Abstract

We present a novel method for achieving in real time a two-dimensional optical wavelet decomposition with white-light illumination. The underlying idea of the suggested method is wavelength multiplexing. The information in the different wavelet components of an input object is transmitted simultaneously in different wavelengths and summed incoherently at the output plane. Experimental results show the utility of the new proposed method.

© 2001 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  2. H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  3. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time Frequency Methods and Phase Space, 2nd ed. (Springer-Verlag, Berlin, 1990).
  4. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
    [CrossRef]
  5. X. J. Lu, A. Katz, E. G. Kanterakis, N. P. Caviris, “Joint transform correlation using wavelet transforms,” Opt. Lett. 17, 1700–1703 (1992).
    [CrossRef] [PubMed]
  6. R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
    [CrossRef]
  7. R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Recogn. Artif. Intell. 1, 273–302 (1987).
    [CrossRef]
  8. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
    [CrossRef]
  9. I. Daubechies, “The wavelet transform time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  10. D. Mendlovic, I. Ouzieli, I. Kiryuschev, E. Marom, “Two-dimensional wavelet transform achieved by computer-generated multireference matched filter and Dammann grating,” Appl. Opt. 34, 8213–8219 (1995).
    [CrossRef] [PubMed]
  11. D. Mendlovic, N. Konforti, “Optical realization of the wavelet transform for two-dimensional objects,” Appl. Opt. 32, 6542–6546 (1993).
    [CrossRef] [PubMed]
  12. M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
    [CrossRef]
  13. J. García, Z. Zalevsky, D. Mendlovic, “Two-dimensional wavelet transform by wavelength multiplexing,” Appl. Opt. 35, 7019–7024 (1996).
    [CrossRef] [PubMed]
  14. D. Mendlovic, “Continuous two-dimensional on-axis optical wavelet transformer and wavelet processor with white-light illumination,” Appl. Opt. 37, 1279–1282 (1998).
    [CrossRef]
  15. Z. Zalevsky, “Experimental implementation of a continuous two-dimensional on-axis optical wavelet transformer with white-light illumination,” Opt. Eng. 37, 1372–1375 (1998).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics, 2nd ed., Electrical Engineering Series (McGraw-Hill, Singapore, 1996).

1998 (2)

Z. Zalevsky, “Experimental implementation of a continuous two-dimensional on-axis optical wavelet transformer with white-light illumination,” Opt. Eng. 37, 1372–1375 (1998).
[CrossRef]

D. Mendlovic, “Continuous two-dimensional on-axis optical wavelet transformer and wavelet processor with white-light illumination,” Appl. Opt. 37, 1279–1282 (1998).
[CrossRef]

1997 (1)

R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
[CrossRef]

1996 (1)

1995 (1)

1993 (1)

1992 (3)

1990 (2)

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

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

1987 (1)

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

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 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]

Bock, B.

M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Caulfield, J.

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

Duell, K. A.

M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Fedor, A.

M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

Ferreira, C.

R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
[CrossRef]

Freeman, M. O.

M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[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]

Gabor, D.

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

García, J.

R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
[CrossRef]

J. García, Z. Zalevsky, D. Mendlovic, “Two-dimensional wavelet transform by wavelength multiplexing,” Appl. Opt. 35, 7019–7024 (1996).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., Electrical Engineering Series (McGraw-Hill, Singapore, 1996).

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).
[CrossRef]

Kanterakis, E. G.

Katz, A.

Kiryuschev, I.

Konforti, N.

Lu, X. J.

Maestre, R. A.

R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
[CrossRef]

Marom, E.

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).
[CrossRef]

Mendlovic, D.

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).
[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]

Sheng, Y.

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]

Zalevsky, Z.

Z. Zalevsky, “Experimental implementation of a continuous two-dimensional on-axis optical wavelet transformer with white-light illumination,” Opt. Eng. 37, 1372–1375 (1998).
[CrossRef]

J. García, Z. Zalevsky, D. Mendlovic, “Two-dimensional wavelet transform by wavelength multiplexing,” Appl. Opt. 35, 7019–7024 (1996).
[CrossRef] [PubMed]

Appl. Opt. (5)

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]

Int. J. Pattern Recogn. Artif. Intell. (1)

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

J. Inst. Electr. Eng. (1)

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

Opt. Commun. (1)

R. A. Maestre, J. García, C. Ferreira, “Pattern recognition using sequential matched filtering of wavelet coefficients,” Opt. Commun. 133, 401–414 (1997).
[CrossRef]

Opt. Eng. (2)

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

Z. Zalevsky, “Experimental implementation of a continuous two-dimensional on-axis optical wavelet transformer with white-light illumination,” Opt. Eng. 37, 1372–1375 (1998).
[CrossRef]

Opt. Lett. (1)

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]

Other (3)

M. O. Freeman, A. Fedor, B. Bock, K. A. Duell, “Optical wavelet processor for producing spatially-localized, ring-wedge-type information,” in Optical Information Processing Systems and Architectures IV, B. Javidi, ed., Proc. SPIE1772, 241–250 (1992).
[CrossRef]

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., Electrical Engineering Series (McGraw-Hill, Singapore, 1996).

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Figures (10)

Fig. 1
Fig. 1

Sketch of the experimental optical setup for obtaining a two-dimensional wavelet transform with white-light illumination.

Fig. 2
Fig. 2

Schematic illustration of the diffractive optical element.

Fig. 3
Fig. 3

Illustration of the color distribution over the plane of the slit. For graphical purposes the three chromatically dispersed wavelet components are not perfectly superimposed. With respect to the visible electromagnetic spectrum, λA=450 nm and λB=750 nm.

Fig. 4
Fig. 4

(a) Chromatic dispersion at the plane of the slit for a particular wavelet component of the input object. For graphical purposes the different replicas of the wavelet component for each wavelength are not perfectly superimposed. (b) Representation of the lateral shift in the wavelet component replicas for the different wavelengths and the slit position.

Fig. 5
Fig. 5

(a) Output plane considering a slit represented by a rect function and (b) the same output plane as that in (a) but using the complementary representation at axes (λ, y5).

Fig. 6
Fig. 6

(a) Output plane considering a slit represented by a delta function and (b) the same output plane as that in (a) but using the complementary representation along the (λ, y5) axes.

Fig. 7
Fig. 7

Input object used for the experimental analysis of the optical setup.

Fig. 8
Fig. 8

Output results obtained for a defined position of the slit in the (a) R channel of the color camera, (b) G channel of the color camera, and (c) B channel of the color camera.

Fig. 9
Fig. 9

Output results obtained for a different position of the slit in the (a) R channel of the color camera and (b) G channel of the color camera.

Fig. 10
Fig. 10

Output result obtained for another different position of the slit in the R channel of the color camera.

Equations (22)

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hab(x)=1a hx-ba,
W(a, b)=-+f(x)hab*(x)dx.
W(a, b)=1a-+-+f(x, y)h*x-b1a, y-b2adxdy,
b=b1b2.
h(x, y)=exp2πif0x2+y2exp-x2+y22.
H(u, v)=2π exp[-2π2(u2+v2-f0)2].
H(u, v)rectu2+v2-f0W,
U(x1, y1; λ)=S(λ)s(x1, y1),
U(x2, y2; λ)=S(λ)s˜x2λz, y2λz×H(x2, y2)exp(-i2πy2/T),
U(x3, y3; λ)=S(λ)W(x3, y3)δx3, y3-λzT
=S(λ)Wx3, y3-λzT,
U(x3, y3; λ)=S(λ)Wx3, y3-λzTrecty3-Y0L,
U(x4, y4; λ)=S(λ)W˜x4λz, y4λzexp(-i2πy4/T)L sincy4Lλzexp(-i2πY0y4)/λz×exp(i2πy4/T)H(x4, y4),
U(x5, y5; λ)=S(λ)Wx5, y5+λzTrecty5+Y0Lδy5-λzThx5λz, y5λz.
U(x5, y5; λ)=S(λ)W(x5, y5)recty5+Y0-λz/TLhx5λz, y5λz.
y5+Y0-λ1zT=L/2  λ1=Tz (y5+Y0-L/2),
y5+Y0-λ2zT=-L/2  λ2=Tz (y5+Y0+L/2),
(Δλ)y5=λ2-λ1=LTz,
λy5=Tz (Y0+y5).
λ(-Δy/2)=TzY0-Δy2,λ(Δy/2)=TzY0+Δy2.
(Δλ)W=λ(Δy/2)-λ(-Δy/2)=ΔyTz,
U(x5, y5; λ)=S(λ)W(x5, y5)δy5+Y0-λzThx5λz, y5λz.

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