Abstract

A two-dimensional wavelet transform is optically performed in real time by use of a new multichannel system that processes the different daughter wavelets separately. The system, which is able to handle every wavelet function, relies on a Dammann grating for generating a multichannel array. All channels are processed in parallel by a conventional two-dimensional correlator. Experimental results applying Morlet-wavelet decomposition are presented.

© 1995 Optical Society of America

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. London 93, 429–457 (1946).
  3. M. Bastiaans, “Gabor expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
    [CrossRef]
  4. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time–Frequency Methods and Phase Space, 2nd ed. (Springer-Verlag, Berlin, 1990).
  5. C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, New York, 1992).
  6. M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).
  7. H. Szu, “Matched filter spectrum shaping for light efficiency,” Appl. Opt. 24, 1426–1431 (1985).
    [CrossRef] [PubMed]
  8. H. Szu, J. Caulfield, “The mutual time–frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
    [CrossRef]
  9. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
    [CrossRef]
  10. X. J. Lu, A. Katz, E. G. Kanterakis, N. P. Caviris, “Joint transform correlation using wavelet transforms,” Opt. Lett. 18, 1700–1703.
  11. R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
    [CrossRef]
  12. M. Porat, Y. Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-10, 452–468 (1988).
    [CrossRef]
  13. H. J. Caulfield, “Wavelet transforms and their relatives,” Photon. Spectra (August1992), p. 73.
  14. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
    [CrossRef]
  15. A. Grossmann, J. Morlet, “Decomposing of Hardy function into a square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
    [CrossRef]
  16. I. Daubechies, “The wavelet transform time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  17. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210– 212 (1992).
    [CrossRef] [PubMed]
  18. H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  19. 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.
  20. P.-X. Wang, J.-W. Tai, Y.-X. Zhang, “Two-dimensional optical wavelet transform in space domain and its performance analysis,” Appl. Opt. 33, 5271–5274 (1994).
    [CrossRef] [PubMed]
  21. D. Mendlovic, N. Konforti, “Optical realization of the wavelet transform for two-dimensional objects,” Appl. Opt. 32, 6542–6546 (1993).
    [CrossRef] [PubMed]
  22. Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
    [CrossRef]
  23. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  24. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  25. J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

1994

1993

1992

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

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

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210– 212 (1992).
[CrossRef] [PubMed]

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

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]

1989

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

1988

M. Porat, Y. Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-10, 452–468 (1988).
[CrossRef]

1987

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

1985

1984

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 a square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
[CrossRef]

1980

M. Bastiaans, “Gabor expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

1977

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1971

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1946

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

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]

Bastiaans, M.

M. Bastiaans, “Gabor expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

Beylkin, G.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Caulfield, H. J.

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

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

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.

Chui, C. K.

C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, New York, 1992).

Coifman, R.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Daubechies, I.

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

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Downs, M. M.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

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,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Grossmann, A.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

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

Jahns, J.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Kanterakis, E. G.

Katz, A.

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Konforti, N.

Li, Y.

Lu, T.

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

Lu, X. J.

Mallat, S.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Martinet, R. K.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

Mendlovic, D.

Meyer, Y.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Morlet, J.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

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

Porat, M.

M. Porat, Y. Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-10, 452–468 (1988).
[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]

Prise, M. E.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Raphael, L.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Roberge, D.

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

Ruskai, M. B.

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

Sheng, Y.

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

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

Streibl, N.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Szu, H.

H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of 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, “Matched filter spectrum shaping for light efficiency,” Appl. Opt. 24, 1426–1431 (1985).
[CrossRef] [PubMed]

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

Tai, J.-W.

Tolimieri, R.

Walker, S. J.

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Wang, P.-X.

Zeevi, Y. Y.

M. Porat, Y. Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-10, 452–468 (1988).
[CrossRef]

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]

IEEE Trans. Pattern Anal. Machine Intell.

M. Porat, Y. Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-10, 452–468 (1988).
[CrossRef]

Int. J. Pattern Recog. Artif. Intell.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recog. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

J. Inst. Elect. Eng. London

D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. London 93, 429–457 (1946).

Opt. Acta

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Eng.

Y. Sheng, T. Lu, D. Roberge, H. J. Caulfield, “Optical N4 implementation of a two-dimensional wavelet transform,” Opt. Eng. 31, 1859–1864 (1992).
[CrossRef]

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

J. Jahns, M. E. Prise, M. M. Downs, S. J. Walker, N. Streibl, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Opt. Lett.

Photon. Spectra

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

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]

Proc. IEEE

M. Bastiaans, “Gabor expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
[CrossRef]

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

SIAM J. Math. Anal.

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

Other

J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time–Frequency Methods and Phase Space, 2nd ed. (Springer-Verlag, Berlin, 1990).

C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, New York, 1992).

M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones & Bartelett, Boston, Mass., 1992).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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 (11)

Fig. 1
Fig. 1

Optical setup for array generation that uses Dammann gratings.

Fig. 2
Fig. 2

Basic period of a 1-D Dammann grating's transparency function with N transition points (here N = 6, thus creating 13 diffraction orders of equal intensities).

Fig. 3
Fig. 3

(a) 2-D Dammann grating for producing an array of 3 × 3 diffraction orders of equal intensity. (b) One period of the Dammann grating, implemented by the crossing of two 1-D gratings with one transition point per period.

Fig. 4
Fig. 4

Optical correlator with Dammann gratings for a 2-D WT.

Fig. 5
Fig. 5

Multireference matched filter, divided into squares, in which each square is encoded by H(x, y) and is scaled by a different dilation factor and a different reference beam (not shown in the figure), thus achieving spatial multiplexing in the output plane.

Fig. 6
Fig. 6

Output response for the correlator shown in Fig. 4 with the MRMF shown in Fig. 5. The central area contains the zero-order information of the output signal.

Fig. 7
Fig. 7

Input pattern (Roseta function).

Fig. 8
Fig. 8

(a) Real part of the Morlet wavelet h(x, y) and (b) its Fourier spectrum H(u, v).

Fig. 9
Fig. 9

Magnification of the multireference matched filter, which contains eight different Morlet-wavelet Fourier transforms.

Fig. 10
Fig. 10

Magnification of one of the eight different Morlet-wavelet Fourier transforms at the MRMF.

Fig. 11
Fig. 11

System's output response with eight different WT orders.

Equations (12)

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

h a b ( x ) = h a ( x b a ) .
W ( a , b ) = + f ( x ) h a b * ( x ) d x ,
W ( ā , b ̅ ) = + + f ( x , y ) h a b ̅ * ( x , y ) d x d y ,
W ( a , b ̅ ) = f ( x , y ) h a b ̅ * ( x , y ) d x d y ,
h a b ̅ ( x , y ) = 1 a h ( x b x a , y b y a ) .
W ( a , b ̅ ) = a H * ( a u , a v ) × exp ( j 2 π u b x , j 2 π v b y ) F ( u , v ) d u d v .
I ( x , y ) = | T ( x , y ) | 2 = n m | δ ( x n x 0 , y m y 0 ) | 2 ,
I ( x , y ) = | a ( x , y ) T ( x , y ) | 2 = n m | a ( x , y ) δ ( x n x 0 , y m y 0 ) | 2 = n m | a ( x n x 0 , y my 0 ) | 2 .
t ( x , y ) = t 1 ( x ) t 2 ( y ) ,
MRMF ( u , v ) = m = N + N n = N + N × ( { H [ a m , n ( u n u 0 , v m v 0 ) ] + exp ( i α m u + i α n v ) } c . c . ) .
h ( x , y ) = exp [ i 2 π f 0 ( x 2 + y 2 ) 1 / 2 ] exp ( x 2 + y 2 2 ) .
H ( u , v ) = 2 π ( exp { 2 π 2 [ ( u 2 + v 2 ) 1 / 2 f 0 ] 2 } ) .

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