Abstract

Superresolution is the process by which the bandwidth of a diffraction-limited spectrum is extended beyond the optical passband. Many algorithms exist that are capable of superresolution; however, most are iterative methods, which are ill suited for real-time operation. One approach that has been virtually ignored is the neural-network approach. We consider the feedforward architecture known as a multilayer perceptron and present results on simulated binary images blurred by a diffraction-limited, circular-aperture optical transfer function and sampled at the Nyquist rate. To avoid aliasing, the network performs as a nonlinear spatial interpolator while simultaneously extrapolating in the frequency domain.

© 2000 Optical Society of America

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  14. D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.
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    [CrossRef]
  16. T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
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  17. J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
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    [CrossRef]
  20. Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
    [CrossRef]
  21. W. Zhang, K. Itoh, J. Tanida, Y. Ichioka, “Hopfield model with multistate neurons and its optoelectronic implementation,” Appl. Opt. 30, 195–200 (1991).
    [CrossRef] [PubMed]
  22. J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
    [CrossRef] [PubMed]
  23. Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
    [CrossRef] [PubMed]
  24. M. Bilgen, H. S. Hung, “Neural network for restoration of signals blurred by a random, shift-variant impulse response function,” Opt. Eng. 33, 2723–2727 (1994).
    [CrossRef]
  25. S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
    [CrossRef]
  26. M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
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  27. K. Sivakumar, U. B. Desai, “Image restoration using a multilayer perceptron with a multilevel sigmoidal function,” IEEE Trans. Signal Process. 41, 2018–2021 (1993).
    [CrossRef]
  28. I. Cha, S. A. Kassam, “RBFN restoration of nonlinearly degraded images,” IEEE Trans. Image Proc. 5, 964–975 (1996).
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    [CrossRef]
  31. H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
    [CrossRef]
  32. B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
    [CrossRef]
  33. S. Haykin, Neural Networks: A Comprehensive Foundation (Macmillan, New York, 1994), Chap. 6.

1998

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

1996

I. Cha, S. A. Kassam, “RBFN restoration of nonlinearly degraded images,” IEEE Trans. Image Proc. 5, 964–975 (1996).
[CrossRef]

1995

S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
[CrossRef]

Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
[CrossRef] [PubMed]

T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
[CrossRef]

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

1994

D. O. Walsh, P. A. Nielsen-Delaney, “Direct method for superresolution,” J. Opt. Soc. Am. A 11, 572–579 (1994).
[CrossRef]

M. Bilgen, H. S. Hung, “Neural network for restoration of signals blurred by a random, shift-variant impulse response function,” Opt. Eng. 33, 2723–2727 (1994).
[CrossRef]

M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
[CrossRef] [PubMed]

1993

K. Sivakumar, U. B. Desai, “Image restoration using a multilayer perceptron with a multilevel sigmoidal function,” IEEE Trans. Signal Process. 41, 2018–2021 (1993).
[CrossRef]

1992

J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
[CrossRef] [PubMed]

1991

J. B. Abbiss, B. J. Brames, M. A. Fiddy, “Superresolution algorithms for a modified Hopfield neural network,” IEEE Trans. Signal Process. 39, 1516–1523 (1991).
[CrossRef]

W. Zhang, K. Itoh, J. Tanida, Y. Ichioka, “Hopfield model with multistate neurons and its optoelectronic implementation,” Appl. Opt. 30, 195–200 (1991).
[CrossRef] [PubMed]

1989

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Sig. Syst. 2, 303–314 (1989).
[CrossRef]

1988

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

1987

1984

J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984).
[CrossRef] [PubMed]

1983

1982

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

1975

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1972

Abbiss, J. B.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, “Superresolution algorithms for a modified Hopfield neural network,” IEEE Trans. Signal Process. 39, 1516–1523 (1991).
[CrossRef]

J. B. Abbiss, B. J. Brames, J. S. Bailey, M. A. Fiddy, “Super-resolution and neural computing,” in High Speed Computing, D. P. Casasent, ed., Proc. SPIE880, 100–106 (1988).
[CrossRef]

Bailey, J. S.

J. B. Abbiss, B. J. Brames, J. S. Bailey, M. A. Fiddy, “Super-resolution and neural computing,” in High Speed Computing, D. P. Casasent, ed., Proc. SPIE880, 100–106 (1988).
[CrossRef]

Bilgen, M.

M. Bilgen, H. S. Hung, “Neural network for restoration of signals blurred by a random, shift-variant impulse response function,” Opt. Eng. 33, 2723–2727 (1994).
[CrossRef]

Bilgin, A.

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

Brames, B. J.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, “Superresolution algorithms for a modified Hopfield neural network,” IEEE Trans. Signal Process. 39, 1516–1523 (1991).
[CrossRef]

J. B. Abbiss, B. J. Brames, J. S. Bailey, M. A. Fiddy, “Super-resolution and neural computing,” in High Speed Computing, D. P. Casasent, ed., Proc. SPIE880, 100–106 (1988).
[CrossRef]

Brown, J. W.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, 5th ed. (McGraw-Hill, New York, 1990), Chap. 2 and Chap. 12, pp. 324–327.

Burke, J. J.

Byrne, C. L.

Cha, I.

I. Cha, S. A. Kassam, “RBFN restoration of nonlinearly degraded images,” IEEE Trans. Image Proc. 5, 964–975 (1996).
[CrossRef]

Chellappa, R.

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

Chen, H.

T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
[CrossRef]

Chen, T.

T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
[CrossRef]

Churchill, R. V.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, 5th ed. (McGraw-Hill, New York, 1990), Chap. 2 and Chap. 12, pp. 324–327.

Clarke, L. P.

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Cybenko, G.

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Sig. Syst. 2, 303–314 (1989).
[CrossRef]

Darling, A. M.

Desai, U. B.

K. Sivakumar, U. B. Desai, “Image restoration using a multilayer perceptron with a multilevel sigmoidal function,” IEEE Trans. Signal Process. 41, 2018–2021 (1993).
[CrossRef]

Eichmann, G.

Fiddy, M. A.

J. B. Abbiss, B. J. Brames, M. A. Fiddy, “Superresolution algorithms for a modified Hopfield neural network,” IEEE Trans. Signal Process. 39, 1516–1523 (1991).
[CrossRef]

A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowledge,” J. Opt. Soc. Am. 73, 1466–1469 (1983).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481–1487 (1983).
[CrossRef]

J. B. Abbiss, B. J. Brames, J. S. Bailey, M. A. Fiddy, “Super-resolution and neural computing,” in High Speed Computing, D. P. Casasent, ed., Proc. SPIE880, 100–106 (1988).
[CrossRef]

Figueiredo, M.

M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
[CrossRef] [PubMed]

Fitzgerald, R. M.

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 77–78; Chap. 6, pp. 137–144, 154–165.

Guan, L.

S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
[CrossRef]

Hall, T. J.

Haykin, S.

S. Haykin, Neural Networks: A Comprehensive Foundation (Macmillan, New York, 1994), Chap. 6.

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

Hopfield, J. J.

J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984).
[CrossRef] [PubMed]

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Hung, H. S.

M. Bilgen, H. S. Hung, “Neural network for restoration of signals blurred by a random, shift-variant impulse response function,” Opt. Eng. 33, 2723–2727 (1994).
[CrossRef]

Hunt, B. R.

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Applications of Digital Image Processing XIV, A. G. Tescherm, ed., Proc. SPIE1567, 600–608 (1991).
[CrossRef]

Ichioka, Y.

Itoh, K.

Jain, V. K.

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Jenkins, B. K.

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

Kallergi, M.

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Kassam, S. A.

I. Cha, S. A. Kassam, “RBFN restoration of nonlinearly degraded images,” IEEE Trans. Image Proc. 5, 964–975 (1996).
[CrossRef]

Katsaggelos, A. K.

J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
[CrossRef] [PubMed]

Leitão, J.

M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
[CrossRef] [PubMed]

Li, H. D.

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Li, J. G.

Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
[CrossRef] [PubMed]

Liu, R.

T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Marcellin, M. W.

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

Nadar, M. S.

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

Nielsen-Delaney, P. A.

Paik, J. K.

J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

Perry, S. W.

S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
[CrossRef]

Qian, W.

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Richardson, W. H.

Rumelhart, D. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

Sheppard, D. G.

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

D. G. Sheppard, “Image super-resolution: iterative multiframe algorithms and training of a nonlinear vector quantizer,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1997).

Sivakumar, K.

K. Sivakumar, U. B. Desai, “Image restoration using a multilayer perceptron with a multilevel sigmoidal function,” IEEE Trans. Signal Process. 41, 2018–2021 (1993).
[CrossRef]

Stojancic, M.

Sun, Y.

Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
[CrossRef] [PubMed]

Tanida, J.

Vaid, A.

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

Walsh, D. O.

Williams, R. J.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

Yu, S. Y.

Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
[CrossRef] [PubMed]

Zhang, W.

Zhou, Y. T.

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

Appl. Opt.

Astron. J.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Electron. Lett.

S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

Y. T. Zhou, R. Chellappa, A. Vaid, B. K. Jenkins, “Image restoration using a neural network,” IEEE Trans. Acoust. Speech Signal Process. 36, 1141–1151 (1988).
[CrossRef]

IEEE Trans. Circuits Syst.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

IEEE Trans. Image Proc.

I. Cha, S. A. Kassam, “RBFN restoration of nonlinearly degraded images,” IEEE Trans. Image Proc. 5, 964–975 (1996).
[CrossRef]

IEEE Trans. Image Process.

M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
[CrossRef] [PubMed]

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Process. 7, 119–124 (1998).
[CrossRef]

J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
[CrossRef] [PubMed]

Y. Sun, J. G. Li, S. Y. Yu, “Improvement on performance of modified Hopfield neural network for image restoration,” IEEE Trans. Image Process. 4, 688–692 (1995).
[CrossRef] [PubMed]

IEEE Trans. Neural Netw.

T. Chen, H. Chen, R. Liu, “Approximation capability in c(Rn) by multilayer feedforward networks and related problems,” IEEE Trans. Neural Netw. 6, 25–30 (1995).
[CrossRef]

IEEE Trans. Signal Process.

K. Sivakumar, U. B. Desai, “Image restoration using a multilayer perceptron with a multilevel sigmoidal function,” IEEE Trans. Signal Process. 41, 2018–2021 (1993).
[CrossRef]

J. B. Abbiss, B. J. Brames, M. A. Fiddy, “Superresolution algorithms for a modified Hopfield neural network,” IEEE Trans. Signal Process. 39, 1516–1523 (1991).
[CrossRef]

Int. J. Imaging Syst. Technol.

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Control Sig. Syst.

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Sig. Syst. 2, 303–314 (1989).
[CrossRef]

Opt. Acta

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Eng.

M. Bilgen, H. S. Hung, “Neural network for restoration of signals blurred by a random, shift-variant impulse response function,” Opt. Eng. 33, 2723–2727 (1994).
[CrossRef]

H. D. Li, M. Kallergi, W. Qian, V. K. Jain, L. P. Clarke, “Neural network with maximum entropy constraint for nuclear medicine image restoration,” Opt. Eng. 34, 1431–1440 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[CrossRef] [PubMed]

J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984).
[CrossRef] [PubMed]

Other

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[CrossRef]

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition (MIT Press, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Applications of Digital Image Processing XIV, A. G. Tescherm, ed., Proc. SPIE1567, 600–608 (1991).
[CrossRef]

D. G. Sheppard, “Image super-resolution: iterative multiframe algorithms and training of a nonlinear vector quantizer,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1997).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 77–78; Chap. 6, pp. 137–144, 154–165.

R. V. Churchill, J. W. Brown, Complex Variables and Applications, 5th ed. (McGraw-Hill, New York, 1990), Chap. 2 and Chap. 12, pp. 324–327.

S. Haykin, Neural Networks: A Comprehensive Foundation (Macmillan, New York, 1994), Chap. 6.

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Figures (9)

Fig. 1
Fig. 1

Typical MLP network. Curves inside processing nodes indicate sigmoidal activation functions.

Fig. 2
Fig. 2

MLP Network for 1D binary image superresolution. Output pixels 2 and 3 have been unblurred; pixel 2.5 has been interpolated.

Fig. 3
Fig. 3

Superresolution results of 10–8–3 MLP on 1D binary data.

Fig. 4
Fig. 4

MLP network for 2D binary image superresolution. Output pixels A–D have been unblurred; shaded pixels have been interpolated.

Fig. 5
Fig. 5

Training–test set for 2D binary image superresolution.

Fig. 6
Fig. 6

MLP Results for SEVEN image.

Fig. 7
Fig. 7

MLP Results for NINE image.

Fig. 8
Fig. 8

Spectral correlation coefficient for images SEVEN and NINE.

Fig. 9
Fig. 9

Average ISNR as a function of input SNR for FIVE image.

Tables (1)

Tables Icon

Table 1 MLP Performance for 2D Binary Images

Equations (8)

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gx, y=-+-+hα-x, β-yfα, βdαdβ,
jincr=d2J1πr2r,
OTFρ=2/πcos-1ρ-ρ1-ρ21/2,
y=ϕ n=0N-1 wnxn,
ϕx = 11+exp-βx,
=y-d2,
ISNR = f-g2f-fˆ2,
rm n=u=0M-1v=0M-1 Fu-m, v-nFˆ*u-m, v-nu=0M-1v=0M-1|Fu-m, v-n)|2u=0M-1v=0M-1|Fˆ(u-m, v-n|21/2,

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