Abstract

Real-time wavelet transformations of two-dimensional objects are implemented by use of the conventional coherent correlator with a multireference matched filter. The different daughter wavelets are spatially multiplexed with different reference-beam directions. Two experiments are described, one of them with a spatial light modulator at the input plane in order to enable the real-time property.

© 1993 Optical Society of America

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References

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  1. H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra (August1992), p. 73.
  2. H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of the matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  3. X. J. Lu, A. Katz, E. G. Katerakis, N. P. Caviris, “Joint transform correlation using wavelet transforms,” Opt. Lett. 18, 1700–1703.
  4. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
    [CrossRef]
  5. H. Szu, J. Caulfield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
    [CrossRef]
  6. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
    [CrossRef]
  7. D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).
  8. A. Grossmann, J. Morlet, “Decomposing of Hardy function into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
    [CrossRef]
  9. I. Daubechies, “The wavelet transform time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  10. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).
  11. H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
    [CrossRef]
  12. M. O. Freeman, K. A. Duell, A. Fedor, “Multi-scale optical image processing,” presented at the IEEE International Symposium on Circuits and Systems, Singapore, June 1991.
  13. A. Haar, “Zur thorie der orthogonalen Funktionen-systeme,” Math. Anal. 69, 331–371 (1910).
    [CrossRef]
  14. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
    [CrossRef]
  15. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 674–693 (1989).
    [CrossRef]

1992 (4)

H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra (August1992), p. 73.

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

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
[CrossRef]

1990 (2)

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]

1989 (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 674–693 (1989).
[CrossRef]

1988 (1)

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

1984 (2)

H. Szu, J. Caulfield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
[CrossRef]

A. Grossmann, J. Morlet, “Decomposing of Hardy function into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

1910 (1)

A. Haar, “Zur thorie der orthogonalen Funktionen-systeme,” Math. Anal. 69, 331–371 (1910).
[CrossRef]

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 (August1992), p. 73.

Caulfield, J.

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

H. Szu, J. Caulfield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
[CrossRef]

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]

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

Duell, K. A.

M. O. Freeman, K. A. Duell, A. Fedor, “Multi-scale optical image processing,” presented at the IEEE International Symposium on Circuits and Systems, Singapore, June 1991.

Fedor, A.

M. O. Freeman, K. A. Duell, A. Fedor, “Multi-scale optical image processing,” presented at the IEEE International Symposium on Circuits and Systems, Singapore, June 1991.

Freeman, M. O.

M. O. Freeman, K. A. Duell, A. Fedor, “Multi-scale optical image processing,” presented at the IEEE International Symposium on Circuits and Systems, Singapore, June 1991.

Freysz, E.

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

Gabor, D.

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

Grossmann, A.

A. Grossmann, J. Morlet, “Decomposing of Hardy function into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Haar, A.

A. Haar, “Zur thorie der orthogonalen Funktionen-systeme,” Math. Anal. 69, 331–371 (1910).
[CrossRef]

Katerakis, E. G.

Katz, A.

Lohmann, A. W.

H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
[CrossRef]

Lu, X. J.

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 674–693 (1989).
[CrossRef]

Morlet, J.

A. Grossmann, J. Morlet, “Decomposing of Hardy function into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Pouligny, B.

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

Sheng, Y.

Szu, H.

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

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

H. Szu, J. Caulfield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
[CrossRef]

Szu, H. H.

H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
[CrossRef]

Telfer, B.

H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
[CrossRef]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 41, 909–996 (1988).
[CrossRef]

IEEE Trans. Inf. Theory (1)

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

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 31, 674–693 (1989).
[CrossRef]

Math. Anal. (1)

A. Haar, “Zur thorie der orthogonalen Funktionen-systeme,” Math. Anal. 69, 331–371 (1910).
[CrossRef]

Opt. Eng. (2)

H. H. Szu, B. Telfer, A. W. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
[CrossRef]

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

Opt. Lett. (1)

Photon. Spectra (1)

H. J. Caufield, “Wavelet transforms and their relatives,” Photon. Spectra (August1992), p. 73.

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

H. Szu, J. Caulfield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
[CrossRef]

Proc. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication,” Proc. Inst. Electr. Eng. 93, 429–457 (1946).

SIAM J. Math. Anal. (1)

A. Grossmann, J. Morlet, “Decomposing of Hardy function into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

Other (2)

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

M. O. Freeman, K. A. Duell, A. Fedor, “Multi-scale optical image processing,” presented at the IEEE International Symposium on Circuits and Systems, Singapore, June 1991.

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Figures (8)

Fig. 1
Fig. 1

Optical correlator architecture for a 2-D WT. The MRMF is the multireference matched filter.

Fig. 2
Fig. 2

Typical response for the improved matched filter. The central area contains the zero-order information of the output signal.

Fig. 3
Fig. 3

(a) Mother wavelet function, (b) its impulse response.

Fig. 4
Fig. 4

Magnification of the multireference matched filter with four daughter wavelets.

Fig. 5
Fig. 5

Matched-filter impulse response.

Fig. 6
Fig. 6

(a) Test mask as the input pattern, (b) system output with four different WT orders.

Fig. 7
Fig. 7

Real-time 2-D WT: (a) input pattern, (b) system output.

Fig. 8
Fig. 8

Magnification of the different daughter wavelet responses as shown in Fig. 7: (a) m = 0, (b) m = 1, (c) m = 2, and (d) m = 3.

Equations (10)

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

W ( a , b ) = - f ( x ) h a , b * ( x ) d x ,
h a , b ( x ) = 1 a h ( x - b a ) .
h ( x ) = exp [ - ( x / x 0 ) 2 / 2 ] exp ( i 2 π ω 0 x ) ,
W ( a , b ) = - - f ( x ) h a , b * ( x ) d x 1 d x 2 ,
h a , b ( x ) = 1 a h ( x 1 - b 1 a , x 2 - b 2 a ) .
W d ( m , b ) = - - f ( x ) h m , b * ( x ) d x ,
h m , b ( x ) = 1 a 0 m h ( x 1 - b 1 a 0 m , x 2 - b 2 a 0 m ) .
f ( x ) = 1 C m - - W d ( m , b ) h m , b ( x ) d b .
MRMF ( x ) = m ( { FT [ h m , b ( x ) ] + exp ( i α m x ) } c . c . ) .
MRMF ( x ) = { FT [ m h m , b ( x - b m ) ] + exp ( i α x ) } c . c .

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