Abstract

A novel scheme for optical realization of wavelet transform for a one-dimensional signal is described. Using commercially available components, the proposed system can perform wavelet transform in real time. Some preliminary experimental results are demonstrated.

© 1992 Optical Society of America

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References

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  1. M. J. Bastiaans, Opt. Commun. 25, 26 (1978).
    [CrossRef]
  2. Y. Zhang, Y. Li, Opt. Lett. 16, 1031 (1991).
    [CrossRef] [PubMed]
  3. K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
    [CrossRef]
  4. J. F. Walkup, T. F. Krile, Appl. Opt. 16, 746 (1977).
    [CrossRef] [PubMed]
  5. H. H. Szu, J. A. Blodgett, AIP Conf. Proc. 65, 292 (1981).
  6. A. Grossmann, J. Morlet, SIAM J. Appl. Math. 15, 723 (1984).
    [CrossRef]
  7. S. G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37, 2091 (1989).
    [CrossRef]
  8. S. G. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 674 (1989).
    [CrossRef]

1991

1989

S. G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37, 2091 (1989).
[CrossRef]

S. G. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 674 (1989).
[CrossRef]

1984

A. Grossmann, J. Morlet, SIAM J. Appl. Math. 15, 723 (1984).
[CrossRef]

1982

K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
[CrossRef]

1981

H. H. Szu, J. A. Blodgett, AIP Conf. Proc. 65, 292 (1981).

1978

M. J. Bastiaans, Opt. Commun. 25, 26 (1978).
[CrossRef]

1977

Bartlet, H. O.

K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, Opt. Commun. 25, 26 (1978).
[CrossRef]

Blodgett, J. A.

H. H. Szu, J. A. Blodgett, AIP Conf. Proc. 65, 292 (1981).

Brenner, K. H.

K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
[CrossRef]

Grossmann, A.

A. Grossmann, J. Morlet, SIAM J. Appl. Math. 15, 723 (1984).
[CrossRef]

Krile, T. F.

Li, Y.

Lohmann, A. W.

K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
[CrossRef]

Mallat, S. G.

S. G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37, 2091 (1989).
[CrossRef]

S. G. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 674 (1989).
[CrossRef]

Morlet, J.

A. Grossmann, J. Morlet, SIAM J. Appl. Math. 15, 723 (1984).
[CrossRef]

Szu, H. H.

H. H. Szu, J. A. Blodgett, AIP Conf. Proc. 65, 292 (1981).

Walkup, J. F.

Zhang, Y.

AIP Conf. Proc.

H. H. Szu, J. A. Blodgett, AIP Conf. Proc. 65, 292 (1981).

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

S. G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37, 2091 (1989).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

S. G. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 674 (1989).
[CrossRef]

Opt. Commun.

K. H. Brenner, H. O. Bartlet, A. W. Lohmann, Opt. Commun. 42, 32 (1982).
[CrossRef]

M. J. Bastiaans, Opt. Commun. 25, 26 (1978).
[CrossRef]

Opt. Lett.

SIAM J. Appl. Math.

A. Grossmann, J. Morlet, SIAM J. Appl. Math. 15, 723 (1984).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Proposed optical wavelet processing system with its top and side views. LS, laser source; SP, signal plane; WM, wavelet mask; WP, wavelet transform plane; IWM, inverse wavelet mask; RSP, reconstructed signal plane; L, lens. The focal length for lenses L4, L6, L7, and L9 is 150 mm; the focal length for all other lenses is 300 mm.

Fig. 2
Fig. 2

Experimental results for (a) a step function and (b) a linearly chirped periodic gate function, where the original signals are shown in the first row of each photograph.

Fig. 3
Fig. 3

Experimentally reconstructed linearly chirped periodic gate function with its profile shown on the top.

Equations (11)

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W f ( s , u ) = + f ( x ) ψ s ( x u ) d x ,
ψ s ( x ) = s ψ ( s x ) .
Ψ ( ω ) = e j ( ω / 2 ) ω N [ Σ 2 N ( ω 2 + π ) ] 1 / 2 [ Σ 2 N ( ω ) Σ 2 N ( ω 2 ) ] 1 / 2 ,
Σ N ( ω ) = k = + 1 ( ω + 2 k π ) N .
Ψ s ( ω ) = 1 s Ψ ( ω s ) .
f ( x ) = + 0 + W f ( s , u ) ψ s ( x u ) d s d u ,
f ( x ) = j = + + W f ( 2 j , u ) ψ 2 j ( x u ) d u .
f ( x ) = + A f 2 ( N + 1 ) ( u ) ϕ 2 ( N + 1 ) ( x u ) d b + j = N 0 + W f ( 2 j , u ) ψ 2 j ( x u ) d u ,
Ψ ( ω ) = e j ( ω / 2 ) H ( ω 2 + π ) Φ ( ω 2 ) ,
H ( ω ) = n = + [ + ϕ 2 1 ( u ) ϕ ( u n ) d u ] e j n ω .
Σ 8 ( ω ) = 5 + 30 ( cos ω 2 ) 2 + 30 ( sin ω 2 ) 2 ( cos ω 2 ) 2 + 2 ( sin ω 2 ) 4 ( cos ω 2 ) 2 + 70 ( cos ω 2 ) 4 + 2 3 ( sin ω 2 ) 6 105 ( sin ω 2 ) 8 .

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