Abstract

Optical architectures for fully connected and limited-fan-out space-variant weighted interconnections based on diffractive optical elements for fixed-connection multilayer neural networks are investigated and compared in terms of propagation lengths, system volumes, connection densities, and interconnection cross talk. For a small overall system volume the limited-fan-out architecture can accommodate a much larger number of input and output nodes. However, the interconnection cross talk of the limited-fan-out space-variant architecture is relatively high owing to noise from the diffractive-optical-element reconstructions. Therefore a cross-talk reduction technique based on a modified design procedure for diffractive optical elements is proposed. It rearranges the reconstruction pattern of the diffractive optical elements such that less noise lands on each detector region. This technique is verified by the simulation of one layer of an interconnection system with 128 × 128 input nodes, 128 × 128 output nodes, and weighted connections that fan out from each input node to the nearest 5 × 5 array of output nodes. In addition to a significant cross-talk reduction, this technique can reduce the propagation length and system volume.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. A. Arbib, ed., The Handbook of Brain Theory and Neural Networks (MIT Press, Cambridge, 1995).
  2. C. Mead, Analog VLSI and Neural Systems (Addison-Wesley, New York, 1989).
    [CrossRef]
  3. Special issue on neural networks, Appl. Opt. 26, December1987.
  4. Special issue on neural networks, Appl. Opt. 32, March1993.
  5. H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1582 (1989).
    [CrossRef]
  6. S. Kakizaki, P. Horan, “Limitations of optical lateral intraconnection of smart pixel arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 201–203.
  7. D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.
  8. G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
    [CrossRef]
  9. W. B. Veldkamp, “Wireless focal planes: on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
    [CrossRef]
  10. A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
    [CrossRef]
  11. C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.
  12. C. C. Huang, B. K. Jenkins, C. B. Kuznia, “Weighted space-variant local interconnections based on micro-optic components: crosstalk analysis and reduction,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 280–282.
  13. C. H. Wang, B. K. Jenkins, J. M. Wang, “Visual cortex operations and their implementation using the incoherent optical neuron model,” Appl. Opt. 32, 1876–1887 (1993).
    [CrossRef] [PubMed]
  14. D. C. Van Essen, C. H. Anderson, “Information processing strategies and pathways in the primate retina and visual cortex,” in Introduction Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, C. Lau, eds., (Academic, San Diego, Calif., 1990), Chap. 3.
  15. R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1990).
  16. P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
    [CrossRef]
  17. M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
    [CrossRef] [PubMed]
  18. M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).
  19. M. Feldman, “Diffractive optics move into the commercial arena,” Laser Focus World 30, 143–151 (Oct.1994).
  20. M. W. Farn, “Modeling of diffractive optics,” in Diffractive Optics, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.
  21. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  22. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  23. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  24. F. McCormick, “Free-space optical interconnects for 3-D optoelectronic computing,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September1995, paper ThCC1.
  25. M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.
  26. K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.
  27. A. Goldstein, B. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–24 October 1996 (Optical Society of America, Washington, D.C., 1996), paper ThKK2.
  28. B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
    [CrossRef] [PubMed]
  29. P. Keller, A. Gmitro, “Computer-generated holograms for optical neural networks: on-axis versus off-axis geometry,” Appl. Opt. 32, 1304–1310 (1993).
    [CrossRef] [PubMed]
  30. M. Feldman, C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  31. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
    [CrossRef] [PubMed]
  32. C. H. Wang, B. K. Jenkins, “Subtracting incoherent optical neuron model: analysis, experiment, and applications,” Appl. Opt. 29, 2171–2186 (1990).
    [CrossRef] [PubMed]
  33. J. F. Lin, A. A. Sawchuk, “Design of diffractive optical elements with optimization of the signal-to-noise ratio and without a dummy area,” Appl. Opt. 36, 3155–3164 (1997).
    [CrossRef] [PubMed]

1997 (1)

1994 (2)

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

M. Feldman, “Diffractive optics move into the commercial arena,” Laser Focus World 30, 143–151 (Oct.1994).

1993 (4)

1992 (4)

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (2)

1989 (2)

1987 (1)

Special issue on neural networks, Appl. Opt. 26, December1987.

1984 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Ananthanarayanan, K.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Anderson, C. H.

D. C. Van Essen, C. H. Anderson, “Information processing strategies and pathways in the primate retina and visual cortex,” in Introduction Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, C. Lau, eds., (Academic, San Diego, Calif., 1990), Chap. 3.

Blum, O.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Brooke, M.

D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.

Brown, A.

D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.

Carson, R.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Cartland, R. F.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Caulfield, H. J.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1582 (1989).
[CrossRef]

Chavel, P.

Chen, C.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Craft, D.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Dames, M. P.

De Valois, K. K.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1990).

De Valois, R. L.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1990).

DeMars, S.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Dowling, R. J.

Du, T.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Esener, S. C.

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Farn, M. W.

M. W. Farn, “Modeling of diffractive optics,” in Diffractive Optics, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.

Feldman, M.

M. Feldman, “Diffractive optics move into the commercial arena,” Laser Focus World 30, 143–151 (Oct.1994).

M. Feldman, C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[CrossRef] [PubMed]

Forchheimer, R.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gmitro, A.

P. Keller, A. Gmitro, “Computer-generated holograms for optical neural networks: on-axis versus off-axis geometry,” Appl. Opt. 32, 1304–1310 (1993).
[CrossRef] [PubMed]

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

Goldstein, A.

A. Goldstein, B. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–24 October 1996 (Optical Society of America, Washington, D.C., 1996), paper ThKK2.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Guest, C.

Horan, P.

S. Kakizaki, P. Horan, “Limitations of optical lateral intraconnection of smart pixel arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 201–203.

Huang, C.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Huang, C. C.

C. C. Huang, B. K. Jenkins, C. B. Kuznia, “Weighted space-variant local interconnections based on micro-optic components: crosstalk analysis and reduction,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 280–282.

Ichikawa, H.

Jaakkola, T.

Jenkins, B.

A. Goldstein, B. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–24 October 1996 (Optical Society of America, Washington, D.C., 1996), paper ThKK2.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Jenkins, B. K.

C. H. Wang, B. K. Jenkins, J. M. Wang, “Visual cortex operations and their implementation using the incoherent optical neuron model,” Appl. Opt. 32, 1876–1887 (1993).
[CrossRef] [PubMed]

C. H. Wang, B. K. Jenkins, “Subtracting incoherent optical neuron model: analysis, experiment, and applications,” Appl. Opt. 29, 2171–2186 (1990).
[CrossRef] [PubMed]

B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
[CrossRef] [PubMed]

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

C. C. Huang, B. K. Jenkins, C. B. Kuznia, “Weighted space-variant local interconnections based on micro-optic components: crosstalk analysis and reduction,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 280–282.

Jokerst, N. M.

D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.

Kakizaki, S.

S. Kakizaki, P. Horan, “Limitations of optical lateral intraconnection of smart pixel arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 201–203.

Karim, Z.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Keller, P.

P. Keller, A. Gmitro, “Computer-generated holograms for optical neural networks: on-axis versus off-axis geometry,” Appl. Opt. 32, 1304–1310 (1993).
[CrossRef] [PubMed]

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

Kilcoyne, S.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Kinser, J.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1582 (1989).
[CrossRef]

Krishnamoorthy, A. V.

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Kuisma, S.

Kuznia, C.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Kuznia, C. B.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

C. C. Huang, B. K. Jenkins, C. B. Kuznia, “Weighted space-variant local interconnections based on micro-optic components: crosstalk analysis and reduction,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 280–282.

Kyriakakis, C.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Lear, K.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Lin, J. F.

Lovejoy, M.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Madhukar, A.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Marsden, G. C.

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

McCormick, F.

F. McCormick, “Free-space optical interconnects for 3-D optoelectronic computing,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September1995, paper ThCC1.

McKee, P.

Mead, C.

C. Mead, Analog VLSI and Neural Systems (Addison-Wesley, New York, 1989).
[CrossRef]

Miller, J. M.

Noponen, E.

Piazzolla, S.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Rogers, S. K.

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1582 (1989).
[CrossRef]

Sawchuk, A.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Sawchuk, A. A.

J. F. Lin, A. A. Sawchuk, “Design of diffractive optical elements with optimization of the signal-to-noise ratio and without a dummy area,” Appl. Opt. 36, 3155–3164 (1997).
[CrossRef] [PubMed]

B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
[CrossRef] [PubMed]

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Schneider, R.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Strand, T. C.

Su, D.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Taghizadeh, M. R.

Tanguay, A.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

Tanguay, A. R.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Turunen, J.

Van Essen, D. C.

D. C. Van Essen, C. H. Anderson, “Information processing strategies and pathways in the primate retina and visual cortex,” in Introduction Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, C. Lau, eds., (Academic, San Diego, Calif., 1990), Chap. 3.

Vasara, A.

Vawter, G.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Veldkamp, W. B.

W. B. Veldkamp, “Wireless focal planes: on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
[CrossRef]

von der Malsburg, C.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

Wang, C. H.

Wang, J. M.

Warren, M.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Wendt, J.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

Westerholm, J.

Wills, D. S.

D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.

Wood, D.

Wyrowski, F.

Yayla, G.

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

Appl. Opt. (10)

Special issue on neural networks, Appl. Opt. 26, December1987.

Special issue on neural networks, Appl. Opt. 32, March1993.

P. Keller, A. Gmitro, “Design and analysis of fixed planar holographic interconnects for optical neural networks,” Appl. Opt. 32, 5517–5526 (1992).
[CrossRef]

B. K. Jenkins, P. Chavel, R. Forchheimer, A. A. Sawchuk, T. C. Strand, “Architectural implications of a digital optical processor,” Appl. Opt. 23, 3465–3474 (1984).
[CrossRef] [PubMed]

C. H. Wang, B. K. Jenkins, “Subtracting incoherent optical neuron model: analysis, experiment, and applications,” Appl. Opt. 29, 2171–2186 (1990).
[CrossRef] [PubMed]

M. P. Dames, R. J. Dowling, P. McKee, D. Wood, “Efficient optical elements to generate intensity weighted spot arrays: design and fabrication,” Appl. Opt. 30, 2685–2691 (1991).
[CrossRef] [PubMed]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

P. Keller, A. Gmitro, “Computer-generated holograms for optical neural networks: on-axis versus off-axis geometry,” Appl. Opt. 32, 1304–1310 (1993).
[CrossRef] [PubMed]

C. H. Wang, B. K. Jenkins, J. M. Wang, “Visual cortex operations and their implementation using the incoherent optical neuron model,” Appl. Opt. 32, 1876–1887 (1993).
[CrossRef] [PubMed]

J. F. Lin, A. A. Sawchuk, “Design of diffractive optical elements with optimization of the signal-to-noise ratio and without a dummy area,” Appl. Opt. 36, 3155–3164 (1997).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

W. B. Veldkamp, “Wireless focal planes: on the road to amacronic sensors,” IEEE J. Quantum Electron. 29, 801–813 (1993).
[CrossRef]

IEEE Trans. Neural Net. (1)

A. V. Krishnamoorthy, G. Yayla, G. C. Marsden, S. C. Esener, “A scalable optoelectronic neural system using free-space optical interconnects,” IEEE Trans. Neural Net. 3, 404–413 (1992).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser Focus World (1)

M. Feldman, “Diffractive optics move into the commercial arena,” Laser Focus World 30, 143–151 (Oct.1994).

Opt. Comput. Process. (1)

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Opt. Lett. (1)

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Proc. IEEE (2)

H. J. Caulfield, J. Kinser, S. K. Rogers, “Optical neural networks,” Proc. IEEE 77, 1573–1582 (1989).
[CrossRef]

G. Yayla, A. V. Krishnamoorthy, G. C. Marsden, S. C. Esener, “A prototype 3D optically interconnected neural network,” Proc. IEEE 82, 1749–1762 (1994).
[CrossRef]

Other (14)

M. W. Farn, “Modeling of diffractive optics,” in Diffractive Optics, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

S. Kakizaki, P. Horan, “Limitations of optical lateral intraconnection of smart pixel arrays,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 201–203.

D. S. Wills, N. M. Jokerst, M. Brooke, A. Brown, “A two layer image processing architecture incorporating integrated focal plane detectors and through-wafer optical interconnect,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 19–22.

C. Kyriakakis, Z. Karim, A. R. Tanguay, R. F. Cartland, A. Madhukar, S. Piazzolla, B. K. Jenkins, C. B. Kuznia, A. A. Sawchuk, C. von der Malsburg, “Photonic implementations of neural networks,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 128–130.

C. C. Huang, B. K. Jenkins, C. B. Kuznia, “Weighted space-variant local interconnections based on micro-optic components: crosstalk analysis and reduction,” in Optical Computing, Vol. 10 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 280–282.

D. C. Van Essen, C. H. Anderson, “Information processing strategies and pathways in the primate retina and visual cortex,” in Introduction Neural and Electronic Networks, S. F. Zornetzer, J. L. Davis, C. Lau, eds., (Academic, San Diego, Calif., 1990), Chap. 3.

R. L. De Valois, K. K. De Valois, Spatial Vision (Oxford U. Press, New York, 1990).

F. McCormick, “Free-space optical interconnects for 3-D optoelectronic computing,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September1995, paper ThCC1.

M. Warren, T. Du, K. Lear, S. Kilcoyne, R. Carson, J. Wendt, G. Vawter, M. Lovejoy, O. Blum, D. Craft, R. Schneider, “Free-space optical interconnect for stacked multi-chip modules based on vertical-cavity laser-arrays with integrated diffractive microlenses,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC2.

K. Ananthanarayanan, C. Chen, S. DeMars, C. Huang, D. Su, C. Kuznia, C. Kyriakakis, Z. Karim, B. Jenkins, A. Sawchuk, A. Tanguay, “Multilayer electronic/photonic multichip modules with vertical optical interconnections,” paper presented at the Optical Society of America Annual Meeting, Portland, Oregon, 10–15 September 1995, paper ThCC7.

A. Goldstein, B. Jenkins, “Neural-network object recognition algorithm for real-time implementation on 3-D photonic multichip modules,” paper presented at the Optical Society of America Annual Meeting, Rochester, N.Y., 20–24 October 1996 (Optical Society of America, Washington, D.C., 1996), paper ThKK2.

M. A. Arbib, ed., The Handbook of Brain Theory and Neural Networks (MIT Press, Cambridge, 1995).

C. Mead, Analog VLSI and Neural Systems (Addison-Wesley, New York, 1989).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Layout and phase distribution (enlarged square) of a four-phase-level DOE. T represents the period length, C represents the compression ratio, and Δ represents the width of a phase element. In this design T = 16Δ.

Fig. 2
Fig. 2

With nonidealities of a lens taken into consideration, a 1-D reconstruction pattern of a DOE consists of signals, SS’s, SDO’s, and NRN. The overall envelope (because of the finite size of the phase elements) is shown by the dashed curve. The example shown has 8 × 8 phase elements in one period of a DOE. The three signal orders shown represent the -1, 0, and +1 diffraction orders.

Fig. 3
Fig. 3

Schematic diagram that represents optoelectronic implementation of multilayer feed-forward neural networks based on the interconnection architectures considered in this paper (not drawn to scale). Each layer contains an optoelectronic chip, collimation optics, a sub-DOE array, and reconstruction optics. The light modulator (or emitter) on the back side of each optoelectronic node is not shown.

Fig. 4
Fig. 4

Optical architecture for fully connected S-V interconnection systems (shown in one dimension, not drawn to scale). In the input plane is an array of sub-DOE’s. In the output plane is an array of detectors. The solid lines represent the principal rays of the diffraction orders; f represents the propagation length from the lens to the detector array; the space between detectors is for electronic circuitry.

Fig. 5
Fig. 5

Optical architecture for limited-fan-out S-V interconnection systems (shown in one dimension, not drawn to scale). In the input plane is an array of sub-DOE’s; in the output plane is an array of detectors. The solid lines represent the principal rays of the diffraction orders. The example shown has a 3 × 3 fan-out for each sub-DOE. f represents the propagation length from the lens array to the detector array. The space between detectors is for electronic circuitry.

Fig. 6
Fig. 6

Cross talk of a fully connected S-V interconnection system as a function of N (the number of nodes in each dimension) for a normalized detector width of K = 1 and a compression ratio of C = 2, 3, 4.

Fig. 7
Fig. 7

Cross talk of a fully connected S-V interconnection system as a function of the compression ratio C for a normalized detector width of K = 0.5, 1, 2, 4, assuming the number of nodes in each dimension is N = 100.

Fig. 8
Fig. 8

Sources of interconnection cross talk for the limited-fan-out S-V interconnection system (shown in one dimension, not drawn to scale). The solid lines represent the desired connections, and the dashed lines represent the noise connections (SS’s or SDO’s). The example shown has a 3 × 3 fan-out for each sub-DOE. Only two sets of sub-DOE reconstructions are shown.

Fig. 9
Fig. 9

Cross talk of the limited-fan-out S-V interconnection system (with 8 × 8 phase elements in one period of the sub-DOE’s) as a function of the compression ratio C for a normalized detector width of K = 0.5, 1, 2, 4. The sum of the major components, βSS + βSDO, is shown by the dashed line.

Fig. 10
Fig. 10

Reconstruction pattern (shown in one dimension) from a DOE (with 16 × 16 phase elements in one period) for a cross-talk reduction parameter of Y = 2. Note that there is one SDO between each pair of detectors, but SS’s still fall on the detectors.

Fig. 11
Fig. 11

Schematic diagram of the modified Gerchberg–Saxton algorithm.

Fig. 12
Fig. 12

(a), (b) Fused-silica binary-level DOE’s with a 4 μm × 4 μm minimum feature size and 64 μm × 64 μm periods in an aperture of 256 μm × 256 μm. (c) The central 3 × 5 diffraction orders of the reconstruction of the DOE shown in (a). This DOE was designed with Y = 1 so that all diffraction orders are aligned with detector locations (the white squares outline a possible detector layout scheme for receiving weighted diffraction orders). The central 3 × 3 detector locations receive signals, and the other detector locations receive SDO’s. (d) The central 7 × 13 diffraction orders for the DOE shown in (b). This DOE was designed with Y = 3 so that there are two SDO’s between each pair of adjacent detector locations in each dimension.

Fig. 13
Fig. 13

Cross talk of the limited-fan-out S-V interconnection system (with 16 × 16 phase elements in one period of the sub-DOE’s) for a cross-talk reduction parameter of Y = 2 as a function of the compression ratio C and a normalized detector width of K = 0.5, 1, 2, 4 (from bottom to top). The sum of the major components, βSS + βSDO, is shown by the dashed line.

Fig. 14
Fig. 14

Reconstruction pattern (shown in one dimension) from a DOE (with 22 × 22 phase elements in one period) for a cross-talk reduction parameter of Y = 3. Note that there are two SDO’s between each pair of detectors and SS’s do not fall on detectors.

Fig. 15
Fig. 15

Major cross-talk components (βSS + βSDO, βSS, and βSDO) of the limited-fan-out S-V interconnection system as a function of the cross-talk reduction parameter Y.

Fig. 16
Fig. 16

Cross talk of the limited-fan-out S-V interconnection system (with 22 × 22 phase elements in one period of the sub-DOE’s) for a cross-talk reduction parameter of Y = 3 as a function of the compression ratio C and a normalized detector width of K = 0.5, 1, 2, 4 (from bottom to top). The sum of the major components, βSS + βSDO, is shown by the dashed line.

Fig. 17
Fig. 17

Cross talk of the limited-fan-out S-V interconnection system (with 16 × 16 phase elements in one period of the sub-DOE’s) for a cross-talk reduction parameter of Y = 3 as a function of the compression ratio C and a normalized detector width of K = 0.5, 1, 2, 4 (from bottom to top). The sum of the major components, βSS + βSDO, is shown by the dashed line.

Tables (6)

Tables Icon

Table 1 Fully Connected Versus Limited-Fan-Out S-V Systems

Tables Icon

Table 2 Comparison of Fully Connected and Limited-Fan-Out S-V Systems with Parameters of N = 100, λ = 850 nm, Δ = 2 μm, B = 2, C = 2, and M = 10

Tables Icon

Table 3 Limited-Fan-Out S-V Systems with the Cross-Talk Reduction Parameter Y

Tables Icon

Table 4 Limited-Fan-Out Systems with Different Values of the Oversampling Ratio B and the Cross-Talk Reduction Parameter Y for N = 128, M = 5, λ = 850 nm, Δ = 2 μm, S = 96 μm, and P = 32 μm

Tables Icon

Table 5 Same Example and Parameters as Given in Table 4 Except with a Larger Sub-DOE Width of S = 192 μm

Tables Icon

Table 6 Parameters and Average Performances for Each Set of the Nine Sub-DOE’s Designed with Different Values of Y

Equations (60)

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

A r ,   s = J - 2 sinc r J · sinc s J p = 0 J - 1 q = 0 J - 1 h p ,   q × exp - j 2 π J rp + sq ,
sinc x = sin π x π x .
I ( x f ,   y f ) = A r ,   s j λ f · sinc CT λ f x f - r λ f T × sinc CT λ f y f - s λ f T 2 ,
S W = 2 λ f CT .
S S = λ f T .
θ r ,   θ s = tan - 1 r λ T ,   tan - 1 s λ T .
I k 1 BJ 0 + u ,   k 2 BJ 0 + v I u ,   v = u BJ 0 k 1 + u BJ 0 2 v BJ 0 k 2 + v BJ 0 2 .
λ f T = CT .
Z = f = C λ BN Δ 2 .
Z = f = C λ BM Δ 2 .
β l 1 l 0 .
I m , n i , j = V i , j W m , n i , j   sinc 2 CT λ f x f - m λ f T × sinc 2 CT λ f y f - n λ f T ,
I ( 0,0 ) ( x f ,   y f ) = i = - N 2 N 2   - 1 j = - N 2 N 2   - 1   V i , j W 0,0 i , j × sinc 2 CTx f λ f sinc 2 CTy f λ f ,
l ˜ 0 = - P 2 P 2 - P 2 P 2 i = - N 2 N 2   - 1 j = - N 2 N 2   - 1   V i , j W 0,0 i , j × sinc 2 CTx f λ f sinc 2 CTy f λ f d x f d y f .
l 0 = N 2 V max W   - P 2 P 2 - P 2 P 2 sinc 2 CTx f λ f × sinc 2 CTy f λ f d x f d y f .
I ( m , n ) ( x f ,   y f ) = i = - N 2 N 2   - 1 j = - N 2 N 2   - 1   V i , j W m , n i , j × sinc 2 CT λ f x f - m λ f T × sinc 2 CT λ f y f - n λ f T .
l ˜ 1 = m = - N 2 m , n 0,0 N 2   - 1 n = - N 2 N 2   - 1 - P 2 P 2 - P 2 P 2 i = - N 2 N 2   - 1 j = - N 2 N 2   - 1 × V i , j W m , n i , j   sinc 2 CT λ f x f - m λ f T × sinc 2 CT λ f y f - n λ f T d x f d y f .
l 1 = m = - N 2 m , n 0,0 N 2   - 1 n = - N 2 N 2   - 1   N 2 V max W   - P 2 P 2 - P 2 P 2 × sinc 2 CT λ f x f - m λ f T × sinc 2 CT λ f y f - n λ f T d x f d y f .
l 0 = DN 2 V max W   - K K - K K sinc 2 ( x sinc 2 ( y ) d x d y ,
l 1 = D   m = - N 2 ( m , n ) ( 0 , 0 ) N 2   - 1 n = - N 2 N 2   - 1   N 2 V max W   - K K - K K × sinc 2 ( x - mC ) sinc 2 ( y - nC ) d x d y ,
S S = 2 BN Δ = 2 T ,
I ( 0,0 ) Signal ( x f ,   y f ) = i , j X 1   V max W 0,0 i , j × sinc 2 CTx f λ f sinc 2 CTy f λ f ,
I 0,0 SS ( x f ,   y f ) = i , j X 2   V max W 0,0 i , j SS × sinc 2 CTx f λ f sinc 2 CTy f λ f ,
I 0,0 SDO ( x f ,   y f ) = i , j X 3   V max W 0,0 i , j SDO × sinc 2 CTx f λ f sinc 2 CTy f λ f .
I ( 0,0 ) ( x f ,   y f ) = I ( 0,0 ) Signal ( x f ,   y f ) + I ( 0,0 ) SS ( x f ,   y f ) + I ( 0,0 ) SDO ( x f ,   y f ) .
l 0 = D   - K K - K K i , j X 1   V max W 0,0 i , j   sinc 2 x sinc 2 y d x d y .
l 1 SS = D   - K K - K K i , j X 2   V max W 0,0 i , j SS × sinc 2 x sinc 2 y d x d y .
l 1 SDO = D   - K K - K K i , j X 3   V max W 0,0 i , j SDO × sinc 2 x sinc 2 y d x d y .
l 1 tail = D   m = - N 2 m , n 0,0 N 2   - 1 n = - N 2 N 2   - 1 - K K - K K   I m , n x ,   y d x d y .
β SS = l 1 SS l 0 ,
β SDO = l 1 SDO l 0 ,
β tail = l 1 tail l 0 ,
β NRN = l 1 NRN l 0 .
A r ,   s = J - 2 sinc r J sinc s J DFT h p ,   q ,
I r ,   s = | A r ,   s | 2 .
h p ,   q = DFT - 1 J 2 A r ,   s sinc r J sinc s J ,
I 0 r ,   s = W u ,   v 0 for r ,   s = Yu ,   Yv otherwise ,
A r ,   s = I 0 r ,   s 1 / 2 exp j 2 π Φ r ,   s .
S W = 2 BM Δ Y = 2 T Y ,
I k 1 BM + Yu ,   k 2 BM + Yv I Yu ,   Yv = Yu BM k 1 + Yu BM 2 × Yv BM k 2 + Yv BM 2 .
θ Yu ,   θ Yv = tan - 1 Yu λ T ,   tan - 1 Yv λ T .
Y λ f T = S .
Z = S BM Δ λ Y .
SRMSE = min SF k = 1 9 i = 1 5 j = 1 5 t ij k - SF · η ij k 2 J 2 1 / 2 ,
C λ BN Δ 2
1 B λ Δ
B Δ λ
C 3 B 4 N 6 Δ 4 λ
λ C 3 B 4 Δ 4 1 N 2
C λ BM Δ 2
1 B λ Δ N M
MB Δ λ
C 3 B 4 N 2 M 4 Δ 4 λ
λ C 3 B 4 Δ 4 1 M 2
N 2 S 2 Δ 2
S BM Δ λ Y
Y B λ Δ N M
M B Y Δ λ
S 3 BM Δ N 2 λ Y
λ MY S 3 B Δ

Metrics