Abstract

The model of a simple perceptron using phase-encoded inputs and complex-valued weights is proposed. The aggregation function, activation function, and learning rule for the proposed neuron are derived and applied to Boolean logic functions and simple computer vision tasks. The complex-valued neuron (CVN) is shown to be superior to traditional perceptrons. An improvement of 135% over the theoretical maximum of 104 linearly separable problems (of three variables) solvable by conventional perceptrons is achieved without additional logic, neuron stages, or higher order terms such as those required in polynomial logic gates. The application of CVN in distortion invariant character recognition and image segmentation is demonstrated. Implementation details are discussed, and the CVN is shown to be very attractive for optical implementation since optical computations are naturally complex. The cost of the CVN is less in all cases than the traditional neuron when implemented optically. Therefore, all the benefits of the CVN can be obtained without additional cost. However, on those implementations dependent on standard serial computers, CVN will be more cost effective only in those applications where its increased power can offset the requirement for additional neurons.

© 2010 Optical Society of America

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    [CrossRef]
  37. A. A. S. Awwal and H. E. Michel, “Enhancing the discrimination capability of phase only filter,” Asian J. Phys. 8, 381-383 (2000).
  38. H. E. Michel and S. Kunjithapatham, “Processing Landsat TM data using complex-valued neural networks,” Proc. SPIE 4730, 43-53 (2002).
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  39. X. Tao and H. E. Michel, “Processing Landsat TM data using complex-valued NRBF neural network,” in Proceedings of the International Joint Conference on Neural Networks (2005), pp. 3081-8086.
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  41. X. Tao and H. E. Michel, “Classification of multi-spectral satellite image data using improved NRBF neural networks,” Proc. SPIE 5267, 311-320 (2003).
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    [CrossRef]
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    [CrossRef]
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  46. H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
    [CrossRef]

2008 (3)

I. Aizenberg, D. Paliy, J. Zurada, and J. Astola, “Blur identification by multilayer neural network based on multi-valued neurons,” IEEE Trans. Neural Networks 19, 883-898(2008).
[CrossRef]

I. Aizenberg, “Solving the xor and parity N problems using a single universal binary neuron, Soft Computing 12, 215-222(2008).

J. DongX. Zhang, J. Xu, and D. Huang, “40 Gb/s all-optical logic nor and or gates using a semiconductor optical amplifier: experimental demonstration and theoretical analysis,” Opt. Commun. 281, 1710-1715 (2008).
[CrossRef]

2007 (1)

I. Aizenberg and C. Moraga, “Multilayer feedforward neural network based on multi-valued neurons and a backpropagation learning algorithm,” Soft Comput. 11, 169-183(2007).

2006 (2)

J. Kim, J. Kang, T. Kim, and S. Han, “All-optical multiple logic gates with xor, nor, or, and nand functions using parallel SOA-MZI structures: theory and experiment,” J. Lightwave Technol. 24, 3392-3399 (2006).
[CrossRef]

H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
[CrossRef]

2005 (1)

X. Tao and H. E. Michel, “Novel artificial neural networks for remote-sensing data classification,” Proc. SPIE 5781, 127-138(2005).
[CrossRef]

2004 (1)

2003 (2)

X. Tao and H. E. Michel, “Classification of multi-spectral satellite image data using improved NRBF neural networks,” Proc. SPIE 5267, 311-320 (2003).
[CrossRef]

T. Nitta, “Solving the xor problem and the detection of symmetry using a single complex-valued neuron,” Neural Networks 16, 1101-1105 (2003).
[CrossRef]

2002 (2)

I. Aizenberg and C. Butakoff, “Image processing using cellular neural networks based on multi-valued and universal binary neurons,” J. VLSI Signal Process. 32, 169-188 (2002).

H. E. Michel and S. Kunjithapatham, “Processing Landsat TM data using complex-valued neural networks,” Proc. SPIE 4730, 43-53 (2002).
[CrossRef]

2001 (2)

B. Igelnik, M. Tabib-Azar, and S. LeClair, “A net with complex weights,” IEEE Trans. Neural Networks 12, 236-249(2001).
[CrossRef]

E. J. Bayro-Corrochano, “Geometric neural computing,” IEEE Trans. Neural Networks 12, 968-986 (2001).
[CrossRef]

2000 (3)

H. E. Michel and A. A. S. Awwal, “How to train a phase only filter,” Proc. SPIE 4046, 24-33 (2000).
[CrossRef]

A. A. S. Awwal and H. E. Michel, “Enhancing the discrimination capability of phase only filter,” Asian J. Phys. 8, 381-383 (2000).

K. Stubkjaer, “Semiconductor optical amplifier-based all optical gates for high-speed optical processing,” IEEE J. Sel. Top. Quantum Electron. 6, 1428-1435 (2000).
[CrossRef]

1998 (2)

J. I. Khan, “Characteristics of multidimensional holographic associative memory in retrieval with dynamic localizable attention,” IEEE Trans.Neural Networks 9, 389-406(1998).
[CrossRef]

D. M. Weber and D. P. Casasent, “The extended piecewise quadratic neural network,” Neural Networks 11, 837-850(1998).
[CrossRef]

1997 (5)

J. I. Khan and D. Y. Yun, “A parallel, distributed and associative approach for pattern matching with holographic dynamics,” J. Visual Languages Comput. 8, 303-331(1997).
[CrossRef]

M. R. Smith and Y. Hui, “A data extrapolation algorithm using a complex domain neural network,” IEEE Trans. Circuits Syst. 2 Analog Digit. Signal Process. 44, 143-147 (1997).

P. Arena, G. Fortuna, G. Muscato, and M. G. Xibilia, “Multilayer perceptrons to approximate quaternion valued functions,” Neural Networks 10, 335-342 (1997).
[CrossRef]

T. Nitta, “An extension of the back-propagation algorithm to complex numbers,” Neural Networks 10, 1391-1415 (1997).
[CrossRef]

N. N. Aizenberg and I. N. Aizenberg, “Universal binary and multivalued paradigm: conception, learning, applications,” Lect. Notes Comput. Sci. 1240, 463-472 (1997).
[CrossRef]

1996 (1)

S. Thorpe, D. Fize, and C. Marlot, “Speed of processing in the human visual system,” Nature 381, 520-522 (1996).

1995 (4)

J. J. Hopfield and A. V. M. Herz, “Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire networks,” Proc. Natl. Acad. Sci. U.S.A. 92, 6655-6662 (1995).

J. J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33-36 (1995).

F. Theunissen and J. P. Miller, “Temporal encoding in nervous systems,” J. Comput. Neurosci. 2, 149-162 (1995).
[CrossRef]

D. Casasent and S. Natarajan, “A classifier neural network with complex-valued weights and square-law nonlinearities,” Neural Networks 8, 989-998 (1995).
[CrossRef]

1994 (1)

A. Hirose, “Applications of complex-valued neural networks to coherent optical computing using phase-sensitive detection scheme,” Information Sci. Appl. 2, 103-117 (1994).

1993 (1)

A. A. S. Awwal and G. Power, “Object tracking by an opto-electronic inner product complex neural network,” Opt. Eng. 32, 2782-2787 (1993).

1992 (3)

G. M. Georgiou and C. Koutsougeras, “Complex domain backpropagation,” IEEE Trans. Circuits Syst. 2 Analog Digit. Signal Process. 39, 330-334 (1992).

N. Benvenuto and F. Piazza, “On the complex backpropagation algorithm,” IEEE Trans. Signal Process. 40, 967-969(1992).
[CrossRef]

A. Hirose, “Dynamics of fully complex-valued neural networks,” Electron. Lett. 28, 1492-1494 (1992).
[CrossRef]

1991 (2)

H. Leung and S. Haykin, “The complex backpropagation algorithm,” IEEE Trans. Signal Process. 39, 2101-2104 (1991).
[CrossRef]

I. N. Aizenberg, “A universal logic element over the complex field,” Cybern. Syst. Anal. 27, 467-473 (1991).
[CrossRef]

1971 (1)

N. N. Aizenberg, Yu L. Ivas'kiv, D. A. Pospelov, and G. F. Khudyakov, “Multivalued threshold functions in Boolean complex-threshold functions and their generalization,” Cybern. Syst. Anal. 7, 626-635 (1971).

1943 (1)

W. S. McCulloch and W. Pitts, “A logical calculus of ideas immanent in nervous activity,” Bull. Mathematical Biophys. 5, 115-133 (1943).

Aizenberg, I.

I. Aizenberg, “Solving the xor and parity N problems using a single universal binary neuron, Soft Computing 12, 215-222(2008).

I. Aizenberg, D. Paliy, J. Zurada, and J. Astola, “Blur identification by multilayer neural network based on multi-valued neurons,” IEEE Trans. Neural Networks 19, 883-898(2008).
[CrossRef]

I. Aizenberg and C. Moraga, “Multilayer feedforward neural network based on multi-valued neurons and a backpropagation learning algorithm,” Soft Comput. 11, 169-183(2007).

I. Aizenberg and C. Butakoff, “Image processing using cellular neural networks based on multi-valued and universal binary neurons,” J. VLSI Signal Process. 32, 169-188 (2002).

I. Aizenberg, D. Paliy, and J. Astola, “Multilayer neural network based on multi-valued neurons and the blur identification problem,” in Proceedings of the 2006 IEEE Joint Conference on Neural Networks (2006), pp. 1200-1207.

Aizenberg, I. N.

N. N. Aizenberg and I. N. Aizenberg, “Universal binary and multivalued paradigm: conception, learning, applications,” Lect. Notes Comput. Sci. 1240, 463-472 (1997).
[CrossRef]

I. N. Aizenberg, “A universal logic element over the complex field,” Cybern. Syst. Anal. 27, 467-473 (1991).
[CrossRef]

Aizenberg, N. N.

N. N. Aizenberg and I. N. Aizenberg, “Universal binary and multivalued paradigm: conception, learning, applications,” Lect. Notes Comput. Sci. 1240, 463-472 (1997).
[CrossRef]

N. N. Aizenberg, Yu L. Ivas'kiv, D. A. Pospelov, and G. F. Khudyakov, “Multivalued threshold functions in Boolean complex-threshold functions and their generalization,” Cybern. Syst. Anal. 7, 626-635 (1971).

Arena, P.

P. Arena, G. Fortuna, G. Muscato, and M. G. Xibilia, “Multilayer perceptrons to approximate quaternion valued functions,” Neural Networks 10, 335-342 (1997).
[CrossRef]

Astola, J.

I. Aizenberg, D. Paliy, J. Zurada, and J. Astola, “Blur identification by multilayer neural network based on multi-valued neurons,” IEEE Trans. Neural Networks 19, 883-898(2008).
[CrossRef]

I. Aizenberg, D. Paliy, and J. Astola, “Multilayer neural network based on multi-valued neurons and the blur identification problem,” in Proceedings of the 2006 IEEE Joint Conference on Neural Networks (2006), pp. 1200-1207.

Awwal, A. A. S.

H. E. Michel and A. A. S. Awwal, “How to train a phase only filter,” Proc. SPIE 4046, 24-33 (2000).
[CrossRef]

A. A. S. Awwal and H. E. Michel, “Enhancing the discrimination capability of phase only filter,” Asian J. Phys. 8, 381-383 (2000).

A. A. S. Awwal and G. Power, “Object tracking by an opto-electronic inner product complex neural network,” Opt. Eng. 32, 2782-2787 (1993).

H. E. Michel and A. A. S. Awwal, “Enhanced Artificial Neural Networks Using Complex-Numbers,” in IJCNN'99 International Joint Conference on Neural Networks, Vol. 1 (1999), pp. 456-461.

H. E. Michel, A. A. S. Awwal, and D. Rancour, “Artificial neural networks using complex numbers and phase encoded weights--electronic and optical implementations,” in 2006 International Joint Conference on Neural Networks (2006), pp. 1213-1218.

Bagnoud, V.

Bayro-Corrochano, E. J.

E. J. Bayro-Corrochano, “Geometric neural computing,” IEEE Trans. Neural Networks 12, 968-986 (2001).
[CrossRef]

Benvenuto, N.

N. Benvenuto and F. Piazza, “On the complex backpropagation algorithm,” IEEE Trans. Signal Process. 40, 967-969(1992).
[CrossRef]

Bishop, C. M.

W. Maass and C. M. Bishop, Pulsed Neural Networks (MIT Press, 1998).

Butakoff, C.

I. Aizenberg and C. Butakoff, “Image processing using cellular neural networks based on multi-valued and universal binary neurons,” J. VLSI Signal Process. 32, 169-188 (2002).

Casasent, D.

D. Casasent and S. Natarajan, “A classifier neural network with complex-valued weights and square-law nonlinearities,” Neural Networks 8, 989-998 (1995).
[CrossRef]

Casasent, D. P.

D. M. Weber and D. P. Casasent, “The extended piecewise quadratic neural network,” Neural Networks 11, 837-850(1998).
[CrossRef]

Dong, H.

H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
[CrossRef]

Dong, J.

J. DongX. Zhang, J. Xu, and D. Huang, “40 Gb/s all-optical logic nor and or gates using a semiconductor optical amplifier: experimental demonstration and theoretical analysis,” Opt. Commun. 281, 1710-1715 (2008).
[CrossRef]

Dutta, N.

H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
[CrossRef]

Fize, D.

S. Thorpe, D. Fize, and C. Marlot, “Speed of processing in the human visual system,” Nature 381, 520-522 (1996).

Fortuna, G.

P. Arena, G. Fortuna, G. Muscato, and M. G. Xibilia, “Multilayer perceptrons to approximate quaternion valued functions,” Neural Networks 10, 335-342 (1997).
[CrossRef]

Georgiou, G. M.

G. M. Georgiou and C. Koutsougeras, “Complex domain backpropagation,” IEEE Trans. Circuits Syst. 2 Analog Digit. Signal Process. 39, 330-334 (1992).

Han, S.

Haykin, S.

H. Leung and S. Haykin, “The complex backpropagation algorithm,” IEEE Trans. Signal Process. 39, 2101-2104 (1991).
[CrossRef]

Herz, A. V. M.

J. J. Hopfield and A. V. M. Herz, “Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire networks,” Proc. Natl. Acad. Sci. U.S.A. 92, 6655-6662 (1995).

Hirose, A.

A. Hirose, “Applications of complex-valued neural networks to coherent optical computing using phase-sensitive detection scheme,” Information Sci. Appl. 2, 103-117 (1994).

A. Hirose, “Dynamics of fully complex-valued neural networks,” Electron. Lett. 28, 1492-1494 (1992).
[CrossRef]

Hopfield, J. J.

J. J. Hopfield and A. V. M. Herz, “Rapid local synchronization of action potentials: toward computation with coupled integrate-and-fire networks,” Proc. Natl. Acad. Sci. U.S.A. 92, 6655-6662 (1995).

J. J. Hopfield, “Pattern recognition computation using action potential timing for stimulus representation,” Nature 376, 33-36 (1995).

Huang, D.

J. DongX. Zhang, J. Xu, and D. Huang, “40 Gb/s all-optical logic nor and or gates using a semiconductor optical amplifier: experimental demonstration and theoretical analysis,” Opt. Commun. 281, 1710-1715 (2008).
[CrossRef]

Hui, Y.

M. R. Smith and Y. Hui, “A data extrapolation algorithm using a complex domain neural network,” IEEE Trans. Circuits Syst. 2 Analog Digit. Signal Process. 44, 143-147 (1997).

Igelnik, B.

B. Igelnik, M. Tabib-Azar, and S. LeClair, “A net with complex weights,” IEEE Trans. Neural Networks 12, 236-249(2001).
[CrossRef]

Iringentavida, S.

H. E. Michel, D. Rancour, and S. Iringentavida, “CMOS Implementation of phase-encoded complex-valued artificial neural networks,” in Proceedings of the International Conference on VLSI, VLSI'04 (2004), pp. 551-557.

Ivas'kiv, Yu L.

N. N. Aizenberg, Yu L. Ivas'kiv, D. A. Pospelov, and G. F. Khudyakov, “Multivalued threshold functions in Boolean complex-threshold functions and their generalization,” Cybern. Syst. Anal. 7, 626-635 (1971).

Jaques, J.

H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
[CrossRef]

Kang, J.

Khan, J. I.

J. I. Khan, “Characteristics of multidimensional holographic associative memory in retrieval with dynamic localizable attention,” IEEE Trans.Neural Networks 9, 389-406(1998).
[CrossRef]

J. I. Khan and D. Y. Yun, “A parallel, distributed and associative approach for pattern matching with holographic dynamics,” J. Visual Languages Comput. 8, 303-331(1997).
[CrossRef]

Khudyakov, G. F.

N. N. Aizenberg, Yu L. Ivas'kiv, D. A. Pospelov, and G. F. Khudyakov, “Multivalued threshold functions in Boolean complex-threshold functions and their generalization,” Cybern. Syst. Anal. 7, 626-635 (1971).

Kim, J.

Kim, T.

Koutsougeras, C.

G. M. Georgiou and C. Koutsougeras, “Complex domain backpropagation,” IEEE Trans. Circuits Syst. 2 Analog Digit. Signal Process. 39, 330-334 (1992).

Kunjithapatham, S.

H. E. Michel and S. Kunjithapatham, “Processing Landsat TM data using complex-valued neural networks,” Proc. SPIE 4730, 43-53 (2002).
[CrossRef]

LeClair, S.

B. Igelnik, M. Tabib-Azar, and S. LeClair, “A net with complex weights,” IEEE Trans. Neural Networks 12, 236-249(2001).
[CrossRef]

Leung, H.

H. Leung and S. Haykin, “The complex backpropagation algorithm,” IEEE Trans. Signal Process. 39, 2101-2104 (1991).
[CrossRef]

Maass, W.

W. Maass and C. M. Bishop, Pulsed Neural Networks (MIT Press, 1998).

Marlot, C.

S. Thorpe, D. Fize, and C. Marlot, “Speed of processing in the human visual system,” Nature 381, 520-522 (1996).

McCulloch, W. S.

W. S. McCulloch and W. Pitts, “A logical calculus of ideas immanent in nervous activity,” Bull. Mathematical Biophys. 5, 115-133 (1943).

Michel, H. E.

X. Tao and H. E. Michel, “Novel artificial neural networks for remote-sensing data classification,” Proc. SPIE 5781, 127-138(2005).
[CrossRef]

X. Tao and H. E. Michel, “Classification of multi-spectral satellite image data using improved NRBF neural networks,” Proc. SPIE 5267, 311-320 (2003).
[CrossRef]

H. E. Michel and S. Kunjithapatham, “Processing Landsat TM data using complex-valued neural networks,” Proc. SPIE 4730, 43-53 (2002).
[CrossRef]

A. A. S. Awwal and H. E. Michel, “Enhancing the discrimination capability of phase only filter,” Asian J. Phys. 8, 381-383 (2000).

H. E. Michel and A. A. S. Awwal, “How to train a phase only filter,” Proc. SPIE 4046, 24-33 (2000).
[CrossRef]

H. E. Michel, “Enhanced artificial neural network using complex numbers,” Ph.D. dissertation (Wright State University, 1999).

H. E. Michel and A. A. S. Awwal, “Enhanced Artificial Neural Networks Using Complex-Numbers,” in IJCNN'99 International Joint Conference on Neural Networks, Vol. 1 (1999), pp. 456-461.

H. E. Michel, A. A. S. Awwal, and D. Rancour, “Artificial neural networks using complex numbers and phase encoded weights--electronic and optical implementations,” in 2006 International Joint Conference on Neural Networks (2006), pp. 1213-1218.

H. E. Michel, D. Rancour, and S. Iringentavida, “CMOS Implementation of phase-encoded complex-valued artificial neural networks,” in Proceedings of the International Conference on VLSI, VLSI'04 (2004), pp. 551-557.

X. Tao and H. E. Michel, “Data clustering via spiking neural networks through spike timing-dependent plasticity,” in Proceedings of the International Conference on Artificial Intelligence, IC-AI'04, Vol. 1 (2004), pp. 168-173.

X. Tao and H. E. Michel, “Processing Landsat TM data using complex-valued NRBF neural network,” in Proceedings of the International Joint Conference on Neural Networks (2005), pp. 3081-8086.

Miller, J. P.

F. Theunissen and J. P. Miller, “Temporal encoding in nervous systems,” J. Comput. Neurosci. 2, 149-162 (1995).
[CrossRef]

Moraga, C.

I. Aizenberg and C. Moraga, “Multilayer feedforward neural network based on multi-valued neurons and a backpropagation learning algorithm,” Soft Comput. 11, 169-183(2007).

Muscato, G.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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H. Dong, H. Sun, Q. Wang, N. Dutta, and J. Jaques, “80 Gb/s all-optical logic and operation using Mach-Zehnder interferometer with differential scheme,” Opt. Commun. 265, 79-93 (2006).
[CrossRef]

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[CrossRef]

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W. Maass and C. M. Bishop, Pulsed Neural Networks (MIT Press, 1998).

H. E. Michel, A. A. S. Awwal, and D. Rancour, “Artificial neural networks using complex numbers and phase encoded weights--electronic and optical implementations,” in 2006 International Joint Conference on Neural Networks (2006), pp. 1213-1218.

H. E. Michel, D. Rancour, and S. Iringentavida, “CMOS Implementation of phase-encoded complex-valued artificial neural networks,” in Proceedings of the International Conference on VLSI, VLSI'04 (2004), pp. 551-557.

X. Tao and H. E. Michel, “Data clustering via spiking neural networks through spike timing-dependent plasticity,” in Proceedings of the International Conference on Artificial Intelligence, IC-AI'04, Vol. 1 (2004), pp. 168-173.

X. Tao and H. E. Michel, “Processing Landsat TM data using complex-valued NRBF neural network,” in Proceedings of the International Joint Conference on Neural Networks (2005), pp. 3081-8086.

H. E. Michel and A. A. S. Awwal, “Enhanced Artificial Neural Networks Using Complex-Numbers,” in IJCNN'99 International Joint Conference on Neural Networks, Vol. 1 (1999), pp. 456-461.

H. E. Michel, “Enhanced artificial neural network using complex numbers,” Ph.D. dissertation (Wright State University, 1999).

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Figures (12)

Fig. 1
Fig. 1

Aggregation function for the complete set of 2-input Boolean variables.

Fig. 2
Fig. 2

Complex-valued intermediate space for xor problem.

Fig. 3
Fig. 3

Effect of changing threshold T.

Fig. 4
Fig. 4

Effect of adding a bias term.

Fig. 5
Fig. 5

Three by three representation of C and T.

Fig. 6
Fig. 6

Variations on C.

Fig. 7
Fig. 7

Variations on T.

Fig. 8
Fig. 8

Variations on C.

Fig. 9
Fig. 9

Variations on T.

Fig. 10
Fig. 10

Picture of skin-cancer growth.

Fig. 11
Fig. 11

Segmented cancer growth picture.

Fig. 12
Fig. 12

Optical implementation of an artificial neuron.

Tables (6)

Tables Icon

Table 1 Learned Weights, in Radians, for 2-Input-Plus-Bias Complex-Valued Perceptron

Tables Icon

Table 2 Learned Parameters for Cancer Cell Segmentation

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Table 3 Hardware Required to Implement and Processing Times of Optical Artificial Neurons

Tables Icon

Table 4 Performance Comparison for Optical Implementations of 2-Input Boolean Functions

Tables Icon

Table 5 Performance Comparison for Optical Implementations of 3-Input Boolean Functions

Tables Icon

Table 6 Comparison of the CVN and the Perceptron

Equations (22)

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

p i = exp ( i value max _ value π 2 ) .
q = w p .
a = { 0 if | q | < T 1 if | q | T ,
r = | q | 2 .
[ θ 1 new θ 2 new ] = [ θ 1 old θ 2 old ] + η ( d - a ) [ 1 δ r δ θ 1 1 δ r δ θ 2 ] ,
b = λ b e i θ b .
w = ( λ w 1 e i θ 1 λ w 2 e i θ 2 λ b e i θ b ) .
T new = T old - η ( d - a ) T old .
λ b new = λ b old + η ( d - a ) δ r δ λ b .
p = ( e i ψ 1 e i ψ 2 e i ψ n 1 ) ,
w = ( e i θ 1 e i θ 2 e i θ n λ b e i θ b ) ,
q = ( p , w ) = e i ( θ i + ψ i ) + λ b e i θ b ,
r = ( i = 1 n cos ( θ i + ψ i ) + λ b cos ( θ b ) ) 2 + ( i = 1 n sin ( θ i + ψ i ) + λ b sin ( θ b ) ) 2 .
r = 2 i = 1 n - 1 j = i + 1 n cos ( θ i + ψ i θ j ψ j ) + 2 λ b i = 1 n cos ( θ i + ψ i θ b ) + λ b 2 + n .
δ r δ θ i = 2 i = 1 n - 1 j = i + 1 n sin ( θ i + ψ i θ j ψ j ) 2 λ b i = 1 n sin ( θ i + ψ i θ b ) .
r j i = | q j i | 2 = 2 + λ j b 2 i + 2 cos ( θ j , 1 i + ψ j , 1 i θ j , 2 i ψ j , 2 i ) + 2 λ j b i cos ( θ j , 1 i + ψ j , 1 i θ j b i ) + 2 λ j b i cos ( θ j , 2 i + ψ j , 2 i θ j b i ) ,
a j i = { 0 if | q j i | < T j i 1 if | q j i | T j i .
ψ j , k 1 = { 0 if   logic _ data = FALSE π 2 if   logic _ data = TRUE .
ψ j , k 2 = { 0 if | q k 1 | < T k 1 π 2 if | q k 1 | T k 1 .
δ i r j δ ψ j 1 i = 2 sin ( θ j 1 i + ψ j 1 i θ j 2 i ψ j 2 i ) 2 λ j b i sin ( θ j 1 i + ψ j 1 i θ j b i ) ,
T j k new i = T j k old i η ( d j i a j i ) 1 δ j r i δ ψ j 1 i ,
λ j k new i = θ j k old i + η ( d j i a j i ) 1 δ j r i δ λ j 1 i .

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