Abstract

We designed and implemented a composite-wavelet-matched filter that is invariant to continuous-scale changes of the input and is useful for correlation-based pattern recognition of objects whose size is not known exactly. We optimize the adaptive wavelet to extract sparse image features and use the scale–space analysis to determine the wavelet scale for the scale invariance. Experimental results obtained by use of a programmable optical correlator with three liquid-crystal spatial light modulators are demonstrated.

© 1999 Optical Society of America

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References

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  1. H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  2. Y. Sheng, D. Roberge, H. Szu, “Optical wavelet transform,” Opt. Eng. 31, 1840–1845 (1992).
    [CrossRef]
  3. Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
    [CrossRef]
  4. M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
    [CrossRef]
  5. Y. Sheng, D. Roberge, H. Szu, T. Lu, “Optical wavelet matched filters for shift-invariant pattern recognition,” Opt. Lett. 18, 299–301 (1993).
    [CrossRef] [PubMed]
  6. H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
    [CrossRef]
  7. Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
    [CrossRef]
  8. X. Chen, X. Zhang, K. Chen, Q. Li, “Optical wavelet matched filtering with bacteriorhodopsin films,” Appl. Opt. 36, 8413–8416 (1997).
    [CrossRef]
  9. L. S. Jamal-Aldin, R. C. D. Young, C. R. Chatwin, “Application of nonlinearity to wavelet-transformed images to improve correlation filter performance,” Appl. Opt. 36, 9212–9224 (1997).
    [CrossRef]
  10. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  11. Y. Sheng, S. Deschenes, H. J. Caulfield, “Monochromatic electromagnetic wavelets and Huygens principle,” Appl. Opt. 37, 828–833 (1997).
    [CrossRef]
  12. S. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
    [CrossRef]
  13. E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
    [CrossRef]
  14. D. Roberge, Y. Sheng, “Optical wavelet matched filter,” Appl. Opt. 33, 5287–5293 (1994).
    [CrossRef] [PubMed]
  15. D. Roberge, Y. Sheng, “Optical composite wavelet matched filters,” Opt. Eng. 33, 2290–2295 (1994).
    [CrossRef]
  16. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  17. D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
    [CrossRef]
  18. J. Rosen, J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
    [CrossRef] [PubMed]
  19. B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
    [CrossRef]
  20. T. Lindeberg, “Scale–space DOE discrete signals,” IEEE Trans. 12, 234–254 (1990).
  21. L. Gonçalves Neto, D. Roberge, Y. Sheng, “Full range continuous complex modulation using two coupled-mode liquid-crystal televisions,” Appl. Opt. 35, 4567–4576 (1996).
    [CrossRef]
  22. D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
    [CrossRef] [PubMed]
  23. D. Roberge, Y. Sheng, “Optical implementation of real-time correlator with phase-only composite filters,” Opt. Eng. 35, 2541–2547 (1996).
    [CrossRef]
  24. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
    [CrossRef]
  25. M. Petrou, J. Kittler, “Optimal edge detectors for ramp edges,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 483–491 (1991).
    [CrossRef]
  26. L. Gonçalves Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
    [CrossRef]

1997 (3)

1996 (5)

L. Gonçalves Neto, D. Roberge, Y. Sheng, “Full range continuous complex modulation using two coupled-mode liquid-crystal televisions,” Appl. Opt. 35, 4567–4576 (1996).
[CrossRef]

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
[CrossRef]

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

S. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

D. Roberge, Y. Sheng, “Optical implementation of real-time correlator with phase-only composite filters,” Opt. Eng. 35, 2541–2547 (1996).
[CrossRef]

1995 (1)

1994 (3)

D. Roberge, Y. Sheng, “Optical wavelet matched filter,” Appl. Opt. 33, 5287–5293 (1994).
[CrossRef] [PubMed]

D. Roberge, Y. Sheng, “Optical composite wavelet matched filters,” Opt. Eng. 33, 2290–2295 (1994).
[CrossRef]

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

1993 (3)

1992 (4)

H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

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

1991 (1)

M. Petrou, J. Kittler, “Optimal edge detectors for ramp edges,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 483–491 (1991).
[CrossRef]

1990 (1)

T. Lindeberg, “Scale–space DOE discrete signals,” IEEE Trans. 12, 234–254 (1990).

1989 (2)

1988 (1)

D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Adelson, E. H.

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

Caufield, J.

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

Caulfield, H. J.

Chatwin, C. R.

Chen, J.

Chen, K.

Chen, X.

Deschenes, S.

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
[CrossRef]

Dobeck, G. J.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

Dubey, A.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

Ennis, D. J.

Freeman, W. T.

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

Garcia, J. P.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

Gonçalves Neto, L.

Gruber, H.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Heeger, D. J.

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

Ho, H.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

Jamal-Aldin, L. S.

Jared, D. A.

Kittler, J.

M. Petrou, J. Kittler, “Optimal edge detectors for ramp edges,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 483–491 (1991).
[CrossRef]

Konforti, N.

D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Kreibl, M.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Kruger, S.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Li, Q.

Li, Y.

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

Lindeberg, T.

T. Lindeberg, “Scale–space DOE discrete signals,” IEEE Trans. 12, 234–254 (1990).

Lu, T.

Mallat, S.

S. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

Maron, E.

D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Mendlovic, D.

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
[CrossRef]

Mendlvic, D.

D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Onural, L.

Ouzieli, I.

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
[CrossRef]

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
[CrossRef]

Petrou, M.

M. Petrou, J. Kittler, “Optimal edge detectors for ramp edges,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 483–491 (1991).
[CrossRef]

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
[CrossRef]

Roberge, D.

Rosen, J.

Schwarzer, H.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Shamir, J.

Sheng, Y.

Y. Sheng, S. Deschenes, H. J. Caulfield, “Monochromatic electromagnetic wavelets and Huygens principle,” Appl. Opt. 37, 828–833 (1997).
[CrossRef]

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

D. Roberge, Y. Sheng, “Optical implementation of real-time correlator with phase-only composite filters,” Opt. Eng. 35, 2541–2547 (1996).
[CrossRef]

L. Gonçalves Neto, D. Roberge, Y. Sheng, “Full range continuous complex modulation using two coupled-mode liquid-crystal televisions,” Appl. Opt. 35, 4567–4576 (1996).
[CrossRef]

L. Gonçalves Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef]

D. Roberge, Y. Sheng, “Optical composite wavelet matched filters,” Opt. Eng. 33, 2290–2295 (1994).
[CrossRef]

D. Roberge, Y. Sheng, “Optical wavelet matched filter,” Appl. Opt. 33, 5287–5293 (1994).
[CrossRef] [PubMed]

Y. Sheng, D. Roberge, H. Szu, T. Lu, “Optical wavelet matched filters for shift-invariant pattern recognition,” Opt. Lett. 18, 299–301 (1993).
[CrossRef] [PubMed]

H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
[CrossRef]

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

H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

Simoncelli, E. P.

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

Szu, H.

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
[CrossRef]

Y. Sheng, D. Roberge, H. Szu, T. Lu, “Optical wavelet matched filters for shift-invariant pattern recognition,” Opt. Lett. 18, 299–301 (1993).
[CrossRef] [PubMed]

H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

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

Teiwes, S.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Telfer, B.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
[CrossRef]

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
[CrossRef]

Vijaya Kumar, B. V. K.

Wernicke, G.

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

Witherspoon, N.

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

Yang, X.

H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
[CrossRef]

Young, R. C. D.

Zalevsky, Z.

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
[CrossRef]

Zhang, X.

Appl. Opt. (11)

Z. Zalevsky, I. Ouzieli, D. Mendlovic, “Wavelet-transform-based composite filters for invariant pattern recognition,” Appl. Opt. 36, 3141–3149 (1996).
[CrossRef]

D. A. Jared, D. J. Ennis, “Inclusion of filter modulation in synthetic discriminant function construction,” Appl. Opt. 28, 232–239 (1989).
[CrossRef] [PubMed]

J. Rosen, J. Shamir, “Scale invariant pattern recognition with logarithmic radial harmonic filters,” Appl. Opt. 28, 240–244 (1989).
[CrossRef] [PubMed]

H. Szu, Y. Sheng, J. Chen, “The wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

D. Roberge, Y. Sheng, “Optical wavelet matched filter,” Appl. Opt. 33, 5287–5293 (1994).
[CrossRef] [PubMed]

X. Chen, X. Zhang, K. Chen, Q. Li, “Optical wavelet matched filtering with bacteriorhodopsin films,” Appl. Opt. 36, 8413–8416 (1997).
[CrossRef]

L. S. Jamal-Aldin, R. C. D. Young, C. R. Chatwin, “Application of nonlinearity to wavelet-transformed images to improve correlation filter performance,” Appl. Opt. 36, 9212–9224 (1997).
[CrossRef]

Y. Sheng, S. Deschenes, H. J. Caulfield, “Monochromatic electromagnetic wavelets and Huygens principle,” Appl. Opt. 37, 828–833 (1997).
[CrossRef]

L. Gonçalves Neto, D. Roberge, Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944–1950 (1995).
[CrossRef]

L. Gonçalves Neto, D. Roberge, Y. Sheng, “Full range continuous complex modulation using two coupled-mode liquid-crystal televisions,” Appl. Opt. 35, 4567–4576 (1996).
[CrossRef]

IEEE Trans. (1)

T. Lindeberg, “Scale–space DOE discrete signals,” IEEE Trans. 12, 234–254 (1990).

IEEE Trans. Inf. Theory (1)

E. P. Simoncelli, W. T. Freeman, E. H. Adelson, D. J. Heeger, “Shiftable multiscale transforms,” IEEE Trans. Inf. Theory 38, 587–607 (1992).
[CrossRef]

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

M. Petrou, J. Kittler, “Optimal edge detectors for ramp edges,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 483–491 (1991).
[CrossRef]

Opt. Commun. (1)

D. Mendlvic, E. Maron, N. Konforti, “Shift and scale invariant pattern recognition using Mellin radial harmonics,” Opt. Commun. 67, 172–176 (1988).
[CrossRef]

Opt. Eng. (4)

B. Telfer, H. Szu, G. J. Dobeck, J. P. Garcia, H. Ho, A. Dubey, N. Witherspoon, “Adaptive wavelet classification of acoustic backscatter and imagery,” Opt. Eng. 33, 2192–2203 (1994).
[CrossRef]

D. Roberge, Y. Sheng, “Optical composite wavelet matched filters,” Opt. Eng. 33, 2290–2295 (1994).
[CrossRef]

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

D. Roberge, Y. Sheng, “Optical implementation of real-time correlator with phase-only composite filters,” Opt. Eng. 35, 2541–2547 (1996).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. E (1)

H. Szu, X. Yang, B. Telfer, Y. Sheng, “Neural network and wavelet transform for scale invariant data classification,” Phys. Rev. E 48, 1497–1501 (1993).
[CrossRef]

Proc. IEEE (2)

Y. Li, H. Szu, Y. Sheng, J. Caufield, “Wavelet processing and optics,” Proc. IEEE 84, 720–732 (1996).
[CrossRef]

S. Mallat, “Wavelets for a vision,” Proc. IEEE 84, 604–614 (1996).
[CrossRef]

Other (2)

M. Kreibl, H. Schwarzer, S. Teiwes, H. Gruber, S. Kruger, G. Wernicke, “Optical processor for real-time detection of defects in texile webs,” in Optical Pattern Recognition VIII, D. P. Casasent, T. Chao, eds., Proc. SPIE3073, 307–311 (1997).
[CrossRef]

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, B. J. Thompson, Physical Optics Notebook: Tutorials in Fourier Optics, Vol. PM01/HC of SPIE Tutorial Text Series (SPIE Press, Bellingham, Wash., 1989), Chap. 6.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Wavelet transform of (a) a step, (b) a combination of step edges, (c) the correlation between the wavelet transforms of a step edge and that of the combination of step edges.

Fig. 2
Fig. 2

Impulse responses of the CWMF’s that contain blurred contours.

Fig. 3
Fig. 3

Adaptive wavelet with α = 0.3 and θ = 65°.

Fig. 4
Fig. 4

Function R(α, θ) of the letter E (left-hand side) and of an image of an aircraft (right-hand side). This function is useful for determining the coefficients of the adaptive wavelets.

Fig. 5
Fig. 5

Impulse responses of the adaptive CWMF’s for images of (a) the letter E and (b) the Space Shuttle. The scale increment of the training images is Δ = 3 pixels. The size of the images is approximately 30 pixels.

Fig. 6
Fig. 6

Correlation-peak intensities of the adaptive CWMF’s for the letter E (left-hand graph), an image of an aircraft (middle graph), and an image of the Space Shuttle (right-hand graph).

Fig. 7
Fig. 7

Correlation-peak intensities of the adaptive CWMF of (a) the letter E for input images of the letters E and F and (b) an image of the Space Shuttle for input images of the Space Shuttle [Ss(1)] and the Space Shuttle with an open cargo bin [Ss(2)].

Fig. 8
Fig. 8

Optical correlation outputs of the CWMF’s of images of (a) the letter E and (b) the Space Shuttle. The input scenes containing four images are shown above the three-dimensional plots.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Wts, x=1s- tξhξ-xsdξ.
Mp=- xphxdx=0,  p=0, 1, 2, , n,
Wts, 0=1st0M0s+t01! M1s2+t02! M2s3++tn0n! Mnsn+1+Osn+2.
Wts, x=- TuH*suexpi2πxudu.
- FuH*suT*uHsuexpi2πxudu=- Wfs, xWt*s, x-xdx,
Gu=|Hsu|2n=1N αnTnu.
 tmx, yn=1N αnFT-1Tnu|Hsu|2dxdy=cm,
Wts, x=1s- tkξh*ξ-xsdξ=1ks- tξh*ξ-kxksdξ=1k Wtks, kx,
hsx=xs2exp-x22s2,
tx=Afor x00for x<0,
Wts, x=A exp-x22s2.
hsx, y=1s2 hxs, ys=e2πs2xys2exp-12s2x2+y2,
Hsu, sv=-4π2es2uvexp-2π2s2u2+v2
ψsx, y, α=αhsx, y+1-αhsξθ, ηθ,
ξθ=x cos θ-y sin θ,  ηθ=x sin θ+y cos θ.
Rα, θ=1 Wfx, y2dxdy,

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