Abstract

The wavelet transform can be expressed mathematically as a convolution between the input function and a continuous set of scaled wavelet mother functions. Optics has managed to implement only the hybrid wavelet transform in which the set of scaled wavelet mother functions is discrete but the shift is continuous. White-light illumination is used to obtain a two-dimensional, fully continuous, on-axis wavelet transformer. When the illumination source is also spatially incoherent, a complete wavelet processor may be constructed.

© 1998 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. [1949-63] 93, 429–457 (1946).
  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. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds., Wavelets: Time Frequency Methods and Phase Space, 2nd ed. (Springer-Verlag, Berlin, 1990).
  4. X. J. Lu, A. Katz, E. G. Kanterakis, N. P. Caviris, “Joint transform correlator that uses wavelet transforms,” Opt. Lett. 17, 1700–1703 (1992).
    [CrossRef] [PubMed]
  5. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
    [CrossRef]
  6. R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Rec. Artif. Intell. 1(2), 273–302 (1987).
    [CrossRef]
  7. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
    [CrossRef]
  8. I. Daubechies, “The wavelet transform time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  9. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of a wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
    [CrossRef] [PubMed]
  10. Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
    [CrossRef]
  11. A. W. Lohmann, B. Telfer, H. Szu, “Casual analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
    [CrossRef]
  12. D. Mendlovic, N. Konforti, “Optical realization of the wavelet transform for two-dimensional objects,” Appl. Opt. 32, 6542–6546 (1993).
    [CrossRef] [PubMed]
  13. 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]
  14. A. W. Lohmann, D. Mendlovic, “Circular harmonic filters for a rotation-invariant incoherent correlator,” Appl. Opt. 31, 6187–6189 (1992).
    [CrossRef] [PubMed]

1995 (1)

1993 (1)

1992 (7)

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 Rec. Artif. Intell. 1(2), 273–302 (1987).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. [1949-63] 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]

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]

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. Elec. Eng. [1949-63] 93, 429–457 (1946).

Grossmann, A.

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

Kanterakis, E. G.

Katz, A.

Kiryuschev, I.

Konforti, N.

Li, Y.

Lohmann, A. W.

Lu, X. J.

Marom, E.

Martinet, R. K.

R. K. Martinet, J. Morlet, A. Grossmann, “Analysis of sound patterns through wavelet transforms,” Int. J. Pattern Rec. Artif. Intell. 1(2), 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 Rec. Artif. Intell. 1(2), 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]

Roberge, D.

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

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

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

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

Telfer, B.

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

Tolimieri, R.

Zhang, Y.

Appl. Opt. (4)

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 Rec. Artif. Intell. (1)

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

J. Inst. Elec. Eng. [1949-63] (1)

D. Gabor, “Theory of communication,” J. Inst. Elec. Eng. [1949-63] 93, 429–457 (1946).

Opt. Eng. (3)

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

A. W. Lohmann, B. Telfer, H. Szu, “Casual 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. (2)

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 (1)

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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Figures (2)

Fig. 1
Fig. 1

Illustration of the suggested optical setup.

Fig. 2
Fig. 2

Suggested optical setup for the wavelet processor.

Equations (16)

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h ab x = 1 a   h x - b a ,
h x ,   y = exp ( 2 π if 0 x 2 + y 2 ) exp - x 2 + y 2 2 ,
H u ,   v = 2 π   exp [ - 2 π 2 ( u 2 + v 2 - f 0 ) 2 ] .
W ( a ,   b ) = -   f ( x ) h ab * ( x ) d x .
u F ( u ,   v ,   λ ) = S ( λ ) λ f - -   u in ( x ,   y ) exp 2 π i ( xu + yv ) λ f × d x d y = S λ λ f   U in u λ f ,   v λ f ,
u out ( b x ,   b y ,   λ ) = 1 λ f - -   u F u ,   v ,   λ H u ,   v × exp 2 π i ( b x u + b y v ) λ f d u d v = S λ 1 λ f 2 - -   U in u λ f ,   v λ f H u ,   v × exp 2 π i ( b x u + b y v ) λ f d u d v .  
u out ( b x ,   b y ,   λ ) = S ( λ ) - -   u in ( x ,   y ) h × b x - x λ f ,   b y - y λ f d x d y .
W ( λ f ,   b x ,   b y ) = u out ( b x ,   b y ,   λ ) = - -   u in ( x ,   y ) h × b x - x λ f ,   b y - y λ f d x d y ,
f x ,   y = 1 C - - - 1 a 3   W a ,   b x ,   b y h × x - b x a ,   y - b y a d a d b x d b y ,
C = - - | H u ,   v | 2 | uv | d u d v .
H 0 ,   0 = 0 .
W ( λ f ,   b x ,   b y ) = S λ - -   I in x ,   y × h b x - x λ f ,   b y - y λ f 2 d x d y ,
I o x ,   y ,   λ = S λ - -   W λ f ,   b x ,   b y × h x - b x λ f ,   y - b y λ f 2 d b x d b y ,
I o x ,   y = -   S λ - -   W λ f ,   b x ,   b y × h x - b x a ,   y - b y a 2 d λ d b x d b y .
S λ = 1 C ( λ f ) 3
I o x ,   y = 1 C - - - 1 a 3   W a ,   b x ,   b y × h x - b x a ,   y - b y a 2 d a d b x d b y ,

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