Abstract

An architecture capable of performing the outer product operation followed by a weighting of the resultant is presented in this paper. By using a polarization encoding technique, the spatial capacity is elements increased by a factor of four over implementations using other techniques. The use of space-integrating photodiodes as detectors permits the evaluation of second-order polynomials. Spatial multiplexing allows the architecture greater versatility in performing a large number of operations in parallel. The primary application discussed here using this architecture is a quadratic neural network.

© 1990 Optical Society of America

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References

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  1. R. Athale, J. Lee, “Optical Processing Using Outer-Product Concepts,” Proc. IEEE 72, 931–941 (1984).
    [CrossRef]
  2. D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
    [CrossRef]
  3. S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).
  4. G. D. Boreman, E. R. Raudenbush, “Modulation Depth Characteristics of a Liquid Crystal Television Modulator,” Appl. Opt. 27, 2940–2943 (1988).
    [CrossRef] [PubMed]
  5. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1987).
  6. J. W. Goodman, L. M. Woody, “Method for Performing Complex-Valued Linear Operations on Complex-Valued Data Using Incoherent Light,” Appl. Opt. 16, 2611–2612 (1977).
    [CrossRef] [PubMed]
  7. M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
    [CrossRef]
  8. D. E. Rumelhart et al., Parallel Distributed Processing (MIT Press, Cambridge, 1987).
  9. J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Hopfield Associative Memory Characteristics using a Single Parameter,” Neural Networks March (1990), in press.
    [CrossRef]
  10. J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Higher Order Associative Memory Characteristics using a Single Parameter,” submitted to Neural Networks.

1989 (1)

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

1988 (3)

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
[CrossRef]

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

G. D. Boreman, E. R. Raudenbush, “Modulation Depth Characteristics of a Liquid Crystal Television Modulator,” Appl. Opt. 27, 2940–2943 (1988).
[CrossRef] [PubMed]

1984 (1)

R. Athale, J. Lee, “Optical Processing Using Outer-Product Concepts,” Proc. IEEE 72, 931–941 (1984).
[CrossRef]

1977 (1)

Athale, R.

R. Athale, J. Lee, “Optical Processing Using Outer-Product Concepts,” Proc. IEEE 72, 931–941 (1984).
[CrossRef]

Bigner, B. J.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

Bochove, E. J.

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

Boreman, G. D.

Goodman, J. W.

Hong, J.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
[CrossRef]

Johnson, K. M.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

Kranzdorf, M.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

Krile, T. F.

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Higher Order Associative Memory Characteristics using a Single Parameter,” submitted to Neural Networks.

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Hopfield Associative Memory Characteristics using a Single Parameter,” Neural Networks March (1990), in press.
[CrossRef]

Lee, J.

R. Athale, J. Lee, “Optical Processing Using Outer-Product Concepts,” Proc. IEEE 72, 931–941 (1984).
[CrossRef]

Lin, S. H.

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

Park, C. H.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
[CrossRef]

Psaltis, D.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
[CrossRef]

Raudenbush, E. R.

Rumelhart, D. E.

D. E. Rumelhart et al., Parallel Distributed Processing (MIT Press, Cambridge, 1987).

Walkup, J. F.

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Higher Order Associative Memory Characteristics using a Single Parameter,” submitted to Neural Networks.

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Hopfield Associative Memory Characteristics using a Single Parameter,” Neural Networks March (1990), in press.
[CrossRef]

Wang, J. H.

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Higher Order Associative Memory Characteristics using a Single Parameter,” submitted to Neural Networks.

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Hopfield Associative Memory Characteristics using a Single Parameter,” Neural Networks March (1990), in press.
[CrossRef]

Woody, L. M.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1987).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1987).

Zhang, L.

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

Appl. Opt. (2)

Neural Networks (1)

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementations,” Neural Networks, 1, 149–163 (1988).
[CrossRef]

Opt. Engineering (1)

M. Kranzdorf, B. J. Bigner, L. Zhang, K. M. Johnson, “Optical Connectionist Machine with Polarization Based Bipolar Weight Values,” Opt. Engineering, 28, 844–848 (1989).
[CrossRef]

Proc. IEEE (1)

R. Athale, J. Lee, “Optical Processing Using Outer-Product Concepts,” Proc. IEEE 72, 931–941 (1984).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

S. H. Lin, T. F. Krile, E. J. Bochove, J. F. Walkup, “Electrooptical Implementations of Programmable Quadratic Neural Networks,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 100–106 (1988).

Other (4)

D. E. Rumelhart et al., Parallel Distributed Processing (MIT Press, Cambridge, 1987).

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Hopfield Associative Memory Characteristics using a Single Parameter,” Neural Networks March (1990), in press.
[CrossRef]

J. H. Wang, T. F. Krile, J. F. Walkup, “Determination of Higher Order Associative Memory Characteristics using a Single Parameter,” submitted to Neural Networks.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1987).

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Figures (17)

Fig. 1
Fig. 1

Experimental architecture used to determine polarization properties of a single modulator.

Fig. 2
Fig. 2

Intensity vs polarization angle of the analyzer showing the twisted nematic effect of the liquid crystal TV.

Fig. 3
Fig. 3

Experimental architecture used to determine polarization properties of two cascaded LCTVs without a polarizer in between the LCTVs.

Fig. 4
Fig. 4

Output intensity vs polarization angle of the analyzer for the cascaded LCTV system.

Fig. 5
Fig. 5

Contrast ratio between the white–white interaction and the other interactions versus polarization angle of the output analyzer.

Fig. 6
Fig. 6

Architecture for polynomial processing. Symbols: LCTV = liquid crystal television, PBS = polarizing beam splitter, and LPF = lowpass filter.

Fig. 7
Fig. 7

Encoding procedure.

Fig. 8
Fig. 8

Truth table showing logic used in outer product processor.

Fig. 9
Fig. 9

(a) Representation of x = [1, −1]; (b) outer product result.

Fig. 10
Fig. 10

Images produced by passing outer product result through a beam splitter.

Fig. 11
Fig. 11

Logic used in performing the weighting.

Fig. 12
Fig. 12

Example of weighted outer product using the encoding described.

Fig. 13
Fig. 13

Architecture used to show decision levels and performance characteristics of the processor; LCTV 1 ~outer product image, and LCTV 2 ~weighting image.

Fig. 14
Fig. 14

Decision levels in the four channels.

Fig. 15
Fig. 15

The quadratic neuron implemented using an outer product processor.

Fig. 16
Fig. 16

Actual solutions to the ex-or problem obtained using a quadratic neural network.

Fig. 17
Fig. 17

Probability of convergence (gain constant = 0.5) with respect to threshold.

Equations (22)

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

X = x x T ,
x T = ( x 1 x 2 x n ) .
X = [ x 1 2 x 1 x 2 x 1 x n x 2 x 1 x 2 2 x 2 x n · · · x n x 1 x n x 2 x n 2 ] ,
W = [ w 11 w 12 w 1 n w 21 w 22 w 2 n · · · w n 1 w n 2 w n n ] .
p ( x ) = w 11 x 1 2 + w 12 x 1 x 2 + + w n m x n 2
p ( x ) = x T Wx .
contrast ratio = I max - I min I max + I min ,
x ( x 1 x 2 · · · x n ) = ( x 1 + x 2 + · · · x n + ) - ( x 1 - x 2 - · · · x n - )
X = [ 1 - 1 - 1 1 ] .
p ( x ) = 0 × x 1 2 + 1 × x 1 x 2 + 1 × x 2 x 1 - 1 × x 2 2 ,
I WHITE + ( n 2 - 1 ) I GRAY > n 2 I GRAY ,
I WHITE > I GRAY .
Δ I = I WHITE - I GRAY ,
p ( x ) = ( X + - X - ) ( W + - W - ) ,
y i = f h { i = 1 n j = 1 n w i j x i x j - b i } ,
p ( x ) = x 1 x 2 + 0.5 ( top left )
p ( x ) = x 1 x 2 - 0.5 ( top right )
p ( x ) = x 1 2 + x 2 2 - x 1 x 2 - 1.5 ( bottom left )
p ( x ) = x 1 2 + x 2 2 + x 1 x 2 - 1.5 ( bottom right )
w i j t + 1 = w i j t + η ( D i - y i ) x i x j
b i t + 1 = b i t - η ( D i - y i )
p ( x ) = [ x 1 x 2 x n 1 ] × [ w 11 w 12 w 1 n 0 w 21 w 22 w 2 n 0 · · · · · · w n 1 w n 2 w n n 0 0 0 0 b ] [ x 1 x 2 · · · x n 1 ] .

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