Abstract

A new neural network architecture for optical computing is proposed. This architecture can be used for implementing a simple system based on a modified theory of the associative memories. It is shown that the memorized patterns can be recalled perfectly by an experimental system using two microchannel spatial light modulators. The modified theory and principle of operation of the optical associatron system are discussed, and the system itself and basic experimental results are also described.

© 1989 Optical Society of America

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  1. K. Nakano, “Associatron—A Model of Associative Memory,” IEEE Trans. Syst. Man Cybern. SMC-2, 380 (1972).
    [CrossRef]
  2. T. Kohonen, “Correlation Matrix Memories,” IEEE Trans. Comput. C-21, 353 (1972).
    [CrossRef]
  3. S. Amari, “Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements,” IEEE Trans. Comput. C-21, 1197 (1972).
    [CrossRef]
  4. J. A. Anderson, “A Simple Neural Networks Generating Interactive Memory,” Math. Biosci. 14, 197 (1972).
    [CrossRef]
  5. H. Wigstrom, “A Neuron Model with Learning Capability and its Relation to Mechanism of Association,” Kybernetik 12, 204 (1973).
    [CrossRef] [PubMed]
  6. B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec., 96 (1960).
  7. T. Kohonen, “An Adaptive Associative Memory Principle,” IEEE Trans. Comput. C-23, 444 (1974).
    [CrossRef]
  8. T. Kohonen, Self-Organization and Associative Memory (Springer-Verlag, Berlin, 1984).
  9. S. Amari, “Neural Theory of Association and Concept-Formation,” Biol. Cybern. 26, 175 (1977).
    [CrossRef] [PubMed]
  10. A. D. Fisher, C. L. Giles, “Optical Adaptive Associatron Computer Architectures,” in Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85 (1985), p. 342.
  11. A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).
  12. N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical Implementation of Hopfield Model,” Appl. Opt. 24, 1469 (1985).
    [CrossRef] [PubMed]
  13. J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci. U.S.A. 79, 2554 (1982).
    [CrossRef]
  14. N. H. Farhat, D. Psaltis, “Optical Implementation of Associative Memory Based on Models of Neural Networks,” in Optical Signal Processing, J. L. Horner, Ed. (Academic Press, San Diego, 1987), p. 129.
  15. T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).
  16. C. Warde, J. Thackara, “Operating Modes of the Microchannel Spatial Light Modulator,” Opt. Eng. 22, 695 (1983).
    [CrossRef]
  17. J. A. McEwan, A. D. Fisher, J. N. Lee, “Four Special Functions of a Microchannel Spatial Light Modulator,” in Postdeadline Papers, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1985), paper 1.

1986

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

1985

A. D. Fisher, C. L. Giles, “Optical Adaptive Associatron Computer Architectures,” in Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85 (1985), p. 342.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).

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

1983

C. Warde, J. Thackara, “Operating Modes of the Microchannel Spatial Light Modulator,” Opt. Eng. 22, 695 (1983).
[CrossRef]

1982

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef]

1977

S. Amari, “Neural Theory of Association and Concept-Formation,” Biol. Cybern. 26, 175 (1977).
[CrossRef] [PubMed]

1974

T. Kohonen, “An Adaptive Associative Memory Principle,” IEEE Trans. Comput. C-23, 444 (1974).
[CrossRef]

1973

H. Wigstrom, “A Neuron Model with Learning Capability and its Relation to Mechanism of Association,” Kybernetik 12, 204 (1973).
[CrossRef] [PubMed]

1972

K. Nakano, “Associatron—A Model of Associative Memory,” IEEE Trans. Syst. Man Cybern. SMC-2, 380 (1972).
[CrossRef]

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

S. Amari, “Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements,” IEEE Trans. Comput. C-21, 1197 (1972).
[CrossRef]

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

1960

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec., 96 (1960).

Amari, S.

S. Amari, “Neural Theory of Association and Concept-Formation,” Biol. Cybern. 26, 175 (1977).
[CrossRef] [PubMed]

S. Amari, “Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements,” IEEE Trans. Comput. C-21, 1197 (1972).
[CrossRef]

Anderson, J. A.

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

Farhat, N. H.

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

N. H. Farhat, D. Psaltis, “Optical Implementation of Associative Memory Based on Models of Neural Networks,” in Optical Signal Processing, J. L. Horner, Ed. (Academic Press, San Diego, 1987), p. 129.

Fisher, A. D.

A. D. Fisher, C. L. Giles, “Optical Adaptive Associatron Computer Architectures,” in Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85 (1985), p. 342.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).

J. A. McEwan, A. D. Fisher, J. N. Lee, “Four Special Functions of a Microchannel Spatial Light Modulator,” in Postdeadline Papers, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1985), paper 1.

Fukuda, R. C.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).

Giles, C. L.

A. D. Fisher, C. L. Giles, “Optical Adaptive Associatron Computer Architectures,” in Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85 (1985), p. 342.

Hara, T.

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

Hoff, M. E.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec., 96 (1960).

Hopfield, J. J.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef]

Kato, T.

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

Kohonen, T.

T. Kohonen, “An Adaptive Associative Memory Principle,” IEEE Trans. Comput. C-23, 444 (1974).
[CrossRef]

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

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

Lee, J. N.

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).

J. A. McEwan, A. D. Fisher, J. N. Lee, “Four Special Functions of a Microchannel Spatial Light Modulator,” in Postdeadline Papers, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1985), paper 1.

McEwan, J. A.

J. A. McEwan, A. D. Fisher, J. N. Lee, “Four Special Functions of a Microchannel Spatial Light Modulator,” in Postdeadline Papers, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1985), paper 1.

Nakano, K.

K. Nakano, “Associatron—A Model of Associative Memory,” IEEE Trans. Syst. Man Cybern. SMC-2, 380 (1972).
[CrossRef]

Ooi, Y.

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

Paek, E.

Prata, A.

Psaltis, D.

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

N. H. Farhat, D. Psaltis, “Optical Implementation of Associative Memory Based on Models of Neural Networks,” in Optical Signal Processing, J. L. Horner, Ed. (Academic Press, San Diego, 1987), p. 129.

Suzuki, Y.

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

Thackara, J.

C. Warde, J. Thackara, “Operating Modes of the Microchannel Spatial Light Modulator,” Opt. Eng. 22, 695 (1983).
[CrossRef]

Warde, C.

C. Warde, J. Thackara, “Operating Modes of the Microchannel Spatial Light Modulator,” Opt. Eng. 22, 695 (1983).
[CrossRef]

Widrow, B.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec., 96 (1960).

Wigstrom, H.

H. Wigstrom, “A Neuron Model with Learning Capability and its Relation to Mechanism of Association,” Kybernetik 12, 204 (1973).
[CrossRef] [PubMed]

Appl. Opt.

Biol. Cybern.

S. Amari, “Neural Theory of Association and Concept-Formation,” Biol. Cybern. 26, 175 (1977).
[CrossRef] [PubMed]

IEEE Trans. Comput.

T. Kohonen, “An Adaptive Associative Memory Principle,” IEEE Trans. Comput. C-23, 444 (1974).
[CrossRef]

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

S. Amari, “Learning Patterns and Pattern Sequences by Self-Organizing Nets of Threshold Elements,” IEEE Trans. Comput. C-21, 1197 (1972).
[CrossRef]

IEEE Trans. Syst. Man Cybern.

K. Nakano, “Associatron—A Model of Associative Memory,” IEEE Trans. Syst. Man Cybern. SMC-2, 380 (1972).
[CrossRef]

IRE WESCON Conv. Rec.

B. Widrow, M. E. Hoff, “Adaptive Switching Circuits,” IRE WESCON Conv. Rec., 96 (1960).

Kybernetik

H. Wigstrom, “A Neuron Model with Learning Capability and its Relation to Mechanism of Association,” Kybernetik 12, 204 (1973).
[CrossRef] [PubMed]

Math. Biosci.

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

Opt. Eng.

C. Warde, J. Thackara, “Operating Modes of the Microchannel Spatial Light Modulator,” Opt. Eng. 22, 695 (1983).
[CrossRef]

Proc. Nat. Acad. Sci. U.S.A.

J. J. Hopfield, “Neural Networks and Physical Systems with Emergent Collective Computational Abilities,” Proc. Nat. Acad. Sci. U.S.A. 79, 2554 (1982).
[CrossRef]

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

A. D. Fisher, R. C. Fukuda, J. N. Lee, “Implementations of Adaptive Associative Optical Computing Elements,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 196 (1985).

T. Hara, Y. Ooi, T. Kato, Y. Suzuki, “Microchannel Spatial Light Modulator with Improved Resolution and Contrast Ratio,” Proc. Soc. Photo-Opt. Instrum. Eng. 613, 153 (1986).

Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85

A. D. Fisher, C. L. Giles, “Optical Adaptive Associatron Computer Architectures,” in Proceedings, IEEE 1985 COMPCON Spring Meeting, Catalog No. CH2135-2/85 (1985), p. 342.

Other

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

J. A. McEwan, A. D. Fisher, J. N. Lee, “Four Special Functions of a Microchannel Spatial Light Modulator,” in Postdeadline Papers, Conference on Lasers and Electro-Optics (Optical Society of America, Washington, DC, 1985), paper 1.

N. H. Farhat, D. Psaltis, “Optical Implementation of Associative Memory Based on Models of Neural Networks,” in Optical Signal Processing, J. L. Horner, Ed. (Academic Press, San Diego, 1987), p. 129.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of a spatial coding method: (a) method for calculating autocorrelation matrix xxT in the learning process; (b) method for calculating the vector–matrix product Mx′ in the recalling process. In these figures, elements of the matrix are numbered in the lexical order as shown in Eq. (13).

Fig. 2
Fig. 2

All-optical associatron system using four MSLMs, three optical shutters, an optical magnifying unit, an optical multi-imaging unit, and an optical local accumulating unit.

Fig. 3
Fig. 3

Experimental optical associatron system based on the electrical and optical hybrid architecture. This system uses two MSLMs, two LED arrays, a PTR array, an electrical analog processing unit, and a computer (HP-1000). (a) Schematic diagram of this system. (b) The optical part of the system.

Fig. 4
Fig. 4

Spatial arrangement of the blocks in the memory matrix.

Fig. 5
Fig. 5

Shading of the MSLM measured by a SIT camera and image processor. The result shows a maximum, minimum, and average intensity of the readout patterns in each block when the MSLM is written by a uniform LED array. The operating conditions are Vb (the voltage of the back surface) = 0.8 kV and TPW (writing time) = 4 s.

Fig. 6
Fig. 6

Writing characteristics of the MSLM. The data show the transient characteristics for the writing mode. Such data—the basis for optimizing blur and peak intensities of a memorized pattern by controlling the writing time—are measured. (a) Halfwidth of the intensity distribution for the readout image of an LED pattern. (b) Peak intensity of the readout image of an LED pattern. These measurements were made with the same readout characteristics as shown in Fig. 5.

Fig. 7
Fig. 7

Learning input patterns and simulated memory matrix for the optical associatron.

Fig. 8
Fig. 8

Readout intensity of a memory matrix memorized in the MSLM. This is measured by a SIT camera and image processor. The result shows a maximum, minimum, and average intensity around the pixels of level 1 in each block. The data of the pixels of level 2 are shown by triangles.

Fig. 9
Fig. 9

Schematic diagram showing an example of the experimental recalling process. The photograph of the memory matrix from a simulated data is involved. The analog pattern through the local accumulating unit is measured by a SIT camera and processed by an image processor, and the output pattern is displayed by the LED monitor.

Fig. 10
Fig. 10

Memorized patterns have been recalled perfectly by the optical associatron system from the distorted recalling input patterns involving noise and defects. The figure shows the input patterns M′ ⊗ Xmlt, the measured analog patterns through the local accumulating operation, and the displayed patterns of the LED monitor for three ideal recalling input patterns and seven distorted recalling input patterns.

Fig. 11
Fig. 11

Example of the learning process for the recalling input patterns LR, UL, and CC. The learning pattern shows the recalling input pattern from MSLM 1. The memory matrix shows the temporary memorized pattern of MSLM 1. The output pattern shows the temporary output of the LED monitor.

Fig. 12
Fig. 12

Example of the orthogonal learning by the adaptive method for nonorthogonal input patterns such as the characters A, b, and C.

Equations (32)

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

x = ( x 1 , x 2 , , x r ) T ,
y = ϕ out [ ϕ M ( M ) * ϕ in ( x ) ] ,
ϕ out ( x ) = [ ϕ out ( x i ) ] , ϕ in ( x ) = [ ϕ in ( x i ) ] , ϕ M ( M ) = [ ϕ M ( M i j ) ] , ( i , j = 1 , 2 , , r ) ,
y i = ϕ out { j = 1 r [ ϕ M ( M i j ) ϕ in ( x j ) ] } , ( i = 1 , 2 , , r ) .
ϕ out ( a ) = ϕ M ( a ) = { - 1 a < 0 , 0 a = 0 , 1 a > 0.
M = x 1 x 1 T + x 2 x 2 T + + x k x k T .
x i 0             ( i = 1 , 2 , , r ) .
M i j 0 ( i , j = 1 , 2 , , r ) .
M t + 1 = ϕ p [ M t + α ( x - x t ) x T ]             ( t = 0 , 1 , ) ,
x t = ϕ out [ ϕ M ( M t ) * ϕ in ( x ) ] ,
ϕ p ( a ) = { a ( a 0 ) , 0 ( a < 0 ) ,
X = [ x 11 , x 12 , , x 1 n x 21 , x 22 , x n 1 , , x n n ] .
x = ( x 11 , x 12 , , x 1 n , x 21 , x 22 , , x n 1 , , x n n ) T .
x = S 2 - 1 ( X ) ,
X = S 1 - 2 ( x ) .
M = X X ,
M = [ x 11 X , x 12 X , , x 1 n X x 21 X , x 22 X , w x n 1 X , , x n n X ] .
x i j X = x i j S 1 - 2 ( x ) = S 1 - 2 ( x i j x ) ( i , j = 1 , 2 , , n ) ,
M = [ S 1 - 2 ( x 11 x ) , S 1 - 2 ( x 12 x ) , , S 1 - 2 ( x 1 n x ) S 1 - 2 ( x 21 x ) , S 1 - 2 ( x 22 x ) , S 1 - 2 ( x n 1 x ) , , S 1 - 2 ( x n n x ) ] = S M - M ( x x T ) .
M = X mag X mlt ,
X mag = S mag ( X ) = [ X 11 , X 12 , , X 1 n X 21 , X 22 , X n 1 , , X n n ] ,
X i j = [ x i j , x i j , , x i j x i j , x i j , x i j , , x i j ]
X mlt = S mlt ( X ) = [ X , X , , X X , X , X , , X ] ,
y = M x .
Y = S 1 - 2 ( y ) = Σ n × n ( M X mlt ) ,
X mlt = S mlt ( X ) ,
X = S 1 - 2 ( x ) ,
Z ij = S 1 - 2 ( z i j ) .
Σ n × n [ Z 11 , Z 12 , , Z 1 n Z 21 , Z 22 , Z n 1 , , Z n n ] = [ Σ Σ z k l 11 , Σ Σ z k l 12 , , Σ Σ z k l 1 n Σ Σ z k l 21 , Σ Σ z k l 22 , Σ Σ z k l n 1 , , Σ Σ z k l n n ] .
Σ n × n ( M X mlt ) = Σ n × n × [ S 1 - 2 ( m 11 ) S 1 - 2 ( x ) , S 1 - 2 ( m 12 ) S 1 - 2 ( x ) , , S 1 - 2 ( m 1 n ) S 1 - 2 ( x ) S 1 - 2 ( m 21 ) S 1 - 2 ( x ) , S 1 - 2 ( m 22 ) S 1 - 2 ( x ) , S 1 - 2 ( m n 1 ) S 1 - 2 ( x ) , , S 1 - 2 ( m n n ) S 1 - 2 ( x ) ] = [ m 11 x , m 12 x , , m 1 n x m 21 x , m 22 x , m n 1 x , , m n n x ] = S 1 - 2 ( M x ) ,
M t + 1 = M t + α X mag X mlt - α S mag { ϕ out [ Σ n × n ( M t X mlt ) ] } X mlt .
Y = ϕ out [ Σ n × n ( M X mlt ) ] .

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