Abstract

An optical implementation of the two-dimensional (2-D) wavelet transform and inverse wavelet transform is performed in real time by the exploitation of a new multichannel system that processes the different daughter wavelets separately. The so-coined wavelet-processor system relies on a multichannel replication array generated that uses a Dammann grating and is able to handle every wavelet function. All channels process in parallel using a conventional 2-D correlator. Experimental results applying the Mexican-hat wavelet-decomposition technique are presented.

© 1996 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. Electr. Eng. Part 3 93, 429–457 (1946).
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  4. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time Frequency Methods and Phase Space, 2nd ed. (Springer-Verlag, Berlin, 1990).
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  6. M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, Wavelets and Their Applications (Jones and Bartelett, Boston, Mass., 1992).
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  9. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
  10. X. J. Lu, A. Katz, E. G. Kanterakis, N. P. Caviris, “Joint transform correlation that uses wavelet transforms,” Opt. Lett. 17, 1700–1702 (1992).
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1995

1994

1992

1990

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

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. 10, 452–468 (1988).

1987

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

1985

1984

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

1980

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

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).

1977

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

1971

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

1967

1946

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 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).

Arneodo, A.

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

Bastiaans, M.

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

Beylkin, G.

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

Caulfield, H. J.

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

Caulfield, J.

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

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

Caviris, N. P.

Chen, J.

Chui, C. K.

C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, San Diego, Calif., 1992).

Coifman, R.

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

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

Daubechies, I.

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

Freysz, E.

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

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 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).

Grossmann, A.

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

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).

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.

Kanterakis, E. G.

Katz, A.

Kiryuschev, I.

Klotz, E.

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

Li, Y.

Lohmann, A. W.

Lu, G.

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 and Bartelett, Boston, Mass., 1992).

Marom, E.

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).

Martinet, R. K.

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

Mendlovic, D.

Meyer, Y.

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

Morlet, J.

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

Ouzieli, I.

Paris, D. P.

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. 10, 452–468 (1988).

Pouligny, B.

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

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 and Bartelett, Boston, Mass., 1992).

Roberge, D.

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

Ruskai, M. B.

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

Sheng, Y.

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

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

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.

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

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

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

H. Szu, “Matched filter spectrum shaping for light efficiency,” Appl. Opt. 24, 1426–1431 (1985).

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

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, J.-M.

Yu, F. T. S.

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. 10, 452–468 (1988).

Zhang, Y.

Appl. Opt.

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. 10, 452–468 (1988).

Int. J. Pattern Recogn. Artif. Intell.

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

J. Inst. Electr. Eng. Part 3

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 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).

Opt. Commun.

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

Opt. Eng.

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

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

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 1992, (August), 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).

Proc. IEEE

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

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

Proc. R. Soc. London Ser. B

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London Ser. B 207, 187–217 (1980).

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, San Diego, Calif., 1992).

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

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

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Figures (11)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

2-D Dammann grating, which was used in the experiment, for producing an array of 3 × 3 diffraction orders of equal intensity.

Fig. 3
Fig. 3

2-D array (3 × 3 pixels) of replicated spectra of the input object. The array is generated at the focal plane of the first correlator.

Fig. 4
Fig. 4

(a) Real part of the wavelet h(x, y) of the Mexican hat and (b) it’s Fourier spectrum H(u, ν).

Fig. 5
Fig. 5

Image of the input object.

Fig. 6
Fig. 6

Magnification of the MRMF used in the experiment; it contains five different Mexican-hat FT’s of a wavelet.

Fig. 7
Fig. 7

Magnification of one of eight different Mexican-hat FT’s of a wavelet at the MRMF.

Fig. 8
Fig. 8

Output response of the second correlator. The inverse WT (in this picture) is achieved in the zero diffraction order by summation of the five inverse frequency WT’s, which appear in the first diffraction order.

Fig. 9
Fig. 9

Inverse WT (reconstructed image) in the zero diffraction order, obtained by summation of all the WT’s.

Fig. 10
Fig. 10

Reconstructed image obtained by summation of the three low-frequency wavelets.

Fig. 11
Fig. 11

Reconstructed image obtained by summation of the three high-frequency wavelets.

Equations (21)

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h a , b ( x ) = 1 a h ( x - b a ) .
W ( a , b ) = - f ( x ) h a , b * ( x ) d x .
f ( x ) = C - 1 - - 1 a 2 W ( a , b ) h a , b ( x ) d a d b ,
C = - H ( ξ ) 2 ξ d ξ .
- h ( x ) d x = 0 .
H a , b ( f ) = - h a , b ( x ) exp ( - j 2 π f x ) d x = a H ( a f ) exp ( - j 2 π f b ) .
W ( a , b ) = a - H * ( a f ) F ( f ) exp ( j 2 π f b ) d f .
W ( a , 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 ν ) F ( u , ν ) exp × ( j 2 π u b x , j 2 π ν b y ) d u d ν .
f ( x , y ) = C - 1 - - - 1 a 2 W ( a , b ) h a , b ( x ) d a d b x d b y ,
f ( x , y ) = C - 1 n = - - - 1 2 2 n W ( 2 n , b ) h 2 n , b ( x ) d b x d b y .
MRMF ( u , ν ) = m = - N N n = - N N ( { H [ a m , n ( u - n u 0 , ν - m ν 0 ) ] + exp ( i α m u + i α n ν ) } c . c . ) .
h ( x ) = ( 1 - x 2 ) exp ( - x 2 2 ) .
H ( f ) = 4 π 2 f 2 exp ( - 2 π f 2 ) .
h ( x , y ) = [ 1 - ( x 2 + y 2 ) ] exp ( - x 2 + y 2 2 ) .
H ( u , ν ) = 4 π 2 ( u 2 + ν 2 ) exp [ - 2 π ( u 2 + ν 2 ) ] .
1 2 2 n - - W ( 2 n , b ) h 2 n , b ( x ) d b x d b y .
f ( x , y ) = C - 1 n = - 1 2 2 n - - W ( 2 n , b ) h 2 n , b ( x ) d b x d b y .
f ( x , y ) = C - 1 n = - K n 2 2 n - - W ( 2 n , b ) h 2 n , b ( x ) d b x d b y ,

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