Abstract

A simple optical implementation for the one-dimensional wavelet transform (WT) is proposed. In contrast with previous WT optical implementations, the obtained WT is continuous along both axes (dilation and shift). An optical implementation to the inverse WT is proposed as well. Thus an optical continuous WT processor can be implemented.

© 1998 Optical Society of America

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References

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  1. H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra 26, 73 (1992).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  12. K. Hirokawa, K. Itoh, Y. Ichioka, “Optical wavelet processor by holographic bipolar encoding and joint-transform correlation,” Appl. Opt. 36, 1023–1026 (1997).
    [CrossRef] [PubMed]
  13. H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  14. J. E. Rau, “Detection of differences in real distributions,” J. Opt. Soc. Am. A 56, 1490–1494 (1966).
    [CrossRef]
  15. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1250 (1966).
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1997

1996

1995

1994

1993

1992

1990

I. Daubechies, “The wavelet transform time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

1987

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Patt. Rec. Art. Intell. 1, 273–302 (1987).
[CrossRef]

1966

J. E. Rau, “Detection of differences in real distributions,” J. Opt. Soc. Am. A 56, 1490–1494 (1966).
[CrossRef]

C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1250 (1966).
[CrossRef] [PubMed]

Argoul, F.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Arneodo, A.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Caufield, H. J.

H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra 26, 73 (1992).

Caviris, N. P.

Chen, J.

Daubechies, I.

I. Daubechies, “The wavelet transform time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Freysz, E.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Goodman, J. W.

Grossmann, A.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Patt. Rec. Art. Intell. 1, 273–302 (1987).
[CrossRef]

Hirokawa, K.

Ichioka, Y.

Itoh, K.

Kanterakis, E. G.

Kiyruschev, I.

Konforti, N.

Li, Y.

Lu, X. J.

Marom, E.

Martinet, R. K.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Patt. Rec. Art. Intell. 1, 273–302 (1987).
[CrossRef]

Mendlovic, D.

Morlet, J.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Patt. Rec. Art. Intell. 1, 273–302 (1987).
[CrossRef]

Ouzieli, I.

Pouligny, B.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Rau, J. E.

J. E. Rau, “Detection of differences in real distributions,” J. Opt. Soc. Am. A 56, 1490–1494 (1966).
[CrossRef]

Roberge, D.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Sheng, Y.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

Szu, H.

Szu, H. H.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Tai, J. W.

Tolimieri, R.

Wang, D. X.

Weaver, C. S.

Zhang, Y.

Zhang, Y. X.

Appl. Opt.

IEEE Trans. Inf. Theory

I. Daubechies, “The wavelet transform time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Int. J. Patt. Rec. Art. Intell.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Patt. Rec. Art. Intell. 1, 273–302 (1987).
[CrossRef]

J. Opt. Soc. Am. A

J. E. Rau, “Detection of differences in real distributions,” J. Opt. Soc. Am. A 56, 1490–1494 (1966).
[CrossRef]

Opt. Eng.

Y. Sheng, D. Roberge, H. H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
[CrossRef]

Opt. Lett.

Photon. Spectra

H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra 26, 73 (1992).

Phys. Rev. Lett.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Other

J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).

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Figures (4)

Fig. 1
Fig. 1

(a) Mexican-hat mother wavelet function. (b) The corresponding function (a, b) for the Mexican-hat wavelet.

Fig. 2
Fig. 2

Implementation of the WT through a 4f correlator.

Fig. 3
Fig. 3

Implementation of the WT through a JTC.

Fig. 4
Fig. 4

Optical wavelet processor.

Equations (12)

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W a ,   b =   f x 1 | a |   h * x - b a d x .
ĥ a ,   b = 1 | a |   h b a
W a ,   b =   f x ĥ * a ,   x - b d x = f b     ĥ a ,   b ,
in x ,   y = δ x f y .
WT a ,   b =   in x ,   y h ˜ a - x ,   b - y d x d y = in a ,   b   *   h ˜ a ,   b ,
h ˜ a ,   b = 1 | a |   h * - b a .
h x = 1 - x 2 exp - x 2 2 .
h ˜ a ,   b = 1 | a | 1 - b a 2 exp - b a 2 2 .
in x ,   y = δ x - x 0 f y + h ˜ * - x - x 0 ,   - y = δ x - x 0 f y + h ˜ * x + x 0 ,   y .
f y =   WT a ,   b 1 a 2 | a |   h y - b a d a d b .
h ˘ x ,   y = 1 x 2 | x |   h - y x
f y = f ˆ 0 ,   y = WT x ,   y   *   h ˘ x ,   y | x = 0 .

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