N.

Stroke, G. W.

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

Sun, G. Z.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

Szu, H. H.

Venkatest, S.

D. Psaltis, J. Hong, S. Venkatest, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189 (1986).

Walkup, J. F.

Yariv, A.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. J. Collier, K. S. Pennington, “Ghost Imaging by Holograms Formed in The Near Field,” Appl. Phys. Lett. 8, 44 (1966).
[CrossRef]

Bull. Math. Biophys. (1)

W. S. McCulloch, W. Pitts, “A Logical Calculus of the Ideas Immanent in Nervous Activity,” Bull. Math. Biophys. 5, 115 (1943).
[CrossRef]

IBM J. Res. Dev. (1)

D. Gabor, “Associative Holographic Memories,” IBM J. Res. Dev. 156 (1969).

IEEE Trans. Comput. (1)

T. Kohonen, “Correlation Matrix Memories,” IEEE Trans. Comput. C-21, 353 (1972).
[CrossRef]

IEEE Trans. Inf. Theory (1)

Y. S. Abu-Mostafa, J. N. St. Jacques, “Information Capacity of the Hopfield Model,” IEEE Trans. Inf. Theory IT-31, 461 (1985).
[CrossRef]

J. Stat. Phys. (1)

S. Grossberg, “On Learning and Energy-Entropy Dependence in Recurrent and Nonrecurrent Signed Networks,” J. Stat. Phys. 1, 319 (1969).
[CrossRef]

Math. Biosci. (1)

J. A. Anderson, “A Simple Neural Network Generating an Interactive Memory,” Math. Biosci. 14, 197 (1972).
[CrossRef]

Nature London (1)

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. (1)

G. W. Stroke, R. Restrick, A. Funkhouser, D. Brumm, “Resolution-Retrieving Compensation of Source Effects by Correlative Reconstruction in High-Resolution Holography,” Phys. Lett. 18, 274 (1965).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

D. Psaltis, J. Hong, S. Venkatest, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189 (1986).

G. J. Dunning, E. Marom, Y. Owechko, B. H. Soffer, “Optical Holographic Associative Memory Using a Phase Conjugate Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 205 (1986).

Other (5)

T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, New York, 1984).

For a review of this field, see Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.

For a review of the associative properties of holograms, see R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

D. Psaltis, C. H. Park, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16 Apr. 1986.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, in Proceedings, Neural Networks for Computing Conference, Snowbird, UT, 13–16, Apr. 1986.

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Figures (10)

Fig. 1
Fig. 1

Recording of data in the holographic analog to the outer product model of associative memory.

Fig. 2
Fig. 2

Readout of data in the holographic analog to the outer-product model of associative memory.

Fig. 3
Fig. 3

Output of associative memory represented as vectors in N-dimensional state space.

Fig. 4
Fig. 4

Recording of data in the reference-based associative memory.

Fig. 5
Fig. 5

Readout of data in the reference-based associative memory.

Fig. 6
Fig. 6

Configuration used in associative memory experiments.

Fig. 7
Fig. 7

Comparison of storage capacity of reference-based associative memory with the outer-product model for various nonlinearities n in the correlation domain. Error-free input objects assumed.

Fig. 8
Fig. 8

(a) Output of reference-based associative memory for unshifted input âmo. (b) Output of reference-based associative memory for input âmo shifted by the maximum amount allowed without ambiguity or crosstalk from other objects.

Fig. 9
Fig. 9

Reconstruction of gray-scale image from partial input: (a) image stored in memory; (b) incomplete input image; (c) associated output image (inversion due to mirror reflection).

Fig. 10
Fig. 10

Reconstruction of complete objects from partial input objects when multiple objects are stored: (a) images stored in memory shown displaced; (b) superimposed images shown as recorded; (c) partial input images; (d) corresponding associated output images (inversions due to mirror reflection).

Equations (22)

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T i j = m V i m V j m ,             0 for i = j .
V ˚ i m o = N . L . ( j T i j V ^ j m o ) ,
V ˚ i m o = N . L . [ j m V i m V j m V ^ j m o ] = N . L . [ V i m o j V j m o V ^ j m o + m m o V i m j V j m V ^ j m o ] = N . L . [ ( 2 N m o - N ) V i m o _ + m m o ( V m , V ^ m o ) V i m ] ,
η i = E [ V ˚ i m o ] = ( 2 N m o - N ) V i m o , σ i 2 = E [ ( V ˚ i m o - η i ) 2 ] = ( M - 1 ) ( N - 1 ) ,
SNR = η i σ = 2 N m o - N [ ( M - 1 ) ( N - 1 ) ] 2 β - 1 N / M ,
a ˚ m o = ( a ^ m o a m o ) * a m o + m m o ( a ^ m o a m ) * a m ,
a ˚ i m o = k C k m o a i - k m o + m m o k C k m a i - k m = C 0 m o a i m o ( signal ) + k 0 C k m o a i - k m o ( autocorrelation     noise ) + m m o k C k m a i - k m ( cross - correlation noise ) ,
C k m = j a ^ j - k m o a i m
SNR = C 0 m o [ k 0 C k m o 2 + m 0 k C k m 2 ] .
C k m = N δ ( k ) + 2 ( N - k ) / 3 , if m = m o ( autocorrelation ) , = 2 ( N - k ) / 3 , if m m o ( cross - correlation ) ,
SNR = ( 3 N / 2 + 1 ) β M N - 1 3 / 2 β M             ( N 1 ) ,
a ˚ m o = F [ m ( f { m ( a ^ m o a m ) * b m } ) b m * a m ] ,
b m = δ ( x - x m ) .
a ˚ m o ( x ) = F { m m f [ C m ( x - x m + x m ) ] * a m ( x ) } ,
C i m = j a ^ j - i m o a j m ,
a ˚ i m o = F [ m m k f ( C i - i m + i m - k m ) a k m ] ,
a ˚ i m o = F [ m m p f ( C p m ) a i - i m + i p - p m ] .
a ˚ i m o = F [ m p f ( C p m ) a i - p m ] .
SNR = f ( C 0 m o ) [ p 0 f ( C p m o ) 2 + m m o p f ( C p m ) 2 ] .
SNR = ( 3 N / 2 + 1 ) n β 2 M N / ( n + 1 ) - 1 n + 1 2 ( 3 2 ) n / 2 β N ( n - 1 ) / 2 M ,             N 1 ,
T i j 1 j 2 j n = m V i m V j 1 m V j 2 m V j n m .
V i m o = N . L . [ j 1 j 2 T i j 1 j 2 j n V ^ j 1 m o V ^ j 2 m o V ^ j n m o ] j n = N . L . [ ( V m o , V ^ m o ) n V i m o + m m o ( V m , V ^ m o ) n V i m ] .

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