Abstract

Superresolution is the process of extending the spectrum of a diffraction-limited image beyond the optical passband. We consider the neural-network approach to accomplish superresolution and present results on simulated gray-scale images degraded by diffraction blur and additive noise. Images are assumed to be sampled at the Nyquist rate, which requires spatial interpolation for avoiding aliasing, in addition to frequency-domain extrapolation. A novel, to our knowledge, use of vector quantization for the generation of training data sets is also presented. This is accomplished by training of a nonlinear vector quantizer, whose codebooks are subsequently used in the supervised training of the neural network with backpropagation.

© 2000 Optical Society of America

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References

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  1. C. A. Dávila, B. R. Hunt, “Superresolution of binary images with a nonlinear interpolative neural network,” Appl. Opt. 39, 2291–2299 (2000).
    [CrossRef]
  2. 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.
  3. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 77–78, and Chap. 6, pp. 137–144, 154–165.
  4. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  5. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
    [CrossRef]
  6. B. R. Hunt, “Imagery super-resolution: emerging prospects,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. SPIE1567, 600–608 (1991).
    [CrossRef]
  7. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  8. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  9. 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]
  10. D. O. Walsh, P. A. Nielsen-Delaney, “Direct method for superresolution,” J. Opt. Soc. Am. A 11, 572–579 (1994).
    [CrossRef]
  11. 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, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.
  12. S. Haykin, Neural Networks: a Comprehensive Foundation (Macmillan, New York, 1994), Chap. 6.
  13. G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Sig. Syst. 2, 303–314 (1989).
    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. J. K. Paik, A. K. Katsaggelos, “Image restoration using a modified Hopfield network,” IEEE Trans. Image Process. 1, 49–63 (1992).
    [CrossRef] [PubMed]
  18. 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]
  19. 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]
  20. S. W. Perry, L. Guan, “Neural network restoration of images suffering space-variant distortion,” Electron. Lett. 31, 1358–1359 (1995).
    [CrossRef]
  21. M. Figueiredo, J. Leitão, “Sequential and parallel image restoration: neural network implementations,” IEEE Trans. Image Process. 3, 789–801 (1994).
    [CrossRef] [PubMed]
  22. 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]
  23. B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
    [CrossRef]
  24. A. Gersho, “Optimal nonlinear interpolative vector quantization,” IEEE Trans. Commun. 38, 1285–1287 (1990).
    [CrossRef]
  25. D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Proc. 7, 119–124 (1998).
    [CrossRef]

2000 (1)

1998 (1)

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

1995 (4)

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

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]

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]

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

1994 (3)

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

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]

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

1993 (1)

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 (1)

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

1991 (2)

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]

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]

1990 (1)

A. Gersho, “Optimal nonlinear interpolative vector quantization,” IEEE Trans. Commun. 38, 1285–1287 (1990).
[CrossRef]

1989 (1)

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

1988 (1)

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]

1975 (1)

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

1974 (2)

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

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

1972 (1)

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]

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 Proc. 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]

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.

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.

Cybenko, G.

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

Dávila, C. A.

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]

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]

Figueiredo, M.

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

Gerchberg, R. W.

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

Gersho, A.

A. Gersho, “Optimal nonlinear interpolative vector quantization,” IEEE Trans. Commun. 38, 1285–1287 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 77–78, and 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]

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, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

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.

C. A. Dávila, B. R. Hunt, “Superresolution of binary images with a nonlinear interpolative neural network,” Appl. Opt. 39, 2291–2299 (2000).
[CrossRef]

D. G. Sheppard, A. Bilgin, M. S. Nadar, B. R. Hunt, M. W. Marcellin, “A vector quantizer for image restoration,” IEEE Trans. Image Proc. 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. Tescher, ed., Proc. SPIE1567, 600–608 (1991).
[CrossRef]

Ichioka, Y.

Itoh, K.

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]

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, 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 Proc. 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 Proc. 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]

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, 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 Proc. 7, 119–124 (1998).
[CrossRef]

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]

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, 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. (2)

Astron. J. (1)

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

Electron. Lett. (1)

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. (1)

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. (1)

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

IEEE Trans. Commun. (1)

A. Gersho, “Optimal nonlinear interpolative vector quantization,” IEEE Trans. Commun. 38, 1285–1287 (1990).
[CrossRef]

IEEE Trans. Image Proc. (1)

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

IEEE Trans. Image Process. (3)

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

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. (1)

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. (2)

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]

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]

Int. J. Imaging Syst. Technol. (1)

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

J. Opt. Soc. Am. (1)

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

Math. Control Sig. Syst. (1)

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

Opt. Acta (1)

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

Opt. Eng. (1)

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]

Other (5)

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, Cambridge, Mass., 1986), Vol. 1, pp. 319–362.

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

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

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.

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

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Figures (19)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Subset of gray-scale images used for training or to test generalization.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

Lena image used to test generalization. Original is 256 × 256; blurred image has been downsampled to 128 × 128 (blurred spectrum has been zero padded to show at same scale as original spectrum).

Fig. 5
Fig. 5

Results of 16–8–9 MLP network on Lena image. Top, sigmoidal output layer; bottom, linear output layer.

Fig. 6
Fig. 6

Spectral correlation coefficient between object and restored images.

Fig. 7
Fig. 7

Generic VQ block diagram.

Fig. 8
Fig. 8

NLIVQ block diagram.

Fig. 9
Fig. 9

Results of 9–8–1L MLP, no postprocessing; 256-VQ training.

Fig. 10
Fig. 10

Three-dimensional renditions of original, band-limited, and restored Lena spectrum, 9–8–1L MLP; 256-VQ training.

Fig. 11
Fig. 11

Postprocessing enhancement block diagram. IFFT, inverse FFT.

Fig. 12
Fig. 12

Results of 9–8–1L MLP with postprocessing; 256-VQ training. Left-hand side, out-of-band quadratic filter (quad); right-hand side, quadratic filter plus Wiener filter for in-band restoration (Q + W).

Fig. 13
Fig. 13

Generalization example of 9–8–1L MLP; 256-VQ training.

Fig. 14
Fig. 14

Restoration of Fig. 13 example.

Fig. 15
Fig. 15

Correlation coefficient images for 9–8–1L barn results; 256-VQ training.

Fig. 16
Fig. 16

Results for 256-NLIVQ restoration.

Fig. 17
Fig. 17

Results of 9–8–1L MLP: noise added prior to interpolation (BSNR = 30 dB).

Fig. 18
Fig. 18

Correlation coefficient image for noisy Lena after Q + W enhancement (BSNR = 30 dB).

Fig. 19
Fig. 19

ISNR reduction as a function of BSNR for 9–8–1L MLP on Lena image. Dashed line (ISNR = 5.92 dB) corresponds to noise-free case.

Equations (15)

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

gx, y=-+-+ hα-x, β-yfα, βdαdβ+ηx, y,
OTFρ=2/πcos-1ρ-ρ1-ρ21/2if ρ10otherwise,
yi=ϕj=0J-1 wijxj,
ϕx=11+exp-x.
=y-d2,
wijn+1=wijn-ηJwijn,
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|Fˆu-m, v-n|21/2,
Fi=f : Ehf=i.
ci*=1Nim=1Nifm,
pˆk, l=1M2m=0N-1n=0N-1 pm, nsincN×a, bexp-jN-1Nπa+b,
a=m-N/Mk,
b=n-N/Ml.
sincNa, b=sinπasinπbsinπa/Nsinπa/N
ci=1Nim=1Nigm

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