Abstract

Higher-order correlations are well known for their use in noise removal, image enhancement, and signal identification. They are generalizations of the well-known second-order correlation. The fractionalization of the second-order correlation provides some interesting features that are related to the shift-variance property of the fractional-Fourier-transform operation. This project proposes the fractionalization of the triple-correlation operation (as well as other higher-order correlations). A suggested definition as well as some applications are given. Computer simulations demonstrate some of the features this operation offers.

© 1998 Optical Society of America

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References

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  1. A. W. Lohmann, B. Wirnitzer, “Triple correlation,” Proc. IEEE 72, 889–901 (1984).
    [Crossref]
  2. A. W. Lohmann, “A new algorithm for image and signal recovery from triple correlation,” Optik 72, 71–76 (1986).
  3. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [Crossref] [PubMed]
  4. A. W. Lohmann, “Pattern recognition based on the triple correlation,” Optik 78, 117–120 (1988).
  5. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [Crossref] [PubMed]
  6. B. Wirnitzer, “Bi-spectrum analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
    [Crossref]
  7. G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. 19, pp. 294–319.
  8. A. W. Lohmann, “Recovery of an object, moving at random in front of a fixed background,” Optik 73, 127–134 (1986).
  9. B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
    [Crossref] [PubMed]
  10. H. Bartelt, B. Wirnitzer, “Shift invariant imaging of photon limited data using bispectral analysis,” Opt. Commun. 53, 13–22 (1985).
    [Crossref]
  11. D. Mendlovic, A. W. Lohmann, D. Mas, G. Shabtay, “Optoelectronic implementation of the triple correlation,” Opt. Lett. 22, 1018–1020 (1997).
    [Crossref] [PubMed]
  12. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [Crossref]
  13. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [Crossref]
  14. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [Crossref]
  15. R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [Crossref] [PubMed]
  16. A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
    [Crossref]
  17. J. Garcia, D. Mendlovic, Z. Zalevsky, A. W. Lohmann, “Space variant simultaneous detection of several objects by the use of multiple anamorphic fractional Fourier transform filters,” Appl. Opt. 35, 3945–3952 (1996).
    [Crossref] [PubMed]

1997 (1)

1996 (2)

1994 (1)

1993 (3)

1988 (1)

A. W. Lohmann, “Pattern recognition based on the triple correlation,” Optik 78, 117–120 (1988).

1986 (2)

A. W. Lohmann, “A new algorithm for image and signal recovery from triple correlation,” Optik 72, 71–76 (1986).

A. W. Lohmann, “Recovery of an object, moving at random in front of a fixed background,” Optik 73, 127–134 (1986).

1985 (2)

B. Wirnitzer, “Bi-spectrum analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
[Crossref]

H. Bartelt, B. Wirnitzer, “Shift invariant imaging of photon limited data using bispectral analysis,” Opt. Commun. 53, 13–22 (1985).
[Crossref]

1984 (2)

1983 (1)

1979 (1)

Bartelt, H.

H. Bartelt, B. Wirnitzer, “Shift invariant imaging of photon limited data using bispectral analysis,” Opt. Commun. 53, 13–22 (1985).
[Crossref]

H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
[Crossref] [PubMed]

Bitran, Y.

Braunecker, B.

Dorsch, R. G.

Garcia, J.

Hauck, R.

Lohmann, A. W.

D. Mendlovic, A. W. Lohmann, D. Mas, G. Shabtay, “Optoelectronic implementation of the triple correlation,” Opt. Lett. 22, 1018–1020 (1997).
[Crossref] [PubMed]

J. Garcia, D. Mendlovic, Z. Zalevsky, A. W. Lohmann, “Space variant simultaneous detection of several objects by the use of multiple anamorphic fractional Fourier transform filters,” Appl. Opt. 35, 3945–3952 (1996).
[Crossref] [PubMed]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[Crossref]

R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
[Crossref] [PubMed]

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[Crossref]

A. W. Lohmann, “Pattern recognition based on the triple correlation,” Optik 78, 117–120 (1988).

A. W. Lohmann, “A new algorithm for image and signal recovery from triple correlation,” Optik 72, 71–76 (1986).

A. W. Lohmann, “Recovery of an object, moving at random in front of a fixed background,” Optik 73, 127–134 (1986).

A. W. Lohmann, B. Wirnitzer, “Triple correlation,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
[Crossref] [PubMed]

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[Crossref] [PubMed]

B. Braunecker, R. Hauck, A. W. Lohmann, “Optical character recognition based on nonredundant correlation measurements,” Appl. Opt. 18, 2746–2753 (1979).
[Crossref] [PubMed]

Mas, D.

Mendlovic, D.

Ozaktas, H. M.

Shabtay, G.

Weigelt, G.

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[Crossref] [PubMed]

G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. 19, pp. 294–319.

Wirnitzer, B.

Zalevsky, Z.

Appl. Opt. (5)

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

H. Bartelt, B. Wirnitzer, “Shift invariant imaging of photon limited data using bispectral analysis,” Opt. Commun. 53, 13–22 (1985).
[Crossref]

A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
[Crossref]

Opt. Lett. (1)

Optik (3)

A. W. Lohmann, “Pattern recognition based on the triple correlation,” Optik 78, 117–120 (1988).

A. W. Lohmann, “A new algorithm for image and signal recovery from triple correlation,” Optik 72, 71–76 (1986).

A. W. Lohmann, “Recovery of an object, moving at random in front of a fixed background,” Optik 73, 127–134 (1986).

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlation,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

Other (1)

G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1991), Vol. 19, pp. 294–319.

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Figures (2)

Fig. 1
Fig. 1

(a) Rectangular signal, (b) rectangular signal embedded in border noise.

Fig. 2
Fig. 2

(a) Triple correlation of a rectangular signal, (b) triple correlation of the signal depicted in Fig. 1(b), (c) fractional triple correlation of the order p=0.2 of a rectangular signal, (d) fractional triple correlation of the order p=0.2 of the signal depicted in Fig. 1(b).

Equations (14)

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l˜p(u)=Fp[l(x)]=C-l(x)expiπ(x2+u2)tan(πp/2)-2πixusin(πp/2)dx,
C=exp-iπ sgn(sin ϕ)4-ϕ2(|sin ϕ|)1/2,
l31(x1, x2)=-l(x)l(x+x1)l(x+x2)dx.
l˜31(u1, u2)=-l31(x1, x2)×exp[-2πi(x1u1+x2u2)]dx1dx2=l˜1(u1)·l˜1(u2)·l˜1(-u1-u2).
l˜3P(u1, u2)=l˜p(u1)·l˜p(u2)·l˜p(-u1-u2),
l3P(x1, x2)=F-p[l˜p(u1)·l˜p(u2)·l˜p(-u1-u2)].
l˜3P(u1, u2)=Fp[l3(x1, x2)],
tanπ2p=πb,
ln1(x1, x2,, xn-1)=l(x)k=1n-1l(x+xk)dx,
l˜1(u)=l(x)exp(-i2πxu)dx,
l˜n1(u1, u2,, un-1)
=l˜1(-u1-u2--un-1)k=1n-1l˜1(uk).
lnp(x1, x2,, xn-1)
=F-pl˜p(-u1-u2--un-1)k=1n-1l˜p(uk).

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