Abstract

An optical implementation of a wavelet transform is presented. Optical Haar wavelets are created by the use of computer-generated holography. Two different holographic techniques are explored: (1) interferogram and (2) detour-phase. A discrete representation of a continuous wavelet transform is obtained by the optical correlation of an image with a Haar mother wavelet. Experimental results are compared with their digital simulations.

© 1994 Optical Society of America

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References

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  1. T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
    [CrossRef]
  2. S. G. Mallat, “A theory for multi-frequency signal decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  3. C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, San Diego, Calif., 1992), pp. 53–87.
  4. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 141–184.
  5. H. H. Szu, B. Hefler, A. Lohmann, “Causal analytical wavelet transform,” Opt. Eng. 31, 1825–1829 (1992).
    [CrossRef]
  6. X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
    [CrossRef]
  7. W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 121–232 (1978).

1992 (3)

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

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

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

1989 (1)

S. G. Mallat, “A theory for multi-frequency signal decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

1978 (1)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 121–232 (1978).

Burns, T. J.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Caufield, H. J.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

Chui, C. K.

C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, San Diego, Calif., 1992), pp. 53–87.

Fielding, K. H.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 141–184.

Hefler, B.

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

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 121–232 (1978).

Lohmann, A.

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

Mallat, S. G.

S. G. Mallat, “A theory for multi-frequency signal decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Pinski, S. D.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Rogers, S. K.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Ruck, D. W.

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Sheng, Y.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

Szu, H. H.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

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

Yang, X.

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

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

S. G. Mallat, “A theory for multi-frequency signal decomposition,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Opt. Eng. (3)

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

X. Yang, H. H. Szu, Y. Sheng, H. J. Caufield, “Optical Haar wavelet transforms of binary images,” Opt. Eng. 31, 1846–1851 (1992).
[CrossRef]

T. J. Burns, K. H. Fielding, S. K. Rogers, S. D. Pinski, D. W. Ruck, “Optical Haar wavelet transform,” Opt. Eng. 31, 1852–1857 (1992).
[CrossRef]

Prog. Opt. (1)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 121–232 (1978).

Other (2)

C. K. Chui, Wavelets: A Tutorial in Theory and Application (Academic, San Diego, Calif., 1992), pp. 53–87.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 141–184.

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Figures (7)

Fig. 1
Fig. 1

Two-dimensional Haar mother wavelet.

Fig. 2
Fig. 2

VanderLugt optical correlator. S, source; P’s, planes; L’s, lenses.

Fig. 3
Fig. 3

Input image (Lenna).

Fig. 4
Fig. 4

Digital simulation by phase-only representation of Haar wavelet transform.

Fig. 5
Fig. 5

Optical Haar wavelet transform result obtained by interferogram technique.

Fig. 6
Fig. 6

Digital simulation by amplitude and phase representation of Haar wavelet transform.

Fig. 7
Fig. 7

Optical Haar wavelet transform result obtained by detour-phase technique.

Equations (4)

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h ( x , y ) = { 1 if 0 x < 0.5 , 0 y < 0.5 or 0.5 x < 1 , 0.5 y < 1 - 1 if 0.5 x < 1 , 0 y < 0.5 or 0 x < 0.5 , 0.5 y < 1 0 otherwise .
t ( x , y ) = 0.5 { 1 + A ( x , y ) cos [ 2 π α x - ϕ ( x , y ) ] } .
t ( x , y ) = 0.5 { 1 + cos [ 2 π α x - ϕ ( x , y ) ] } .
2 π α x - ϕ ( x , y ) = 2 π n k .

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