Abstract

The wavelet transform is a powerful tool for the analysis of short transient signals. We detail the advantages of the wavelet transform over the Fourier transform and the windowed Fourier transform and consider the wavelet as a bank of the VanderLugt matched filters. This methodology is particularly useful in those cases in which the shape of the mother wavelet is approximately known a priori. A two-dimensional optical correlator with a bank of the wavelet filters is implemented to yield the time–frequency joint representation of the wavelet transform of one-dimensional signals.

© 1992 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds. Wavelets, 2nd ed. (Springer-Verlag, Berlin, 1990).
    [CrossRef]
  3. H. Szu, “Matched filter spectrum shaping for light efficiency,” Appl. Opt. 24, 1426–1431 (1985).
    [CrossRef] [PubMed]
  4. H. Szu, J. Caufield, “The mutual time-frequency content of two signals,” Proc. IEEE 72, 902–908 (1984).
    [CrossRef]
  5. H. Szu, B. Telfer, A. Lohmann, “Modified wavelets that accommodate causality,” Opt. Eng. (to be published).
  6. E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
    [CrossRef] [PubMed]
  7. H. Szu, J.A. Blogett, “Self-reference spatiotemporal image restoration technique,” J. Opt. Soc. Am. 72, 1666–1669 (1982).
    [CrossRef]
  8. H. Szu, J. Caulfield, “Optical implementation of wavelet transform in high dimensions,” Appl. Opt. (to be published).
  9. A. Haar, “Zur Theorie der orthogonalen Funktionen-systeme,” Math. Annal. 69, 331–371 (1910).
    [CrossRef]
  10. Y. Meyer, “Principe d’incertitude, bases hibertiennes et algebres d’operation,” Semin. Boubaki 622 (1985).
  11. A. Grossmann, J. Morlet, “Decomposition of Hardy functions into square integrable wavelets of constant shape,” SIAM J. Math. Anal. 15, 723–736 (1984).
    [CrossRef]
  12. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
    [CrossRef]
  13. S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
    [CrossRef]
  14. H. Szu, “Adaptive soliton wavelets,” Opt. Eng. (to be published).
  15. H. Szu, “Adaptive neural net wavelet theory,” Opt. Eng. (to be published).
  16. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).
  17. M. Bastiaans, “Gabor’s expansion of a signal into Gaussian elementary signals,” Proc. IEEE 68, 538–539 (1980).
    [CrossRef]
  18. J. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech and Signal Process. 36, 1169–1179 (1988).
    [CrossRef]
  19. Y. Zan, Y. Li, “Optical determination of Gabor coefficients of transient signals,” Opt. Lett. 16, 1031–1033 (1991).
    [CrossRef]
  20. K. H. Brenner, A. Lohmann, “Optical production of Wigner distributions,” Opt. Commun. 42, 310–314, (1982).
    [CrossRef]
  21. H. Szu, “Two dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).
  22. H. Szu, “Application of Wigner and ambiguity functions to optics,” presented at the International Symposium on Circuits and Systems, San Jose, Calif., 5–7 May 1986.

1991 (1)

1990 (2)

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

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

1989 (1)

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

1988 (1)

J. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech and Signal Process. 36, 1169–1179 (1988).
[CrossRef]

1985 (2)

Y. Meyer, “Principe d’incertitude, bases hibertiennes et algebres d’operation,” Semin. Boubaki 622 (1985).

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

1984 (2)

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

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

1982 (3)

H. Szu, J.A. Blogett, “Self-reference spatiotemporal image restoration technique,” J. Opt. Soc. Am. 72, 1666–1669 (1982).
[CrossRef]

K. H. Brenner, A. Lohmann, “Optical production of Wigner distributions,” Opt. Commun. 42, 310–314, (1982).
[CrossRef]

H. Szu, “Two dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).

1980 (1)

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

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).

1910 (1)

A. Haar, “Zur Theorie der orthogonalen Funktionen-systeme,” Math. Annal. 69, 331–371 (1910).
[CrossRef]

Argoul, S.

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

Arneodo, A.

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

Bastiaans, M.

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

Blogett, J.A.

Brenner, K. H.

K. H. Brenner, A. Lohmann, “Optical production of Wigner distributions,” Opt. Commun. 42, 310–314, (1982).
[CrossRef]

Caufield, J.

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

Caulfield, J.

H. Szu, J. Caulfield, “Optical implementation of wavelet transform in high dimensions,” Appl. Opt. (to be published).

Daubechies, I.

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

Daugman, J.

J. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech and Signal Process. 36, 1169–1179 (1988).
[CrossRef]

Freysz, E.

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

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).

Grossmann, A.

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

Haar, A.

A. Haar, “Zur Theorie der orthogonalen Funktionen-systeme,” Math. Annal. 69, 331–371 (1910).
[CrossRef]

Li, Y.

Lohmann, A.

K. H. Brenner, A. Lohmann, “Optical production of Wigner distributions,” Opt. Commun. 42, 310–314, (1982).
[CrossRef]

H. Szu, B. Telfer, A. Lohmann, “Modified wavelets that accommodate causality,” Opt. Eng. (to be published).

Mallat, S.

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

Meyer, Y.

Y. Meyer, “Principe d’incertitude, bases hibertiennes et algebres d’operation,” Semin. Boubaki 622 (1985).

Morlet, J.

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

Pouligny, B.

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

Szu, H.

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

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

H. Szu, J.A. Blogett, “Self-reference spatiotemporal image restoration technique,” J. Opt. Soc. Am. 72, 1666–1669 (1982).
[CrossRef]

H. Szu, “Two dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).

H. Szu, “Application of Wigner and ambiguity functions to optics,” presented at the International Symposium on Circuits and Systems, San Jose, Calif., 5–7 May 1986.

H. Szu, B. Telfer, A. Lohmann, “Modified wavelets that accommodate causality,” Opt. Eng. (to be published).

H. Szu, J. Caulfield, “Optical implementation of wavelet transform in high dimensions,” Appl. Opt. (to be published).

H. Szu, “Adaptive soliton wavelets,” Opt. Eng. (to be published).

H. Szu, “Adaptive neural net wavelet theory,” Opt. Eng. (to be published).

Telfer, B.

H. Szu, B. Telfer, A. Lohmann, “Modified wavelets that accommodate causality,” Opt. Eng. (to be published).

Zan, Y.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech and Signal Process. (1)

J. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust. Speech and Signal Process. 36, 1169–1179 (1988).
[CrossRef]

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]

IEEE Trans. Pattern Anal. Machine Intell. (1)

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 31, 674–693 (1989).
[CrossRef]

J. Inst. Electr. Eng. Part 3 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. Part 3 93, 429–457 (1946).

J. Opt. Soc. Am. (1)

Math. Annal. (1)

A. Haar, “Zur Theorie der orthogonalen Funktionen-systeme,” Math. Annal. 69, 331–371 (1910).
[CrossRef]

Opt. Commun. (1)

K. H. Brenner, A. Lohmann, “Optical production of Wigner distributions,” Opt. Commun. 42, 310–314, (1982).
[CrossRef]

Opt. Eng. (1)

H. Szu, “Two dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Freysz, B. Pouligny, S. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 745–748 (1990).
[CrossRef] [PubMed]

Proc. IEEE (2)

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

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

Semin. Boubaki (1)

Y. Meyer, “Principe d’incertitude, bases hibertiennes et algebres d’operation,” Semin. Boubaki 622 (1985).

SIAM J. Math. Anal. (1)

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

Other (7)

H. Szu, J. Caulfield, “Optical implementation of wavelet transform in high dimensions,” Appl. Opt. (to be published).

H. Szu, B. Telfer, A. Lohmann, “Modified wavelets that accommodate causality,” Opt. Eng. (to be published).

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

J. M. Combes, A. Grossmann, Ph. Tchamitchian, eds. Wavelets, 2nd ed. (Springer-Verlag, Berlin, 1990).
[CrossRef]

H. Szu, “Adaptive soliton wavelets,” Opt. Eng. (to be published).

H. Szu, “Adaptive neural net wavelet theory,” Opt. Eng. (to be published).

H. Szu, “Application of Wigner and ambiguity functions to optics,” presented at the International Symposium on Circuits and Systems, San Jose, Calif., 5–7 May 1986.

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Figures (4)

Fig. 1
Fig. 1

(a). Basic wavelet function set with the mother wavelet h ( t ) = ( 1 / 2 π σ ) cos ( 2 π f 0 t ) exp ( - t 2 / 2 σ 2 ) and the daughter wavelet h a , b ( t ) = ( 1 / a ) h [ ( t - b ) / a ], where 2πf0 = 5, σ = 1, the dilation factor a = a0m with a0 = 2, and the translation factor b/a = nb0. (b) Fixed Gaussian window as a basic Gabor function set gm,n(t), with g ( t ) = ( 1 / 2 π σ ) exp ( - t 2 / 2 σ 2 ) and gm,n(t) = g(tnt0)exp(jmω0t), where ω0 = π, σ = 1. The real part of the gm,n is shown in the figure.

Fig. 2
Fig. 2

Time–frequency joint representation of (a) the basic wavelet function set shown in Fig. 1(a) and (b) the basic Gabor function set as shown in Fig. 1(b). The scale of the TFJR is equal for the GT and is logarithmic for the WT.

Fig. 3
Fig. 3

Two-dimensional optical correlator with cylindrical FT lenses and a bank of the wavelet filters for the WT of a one-dimensional signal.

Fig. 4
Fig. 4

(a). One-dimensional input signals for the optical correlator shown in Fig. 3. The signals are an approximation to the cosine-Gaussian wavelet function, described in Fig. 1(a), with the Ronchi gratings replacing cos(2πf0) and a slit approximating the Gaussian function. (b) Computer simulation output of correlation between the two daughter wavelets shown in Fig. 1(a) and a bank of filters consisting of two matched filters. The vertical coordinate is the output intensity and the horizontal coordinate is the translation factor b of the signal. (c) Optical output of the correlator of Fig. 3 with a bank of the wavelet filters and the approximate wavelet functions, shown in Fig. 4(a) as the input. The peak in the top half of the plane corresponds with the input s1(t), m = 0 in Fig. 4(a). The peak in the bottom half of the plane corresponds with the input s2(t), m = 1 in Fig. 4(a). When m =0, the dilation a = 1, s1(t) had high frequency. When m = 1 and a = 2, s2(t) had lower frequency and the wavelet was also larger. The correlation output for s2(t) is larger than that for s1(t). The horizontal coordinate corresponds to the shift b of the input signal, and the vertical coordinate represents the dilation a of the signal. The scales of the axis bd in Figs 4(b) and 4(c) are different.

Equations (54)

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t t = ( t - b ) a .
h a , b ( t ) = 1 a 1 / 2 h ( t - b a ) .
W s ( a , b ) = ( h a , b , s ) = h a , b * ( t ) s ( t ) d t = 1 a 1 / 2 h * ( t - b a ) s ( t ) d t ,
H a , b ( f ) = exp ( - j 2 π f t ) h a , b ( t ) d t = a 1 / 2 exp ( - j 2 π f b ) H ( a f ) .
W s ( a , b ) = a 1 / 2 H * ( a f ) exp ( j 2 π f b ) S ( f ) d f = ( H a , b , S ) ,
s ( t ) = 1 C h ( h a , b , s ) h a , b ( t ) d a a 2 d b .
d a a 2 d b ( s 1 , h a , b ) ( h a , b , s 2 ) = C h ( s 1 , s 2 ) ,
C h H ( f ) 2 f d f < .
h ( t ) d t = 0 .
h m , n ( t ) = a 0 - m / 2 h [ a 0 - m ( t - n b 0 a 0 m ) ] ,
W s ( m , n ) = h m , n * ( t ) s ( t ) d t .
h ( t ) 2 d t = 1 ,             h ( t ) 2 t d t = 0 .
h m , n ( t ) 2 t d t = a 0 - m h [ a 0 - m ( t - n b 0 a 0 m ) ] 2 t d t = h ( t ) 2 ( t + n b 0 ) a 0 m d t = n b 0 a 0 m ,
0 H m , n ( ω ) 2 ω d ω = 0 a 0 m H ( a 0 m ω ) 2 ω d ω = a 0 - m k ,             for ω > 0 , - 0 H m , n ( ω ) 2 ω d ω = - 0 a 0 m H ( a 0 m ω ) 2 ω d ω = - a 0 - m k ,             for ω < 0 ,
0 H ( ω ) 2 ω d ω = k ,             - 0 H ( ω ) 2 ω d ω = - k .
FT t { s ( t ) g * ( t ) } = exp ( - j ω t ) s ( t ) g * ( t ) d t ,
A ( ω , τ ) = exp ( - j ω t 0 ) g * ( t 0 - τ 2 ) s ( t 0 + τ 2 ) d t 0 ,
W ( t 0 , ω ) = exp ( - j ω τ ) g * ( t 0 - τ 2 ) s ( t 0 + τ 2 ) d τ .
FT t 1 FT t 2 { s ( t 1 ) g * ( t 2 ) } = FT T { AF } = FT t 0 { WD } ,
C s ( m , n ) = exp ( - j m ω 0 t ) g * ( t - n t 0 ) s ( t ) d t ,
g m , n ( t ) = g ( t - n t 0 ) exp ( j m ω 0 t )
G m , n ( ω = TF t { g m , n ( t ) } = exp [ j ( ω - m ω 0 ) n t 0 ] exp [ - σ 2 ( ω - m ω 0 ) 2 / 2 ] .
T b g ( t ) = g ( t - b ) ;
M ω g ( t ) = exp ( j ω t ) g ( t ) ;
D a g ( t ) = g ( t / a ) a 1 / 2 .
GT :             M ω T b g ( t ) = exp ( j ω t ) g ( t - b ) .
WT :             D a T b h ( t ) = h a , b ( t ) = 1 a 1 / 2 h ( t - b a ) .
W s 1 + s 2 ( a , b ) = W s 1 ( a , b ) + W s 2 ( a , b ) ;
s ( t ) 2 d t = C h W s ( a , b ) 2 d a a 2 d b ,
W s ( t - t 0 ) ( a , b ) = W s ( t ) ( a , b - t 0 ) ;
W α 1 / 2 s ( a t ) ( a , b ) = W s ( t ) ( α a , α b ) .
h ( t ) = cos ( 2 π f 0 t ) 1 ( 2 π ) 1 / 2 σ exp ( - t 2 2 σ 2 ) .
H ( f ) = 1 2 { exp [ - 2 π 2 σ 2 ( f - f 0 ) 2 ] + exp ( - 2 π 2 σ 2 ( f + f 0 ) 2 ] } .
d a a 2 d b h a , b h a , b C h / .
d a a 2 d b s h a , b h a , b χ C h s χ .
s h a , b = W s * ( a , b ) = a 1 / 2 d f exp ( - 2 π f b ) H ( a f ) S * ( f ) , h a , b χ = W x ( a , b ) = a 1 / 2 d f exp ( - 2 π f b ) H * ( a f ) X ( f ) ,
d b exp ( - 2 π f b ) exp ( 2 π f b ) = δ ( f - f ) ,
d f δ ( f - f ) a 1 / 2 H ( a f ) S * ( f ) a 1 / 2 H * ( a f ) X ( f ) = a H ( a f ) 2 S * ( f ) X ( f ) ,
d a a H ( a f ) 2 = d a f a f H ( a f ) 2 = d f f ( H ( f ) 2 ,
C h d f H ( f ) 2 f ,
s χ = d f S * ( f ) X ( f ) .
g p , q ( t ) = exp ( j p t ) g ( t - q ) .
C 3 ( p , q ) = g p , q * ( t ) s ( t ) d t = exp ( - j p t ) g * ( t - q ) s ( t ) d t .
G p , q ( ω ) = g ( t - q ) exp ( j p t ) exp ( - j ω t ) d t = G ( ω - p ) exp ( - j ( ω - p ) q ) .
s ( t ) = 1 2 π C s ( p , q ) g p , q ( t ) d p d q ,
C p , q g p , q ( t ) d p d q = exp ( - j p t 1 ) g * ( t 1 - q ) s ( t 1 ) exp ( j p t ) g ( t - q ) d p d q d t 1 = 2 π δ ( t - t 1 ) g * ( t 1 - q ) g ( t - q ) s ( t 1 ) d t 1 d q = 2 π g ( t - q ) 2 d q s ( t ) = s ( t ) ,
g ( t ) 2 d t = 1.
C s ( m , n ) = exp ( - j m ω 0 t ) g * ( t - n t 0 ) s ( t ) d t ,
1 ( 2 π ) σ exp ( - t 2 σ 2 ) exp ( j ω t ) d t = exp ( - σ 2 ω 2 2 ) .
G p , q ( ω ) = exp [ - j ( ω - m ω 0 ) ] n t 0 exp [ ( - σ 2 ( ω - m ω 0 ) 2 / 2 ) ] .
g ( t ) 2 d t = 1 ,             G ( ω ) 2 d ω = 1 ,
g ( t ) 2 t d t = 0 ,             G ( ω ) 2 ω d ω = 0 ,
G m , n ( ω ) 2 ω d ω = G ( ω - m ω 0 ) 2 ω d ω = m ω 0
g m , n ( t ) 2 t d t = g ( t - n t 0 ) 2 t d t = n t 0 ,

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