Abstract

Perspective projections in the space-frequency plane are analyzed, and it is shown that under certain conditions they can be approximately modeled in terms of the fractional Fourier transform. The region of validity of the approximation is examined. Numerical examples are presented.

© 2000 Optical Society of America

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References

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  1. D. Vernon, Machine Vision: Automated Visual Inspection and Robot Vision (Prentice-Hall, New York, 1991).
  2. A. Low, Introductory Computer Vision and Image Processing (McGraw-Hill, New York, 1991).
  3. G. Woldberg, Digital Image Warping (IEEE Computer Society, Los Alamitos, Calif., 1992).
  4. L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995).
  5. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).
  6. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
    [CrossRef]
  7. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
    [CrossRef]
  8. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3092 (1994).
    [CrossRef]
  9. S. Abe, J. T. Sheridan, “Comment on ‘The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 42, 2373–2378 (1995).
  10. O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
    [CrossRef]
  11. T. Alieva, “Fractional Fourier transform as a tool for investigation of fractal objects,” J. Opt. Soc. Am. A 13, 1189–1192 (1996).
    [CrossRef]
  12. T. Alieva, F. Agullo-Lopez, “Diffraction analysis of random fractal fields,” J. Opt. Soc. Am. A 15, 669–674 (1998).
    [CrossRef]
  13. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994);Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996).
    [CrossRef] [PubMed]
  14. W. X. Cong, N. X. Chen, B. Y. Gu, “Beam shaping and its solution with the use of an optimization method,” Appl. Opt. 37, 4500–4503 (1998).
    [CrossRef]
  15. R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens design problem,” Appl. Opt. 34, 4111–4112 (1995).
    [CrossRef] [PubMed]
  16. D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
    [CrossRef]
  17. M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
    [CrossRef]
  18. J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. 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]
  19. S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
    [CrossRef]
  20. C. J. Kuo, Y. Luo, “Generalized joint fractional Fourier transform correlators: a compact approach,” Appl. Opt. 37, 8270–8276 (1998).
    [CrossRef]
  21. M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
    [CrossRef]
  22. M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
    [CrossRef]
  23. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Chap. 4, pp. 263–342.
  24. S. Mann, S. Haykin, “The chirplet transform: physical considerations,” IEEE Trans. Signal Process. 43, 2745–2761 (1995).
    [CrossRef]
  25. D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
    [CrossRef]
  26. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  27. H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
    [CrossRef]
  28. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
  29. Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
    [CrossRef] [PubMed]
  30. Y. Zhang, B.-Z. Dong, B.-Y. Gu, G.-Z. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
    [CrossRef]

1999

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
[CrossRef]

1998

1997

Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
[CrossRef] [PubMed]

D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

1996

1995

S. Abe, J. T. Sheridan, “Comment on ‘The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 42, 2373–2378 (1995).

R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens design problem,” Appl. Opt. 34, 4111–4112 (1995).
[CrossRef] [PubMed]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

S. Mann, S. Haykin, “The chirplet transform: physical considerations,” IEEE Trans. Signal Process. 43, 2745–2761 (1995).
[CrossRef]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

1994

1980

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Abe, S.

S. Abe, J. T. Sheridan, “Comment on ‘The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 42, 2373–2378 (1995).

Agullo-Lopez, F.

Akay, O.

O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
[CrossRef]

Alieva, T.

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3092 (1994).
[CrossRef]

Arikan, O.

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

Aytür, O.

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

Barshan, B.

Bernardo, L. M.

Boudreaux-Bartels, G. F.

O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
[CrossRef]

Caulfield, J. H.

Chen, N. X.

Cohen, L.

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995).

Cong, W. X.

Dong, B.-Z.

Dorsch, R. G.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens design problem,” Appl. Opt. 34, 4111–4112 (1995).
[CrossRef] [PubMed]

Dragoman, D.

D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

Dragoman, M.

D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

Erden, M. F.

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
[CrossRef]

Garci´a, J.

Granieri, S.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Gu, B. Y.

Gu, B.-Y.

Haykin, S.

S. Mann, S. Haykin, “The chirplet transform: physical considerations,” IEEE Trans. Signal Process. 43, 2745–2761 (1995).
[CrossRef]

Kuo, C. J.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Lohmann, A.

Lohmann, A. W.

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

R. G. Dorsch, A. W. Lohmann, “Fractional Fourier transform used for a lens design problem,” Appl. Opt. 34, 4111–4112 (1995).
[CrossRef] [PubMed]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Chap. 4, pp. 263–342.

Low, A.

A. Low, Introductory Computer Vision and Image Processing (McGraw-Hill, New York, 1991).

Luo, Y.

Mann, S.

S. Mann, S. Haykin, “The chirplet transform: physical considerations,” IEEE Trans. Signal Process. 43, 2745–2761 (1995).
[CrossRef]

Mendlovic, D.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. 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]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Chap. 4, pp. 263–342.

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

Onural, L.

Ozaktas, H. M.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11, 547–559 (1994).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Sheridan, J. T.

S. Abe, J. T. Sheridan, “Comment on ‘The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 42, 2373–2378 (1995).

Sicre, E. E.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Soares, O. D. D.

Trabocchi, O.

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Vernon, D.

D. Vernon, Machine Vision: Automated Visual Inspection and Robot Vision (Prentice-Hall, New York, 1991).

Woldberg, G.

G. Woldberg, Digital Image Warping (IEEE Computer Society, Los Alamitos, Calif., 1992).

Yang, G.-Z.

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, J. H. Caulfield, “Localized, partially space-invariant filtering,” Appl. Opt. 36, 1086–1092 (1997).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
[CrossRef] [PubMed]

J. Garcı́a, D. Mendlovic, Z. Zalevsky, A. 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, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Chap. 4, pp. 263–342.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Zhang, Y.

Adv. Imaging Electron Phys.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications,” Adv. Imaging Electron Phys. 106, 239–291 (1999).
[CrossRef]

Appl. Opt.

IEEE Signal Process. Lett.

O. Akay, G. F. Boudreaux-Bartels, “Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform,” IEEE Signal Process. Lett. 5, 312–314 (1998).
[CrossRef]

IEEE Trans. Signal Process.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3092 (1994).
[CrossRef]

M. F. Erden, M. A. Kutay, H. M. Ozaktas, “Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration,” IEEE Trans. Signal Process. 47, 1458–1462 (1999).
[CrossRef]

M. A. Kutay, H. M. Ozaktas, O. Arikan, L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
[CrossRef]

S. Mann, S. Haykin, “The chirplet transform: physical considerations,” IEEE Trans. Signal Process. 43, 2745–2761 (1995).
[CrossRef]

J. Inst. Math. Appl.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
[CrossRef]

J. Mod. Opt.

S. Abe, J. T. Sheridan, “Comment on ‘The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 42, 2373–2378 (1995).

J. Opt. Soc. Am. A

Opt. Commun.

D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier transform,” Opt. Commun. 141, 5–9 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, A. W. Lohmann, R. G. Dorsch, “Signal spatial-filtering using the localized fractional Fourier transform,” Opt. Commun. 126, 14–18 (1996).
[CrossRef]

S. Granieri, O. Trabocchi, E. E. Sicre, “Fractional Fourier transform applied to spatial filtering in the Fresnel domain,” Opt. Commun. 119, 275–278 (1995).
[CrossRef]

Signal Process.

H. M. Ozaktas, O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
[CrossRef]

Other

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics, XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998), Chap. 4, pp. 263–342.

D. Vernon, Machine Vision: Automated Visual Inspection and Robot Vision (Prentice-Hall, New York, 1991).

A. Low, Introductory Computer Vision and Image Processing (McGraw-Hill, New York, 1991).

G. Woldberg, Digital Image Warping (IEEE Computer Society, Los Alamitos, Calif., 1992).

L. Cohen, Time-Frequency Analysis (Prentice-Hall PTR, Englewood Cliffs, N.J., 1995).

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

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Figures (9)

Fig. 1
Fig. 1

Perspective model: f(x) represents the object distribution on the x axis; g(xp) represents its perspective projection onto the xp axis. The point A with coordinates (-x0, xp0) is the center of projection.

Fig. 2
Fig. 2

(a) Wigner distribution of the original exponential. (b) Wigner distribution of the approximate perspective projection: a chirp.

Fig. 3
Fig. 3

Illustration of the decomposition of the approximation into elementary operations in the space-frequency plane: (a) Original signal, (b) after step 1 (space and frequency shift), (c) after step 2 (fractional Fourier transform), (d) after step 3 (space and frequency shift): approximate perspective projection.

Fig. 4
Fig. 4

(a) Original signal. (b) Exact perspective projection (solid curve) superimposed upon the fractional Fourier approximation (dashed curve).

Fig. 5
Fig. 5

(a) Original signal. (b) Exact perspective projection (solid curve) superimposed upon the fractional Fourier approximation (dashed curve).

Fig. 6
Fig. 6

Perspective model: f(x, y) represents the object distribution on the xy plane; g(xp, yp) represents the object distribution’s perspective projection onto the xpyp plane. The point A with coordinates (-x0, xp0, 0) is the center of projection.

Fig. 7
Fig. 7

(a) Original signal. (b) Exact perspective projection. (c) Fractional Fourier approximation.

Fig. 8
Fig. 8

(a) Wigner distribution of the original signal. (b) Comparison of Wigner distributions underlying error analysis.

Fig. 9
Fig. 9

Dark shading: parameter combinations whose normalized error is less than 10%. Light shading: region in which the error is large. See text for details.

Equations (46)

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

Wf(x, σx)=f(x+x/2)f*(x-x/2)×exp(-i2πσxx)dx.
Wf(x, σx)dσx=|f(x)|2,
Wf(x, σx)dx=|F(σx)|2,
Wf(x, σx)dxdσx=f2=En[f]=signalenergy.
Wf(x, σx)=δ(σx-ξ),
Wf(x, σx)=δ(σx-χx-ξ).
fa(x)=Ka(x, x)f(x)dx,
Ka(x, x)=Aa exp{iπ[cot(aπ/2)x2-2 csc(aπ/2)xx+cot(aπ/2)x2]},
Wfa(x, σx)=Wf(x cos α-σx sin α, x sin α+σx cos α).
xp=xxp0x+x0,
x=x0xpxp0-xp,
g(xp)=fx0xpxp0-xp.
f(x)=F(σx)exp(i2πxσx)dσx,
g(xp)=F(σx)h(xp, σx)dσx,
h(xp, σx)=expi2πσxx0xpxp0-xpdσx.
x¯x¯+x0xp0,
h(xp, σ˜x)=expi2πσxx0xp2(1-κ)3xp02+xp(1-3κ)(1-κ)3xp0+κ3(1-κ)3+.
|κ+2||2xp0(κ-1)|.
δσx+2σ¯x(1-κ)3xp02xp+σ¯x(1-3κ)(1-κ)3xp0
f(x)=F(σx)exp(2iπxσx)dσx.
exp(i2πx¯σx).
1+i2σxx¯+x0xp02x031/2 exp(i2πx¯σx)×expi2πx2σxx¯+x0xp02x03.
1+i2σx(x¯+x0)3xp02x031/2 exp(i2πx¯σx)×expi2πσxx-xx¯p0x¯+x02x¯+x0xp02x03×expi4πσxx¯+x02x02xp0.
exp2iπσxx0xpxp0-xp.
expi2πσxx0xp2(1-κ)3xp02+xp(1-3κ)(1-κ)3xp0+κ3(1-κ)3,
cos(4πx)rectx-46=exp(i4πx)+exp(-i4πx)2×rectx-46
xp=xxp0x+x0,
x=x0xpxp0-xp,
yp=xΔy+2x0y2(x+x0),
y=ypxp0xp0-xp-xpΔy2(x0-xp).
f(x, y)=F(σx, σy)exp(i2πσxx)exp(i2πσyy)dσxdσy.
expi2πσ¯xx0xp2(1-κ)3xp02+xp(1-3κ)(1-κ)3xp0+κ3(1-κ)3,×exp-iπσ¯yΔyxp2(1-κ)3x02+xp(1-3κ)(1-κ)3x0+κ3(1-κ)3,×expi2πσ¯yypxp2xp02(1-κ)3+xp(1-3κ)xp(1-κ)3+3κ2-3κ+1(κ-1)3,
a=-2πarctan2σ¯x(x¯+x0)3xp02x02-(x¯+Δy)3σ¯yΔy2x02,
σ¯x(x¯+x0)3xp0x0-(x¯+Δy)3σ¯yΔy2x02
σ¯yxp2xp02(1-κ)3+xp(1-3κ)xp(1-κ)3+3κ2-3κ+1(κ-1)3.
eup-left,space=ΔσxΔxx0sinαc-αu2cosαc+αu2+1x0sinαc+αu2,
eup-right,space=ΔσxΔxx0sinαc-αu2cosαc+αu2-1x0sinαc+αu2,
elow-left,space=-ΔσxΔxx0sinαc-αd2cosαc-αd2+1x0sinαc+αd2,
elow-right,space=-ΔσxΔxx0sinαc-αd2cosαc+αd2-1x0sinαc+αd2,
eup-left,freq.=-1x0sinαc-αu2sinαc+αu2+ΔxΔσxx0cosαc+αu2,
eup-right,freq.=-1x0sinαc-αu2sinαc+αu2-ΔxΔσxx0cosαc+αu2,
elow-left,freq.=1x0sinαc-αd2sinαc+αd2+ΔxΔσxx0cosαc+αd2,
elow-right,freq.=1x0sinαc-αd2sinαc+αu2-ΔxΔσxx0cosαc+αd2,
αc=arctan2σ¯xx0x¯+x0x03xp0x02,
αu=arctan2σ¯xx0+Δσx2x0x¯+x0x03xp0x02,
αd=arctan2σ¯xx0-Δσx2x0x¯+x0x03xp0x02.

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