Abstract

Optoelectronic wavelet-processor implementations based on Smartt interferometry are described for both one-dimensional (1-D) and two-dimensional (2-D) wavelet transforms. The 1-D processor study focuses on the processor’s capability to perform the wavelet transform when the wavelets are defined in the time domain. The experimental results indicate that the system preserves all the phase information of the selected mother wavelets and thus delivers the true wavelet transform. The 2-D version of the processor is designed for the implementation of the complex wavelet transform. This processor will be valuable for applications, such as image coding-decoding, for which the preservation of phase information is necessary. A pair of prototype processors that incorporates. the 1-D and the 2-D systems has been built and tested to mechanical vibrations. High-quality reconstructed images were also obtained from the experimental data. The proposed systems have great potential for many practical applications.

© 1994 Optical Society of America

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References

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  1. Special issue on wavelet transforms and multiresolution signal analysis, IEEE Trans. on Inf. Theory 38, 529–924 (1992).
  2. Special section on wavelet transforms, Opt Eng. 31, 1823–1916 (1992).
  3. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
    [CrossRef] [PubMed]
  4. X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
    [CrossRef]
  5. R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1974).
  6. S. G. Mallat, “A theory of multifrequency signal decompositions: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 674–693 (1989).
    [CrossRef]

1992 (4)

Special issue on wavelet transforms and multiresolution signal analysis, IEEE Trans. on Inf. Theory 38, 529–924 (1992).

Special section on wavelet transforms, Opt Eng. 31, 1823–1916 (1992).

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

1989 (1)

S. G. Mallat, “A theory of multifrequency signal decompositions: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 674–693 (1989).
[CrossRef]

1974 (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1974).

Caviris, N. P.

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Kanterakis, E. G.

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

Katz, A.

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Li, Y.

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

Lu, X. J.

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

Mallat, S. G.

S. G. Mallat, “A theory of multifrequency signal decompositions: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 674–693 (1989).
[CrossRef]

Smartt, R. N.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1974).

Steel, W. H.

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1974).

Tolimieri, R.

Zhang, Y.

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, “Optical realization of wavelet transform for a one-dimensional signal,” Opt. Lett. 17, 210–212 (1992).
[CrossRef] [PubMed]

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

IEEE Trans. on Inf. Theory (1)

Special issue on wavelet transforms and multiresolution signal analysis, IEEE Trans. on Inf. Theory 38, 529–924 (1992).

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

S. G. Mallat, “A theory of multifrequency signal decompositions: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 674–693 (1989).
[CrossRef]

Jpn. J. Appl. Phys. (1)

R. N. Smartt, W. H. Steel, “Theory and application of point-diffraction interferometers,” Jpn. J. Appl. Phys. 14, 351–356 (1974).

Opt Eng. (1)

Special section on wavelet transforms, Opt Eng. 31, 1823–1916 (1992).

Opt. Commun. (1)

X. J. Lu, A. Katz, E. G. Kanterakis, Y. Li, Y. Zhang, N. P. Caviris, “Image analysis via optical wavelet transform,” Opt. Commun. 92, 337–345 (1992).
[CrossRef]

Opt. Lett. (1)

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the Smartt interferometer.

Fig. 2
Fig. 2

Smartt-interferometer-based optical 1-D wavelet processor. AO, acousto-optic; CL’s, cylindrical lenses; DT’s, 2-D CCD arrays.

Fig. 3
Fig. 3

Filtering mask used in the Smartt-interferometer-based 1-D wavelet processor of Fig. 2.

Fig. 4
Fig. 4

Modified VanderLugt filter. CL’s, cylindrical lenses.

Fig. 5
Fig. 5

Smartt-interferometer-based 2-D wavelet processor: (a) optical wavelet processor, (b) modified filter mask.

Fig. 6
Fig. 6

Proposed videorate opto-electronic wavelet processor. SLM, spatial light modulator; AGC, automatic gain control.

Fig. 7
Fig. 7

Built prototype optoelectronic wavelet processor system.

Fig. 8
Fig. 8

Optical Haar wavelet transforms on a pseudorandom-number sequence made by the processor in Fig. 2.

Fig. 9
Fig. 9

Haar wavelets aligned at their left edges for mask preparation.

Fig. 10
Fig. 10

Miniature tank used in the experiments. (a) Image captured at the output plane of the system without filtering masks installed, (b) low-pass filtering result.

Fig. 11
Fig. 11

Experimental result of 2-D wavelet transforms: (a) generated reference, (b) interference pattern of bandpass channel one, (c) amplitude of the wavelet transform in channel one, (d) interference pattern of bandpass channel two, (e) amplitude of the wavelet transform in channel two, (f) interference pattern of bandpass channel three, (g) amplitude of the wavelet transform in channel three.

Fig. 12
Fig. 12

Images reconstructed electronically by the use of the resulting optical wavelet transforms in Fig. 11: (a) low pass and channel one; (b) low pass, channel one, and channel two; (c) low pass, channel one, channel two, and channel three.

Equations (25)

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W f ( s , u ) = - f ( x ) ψ s ( x - u ) d x ,
ψ s ( x ) = s ψ ( s x ) .
f ( x ) = 1 C ψ - 0 W f ( s , u ) ψ s ( x - u ) d s d u ,
C ψ = 0 Ψ ( ω ) 2 ω d ω .
f ( x ) = - A f 2 - ( N + 1 ) ( u ) ϕ 2 - ( N + 1 ) ( x - u ) d u + j = - N 0 - W f ( 2 j , u ) ψ 2 j ( x - u ) d u ,
A f 2 - ( N + 1 ) ( u ) = - f ( x ) ϕ 2 - ( 2 N + 1 ) ( x - u ) d x ,
ϕ 2 - ( N + 1 ) ( x ) = [ 2 - ( N + 1 ) ] 1 / 2 ϕ [ 2 - ( N + 1 ) x ] .
Ψ ( ω ) = exp [ - j ( ω 2 ) ] H ( ω 2 + π ) Φ ( ω 2 ) ,
H ( ω ) = n = - [ - ϕ 2 - 1 ( u ) ϕ ( u - n ) d u ] exp ( - j n ω ) ,
t ( x i , y i ) = A ( x i , y i ) + [ 1 - A ( x i , y i ) ] rect [ ( x i 2 + y i 2 ) 1 / 2 ] ,
G ( x f , y f ) = F [ S ( x i , y i ) A ( x i , y i ) ] + [ 1 - A ( 0 , 0 ) ] 2 = F [ S ( x i , y i ) A ( x i , y i ) ] 2 + [ 1 - A ( 0 , 0 ) ] 2 + F [ S ( x i , y i ) A ( x i , y i ) ] * [ 1 - A ( 0 , 0 ) ] + F [ S ( x i , y i ) A ( x i , y i ) ] [ 1 - A ( 0 , 0 ) ] * ,
R ( x , y ) = A exp ( - j 2 π α y ) ,
α = sin θ λ .
T w ( x , y ) = B + C [ Ψ y ( x λ f ) exp ( j 2 π β y ) + Ψ y * ( x λ f ) exp ( - j 2 π β y ) ] ,
I DT 1 ( x , y ) = A exp ( j 2 π α y ) + U WT ( x , y ) 2 = A 2 + U WT ( x , y ) 2 + A U WT ( x , y ) exp ( j 2 π α y ) + A U WT * ( x , y ) exp ( - j 2 π α y ) ,
I DT 2 ( x , y ) = U WT ( x , y ) 2 .
t w ( x , y ) = ψ y ( x ) + A w ,
U w ( x f , y f ) = 1 λ f [ ψ y f ( x f λ f ) + A w δ ( x f λ f ) ] ,
I w ( x f , y f ) = t w ( x f , y f ) + U w ( x f , y f ) 2 = A 2 + 1 λ 2 f 2 | Ψ y 0 ( x f λ f ) + A w δ ( x f λ f ) | 2 + A λ f [ Ψ y f ( x f ) + A w δ ( x f λ f ) ] exp ( j 2 π α y f ) + A λ f [ Ψ y f * ( x f λ f ) + A w δ ( x f λ f ) ] exp ( - j 2 π α y f ) .
t ( x , y ) = Ψ ( x , y ) + δ ( 0 , 0 ) .
U ( x 0 , y 0 ) = F [ Ψ ( x , y ) F ( x , y ) ] + F ( 0 , 0 ) ,
I ( x 0 , y 0 ) = F [ Ψ ( x , y ) F ( x , y ) ] + F ( 0 , 0 ) 2 = F [ Ψ ( x , y ) F ( x , y ) 2 + F ( 0 , 0 ) 2 + F * ( 0 , 0 ) F [ Ψ ( x , y ) F ( x , y ) ] + F ( 0 , 0 ) F * [ Ψ ( x , y ) F ( x , y ) ] .
I ( x 0 , y 0 ) = F [ Ψ ( x , y ) F ( x , y ) ] 2 + F ( 0 , 0 ) 2 + 2 F ( 0 , 0 ) F [ Ψ ( x , y ) F ( x , y ) ] .
h ( t ) = rect [ 2 ( t - ¼ ) ] - rect [ 2 ( t - ¾ ) ] ,
H ( ω ) = 4 j exp ( - ω 2 ) 1 - cos ω 2 ω .

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