Abstract

We introduce optical wavelet matched filters that perform the wavelet transforms for edge enhancement and perform correlations between the wavelet coefficients for shift-invariant pattern recognition. These new bandpass matched filters show improved discrimination capability with respect to the conventional matched spatial filter and improved signal-to-noise ratio with respect to the phase-only matched filter.

© 1993 Optical Society of America

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References

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  1. S. G. Mallat, IEEE Trans. Acoust. Speech Signal Process. 37, 2091 (1989).
    [CrossRef]
  2. R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
    [CrossRef]
  3. H. Szu, Y. Sheng, J. Chen, Appl. Opt. 31, 3267 (1992).
    [CrossRef] [PubMed]
  4. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
    [CrossRef] [PubMed]
  5. Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, Opt. Lett. 17, 210 (1992).
    [CrossRef] [PubMed]
  6. S. G. Mallat, W. L. Hwang, IEEE Transe. Inf. Theory 38, 617 (1992).
    [CrossRef]
  7. B. Telfer, H. Szu, Opt. Eng. 31, 1830 (1992).
    [CrossRef]
  8. Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
    [CrossRef]
  9. S. Lowenthal, Y. Belvaux, C. R. Acad. Sci. Ser. B 262, 413 (1966).
  10. A. W. Lohmman, D. P. Paris, Appl. Opt. 7, 651 (1968).
    [CrossRef]
  11. J. L. Horner, P. D. Gianino, Appl. Opt. 23, 812 (1984).
    [CrossRef] [PubMed]

1992 (5)

H. Szu, Y. Sheng, J. Chen, Appl. Opt. 31, 3267 (1992).
[CrossRef] [PubMed]

Y. Zhang, Y. Li, E. G. Kanterakis, A. Katz, X. J. Lu, R. Tolimieri, N. P. Caviris, Opt. Lett. 17, 210 (1992).
[CrossRef] [PubMed]

S. G. Mallat, W. L. Hwang, IEEE Transe. Inf. Theory 38, 617 (1992).
[CrossRef]

B. Telfer, H. Szu, Opt. Eng. 31, 1830 (1992).
[CrossRef]

Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
[CrossRef]

1990 (1)

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

1989 (1)

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

1987 (1)

R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
[CrossRef]

1984 (1)

1968 (1)

1966 (1)

S. Lowenthal, Y. Belvaux, C. R. Acad. Sci. Ser. B 262, 413 (1966).

Argoul, F.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

Arneodo, A.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

Belvaux, Y.

S. Lowenthal, Y. Belvaux, C. R. Acad. Sci. Ser. B 262, 413 (1966).

Caviris, N. P.

Chen, J.

Freysz, E.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

Gianino, P. D.

Grossmann, A.

R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
[CrossRef]

Horner, J. L.

Hwang, W. L.

S. G. Mallat, W. L. Hwang, IEEE Transe. Inf. Theory 38, 617 (1992).
[CrossRef]

Kanterakis, E. G.

Katz, A.

Kronland-Martinet, R.

R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
[CrossRef]

Li, Y.

Lohmman, A. W.

Lowenthal, S.

S. Lowenthal, Y. Belvaux, C. R. Acad. Sci. Ser. B 262, 413 (1966).

Lu, T.

Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
[CrossRef]

Lu, X. J.

Mallat, S. G.

S. G. Mallat, W. L. Hwang, IEEE Transe. Inf. Theory 38, 617 (1992).
[CrossRef]

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

Morlet, J.

R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
[CrossRef]

Paris, D. P.

Pouligny, B.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

Roberge, D.

Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
[CrossRef]

Sheng, Y.

Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
[CrossRef]

H. Szu, Y. Sheng, J. Chen, Appl. Opt. 31, 3267 (1992).
[CrossRef] [PubMed]

Szu, H.

Telfer, B.

B. Telfer, H. Szu, Opt. Eng. 31, 1830 (1992).
[CrossRef]

Tolimieri, R.

Zhang, Y.

Appl. Opt. (3)

C. R. Acad. Sci. Ser. B (1)

S. Lowenthal, Y. Belvaux, C. R. Acad. Sci. Ser. B 262, 413 (1966).

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

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

IEEE Transe. Inf. Theory (1)

S. G. Mallat, W. L. Hwang, IEEE Transe. Inf. Theory 38, 617 (1992).
[CrossRef]

Int. J. Pattern Recogn. Artif. Intell. (1)

R. Kronland-Martinet, J. Morlet, A. Grossmann, Int. J. Pattern Recogn. Artif. Intell. 1, 273 (1987).
[CrossRef]

Opt. Eng. (2)

B. Telfer, H. Szu, Opt. Eng. 31, 1830 (1992).
[CrossRef]

Y. Sheng, T. Lu, D. Roberge, Opt. Eng. 31, 1859 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, Phys. Rev. Lett. 64, 745 (1990).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Optical wavelet transform. Clockwise from top left: input image; output with the 1-D Mexican-hat wavelet with sy = 2; output with the wavelet of Eq. (6) with s = 2; output with the wavelet of Eq. (5) with sx = sy = 2.

Fig. 2
Fig. 2

Two approaches to obtain correlations between two wavelet transform coefficients for pattern recognition: (a) Two steps, (b) one step, where f(x, y) is known, a priori, and t(x, y) is a unknown input pattern.

Fig. 3
Fig. 3

Optical wavelet matched filters, (a) Input, (b) output of the conventional matched filter, (c) output of the wavelet matched filter with the 1-D Mexican-hat wavelet with sy = 5, (d) output with the wavelet of Eq. (5) with sx = sy = 5.

Fig. 4
Fig. 4

Three-dimensional plots of the outputs of four phase-only wavelet matched filters with the dyadic wavelets of Eq. (6) with s = 1/2 (left-hand corner), s = 1 (top corner), s = 2 (bottom corner), and s = 4 (right-hand corner). The input was that of Fig. 3(a) degraded by an additive random noise.

Equations (7)

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W f ( s x , s y , x , y ) = f ( x , y ) * h s ( x , y ) .
h s ( x , y ) = 1 s x s y h ( x s x , y s y ) ,
2 ( g s * f ) ( x , y ) = [ 2 g s ( x , y ) ] * f ( x , y ) = W f ( s x , s y , x , y ) .
W f ( s x , s y , x , y ) = s x s y F ( u , υ ) H * ( s x u , s y υ ) × exp [ i 2 π ( x u + y υ ) ] d u d υ ,
H ( u , υ ) = s x 2 u 2 exp ( 2 π 2 s x 2 u 2 ) s y 2 υ 2 exp ( 2 π 2 s y 2 υ 2 ) ,
H ( u , υ ) = s 2 ( u 2 + υ 2 ) exp [ 2 π 2 s 2 ( u 2 + υ 2 ) ] ,
W t ( s x , s y , x , y ) W f * ( s x , s y , x x , y y ) d x d y = T ( u , υ ) H ( s x u , s y υ ) F * ( u , υ ) H * ( s x u , s y υ ) × exp [ i 2 π ( x u + y υ ) ] d u d υ ,

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