Abstract

The classical Wiener filter, which can be implemented in O(N log N) time, is suited best for space-invariant degradation models and space-invariant signal and noise characteristics. For space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. Optimal filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain Wiener filtering for certain types of degradation and noise while requiring only O(N log N) implementation time. The amount of reduction in error depends on the signal and noise statistics as well as on the degradation model. The largest improvements are typically obtained for chirplike degradations and noise, but other types of degradation and noise may also benefit substantially from the method (e.g., nonconstant velocity motion blur and degradation by inhomegeneous atmospheric turbulence). In any event, these reductions are achieved at no additional cost.

© 1998 Optical Society of America

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References

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    [CrossRef]
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  9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
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    [CrossRef]
  16. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
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    [CrossRef]
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    [CrossRef]
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  29. M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
    [CrossRef]
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  36. M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).
    [CrossRef]

1997 (3)

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]

M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).
[CrossRef]

1996 (4)

M. R. Banham, A. K. Katsaggelos, “Spatially adaptive wavelet-based multiscale image restoration,” IEEE Trans. Image Process. 5, 619–634 (1996).
[CrossRef] [PubMed]

H. M. Ozaktas, N. Erkaya, M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett. 3(2), 40–41 (1996).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

1995 (3)

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

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

1994 (9)

J. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166–3177 (1994).
[CrossRef]

A. W. Lohmann, B. H. Soffer, “Relationships between the Radon-Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
[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, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

L. M. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

1993 (6)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

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

J. Zhang, “The mean field theory in EM procedures for blind Markov random field image restoration,” IEEE Trans. Image Process. 2, 27–40 (1993).
[CrossRef] [PubMed]

1987 (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–245 (1980).
[CrossRef]

1976 (1)

1937 (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Agullo-Lopez, F.

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. M.

L. M. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Anderson, B. D. O.

B. D. O. Anderson, J. B. Moore, Optimal Filtering (Prentice-Hall, New York, 1979).

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]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Aytür, O.

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

Banham, M. R.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).
[CrossRef]

M. R. Banham, A. K. Katsaggelos, “Spatially adaptive wavelet-based multiscale image restoration,” IEEE Trans. Image Process. 5, 619–634 (1996).
[CrossRef] [PubMed]

Barry, D. T.

J. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166–3177 (1994).
[CrossRef]

Barshan, B.

Beck, M.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Bernardo, L. M.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Bozdagi, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Erkaya, N.

H. M. Ozaktas, N. Erkaya, M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett. 3(2), 40–41 (1996).
[CrossRef]

Faridanil, A.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.

Fonollosa, J. R.

J. R. Fonollosa, C. L. Nikias, “A new positive time-frequency distribution,” in Proceedings of the IEEE International Conference on Acoustic Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. IV-301–IV-304.

Jazwinski, A.

A. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).

Katsaggelos, A. K.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).
[CrossRef]

M. R. Banham, A. K. Katsaggelos, “Spatially adaptive wavelet-based multiscale image restoration,” IEEE Trans. Image Process. 5, 619–634 (1996).
[CrossRef] [PubMed]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kutay, M. A.

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. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, N. Erkaya, M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett. 3(2), 40–41 (1996).
[CrossRef]

Lewis, F. L.

F. L. Lewis, Optimal Estimation (Wiley, New York, 1986).

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

Lohmann, A. W.

Lopez, V.

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

Moore, J. B.

B. D. O. Anderson, J. B. Moore, Optimal Filtering (Prentice-Hall, New York, 1979).

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–245 (1980).
[CrossRef]

Nikias, C. L.

J. R. Fonollosa, C. L. Nikias, “A new positive time-frequency distribution,” in Proceedings of the IEEE International Conference on Acoustic Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. IV-301–IV-304.

Onural, L.

Ozaktas, H. M.

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]

M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

H. M. Ozaktas, N. Erkaya, M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett. 3(2), 40–41 (1996).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

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

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
[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]

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.

Raymer, M. G.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Sahin, A.

M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

A. Sahin, “Two-dimensional fractional Fourier transformation and its optical implementation,” Master’s thesis (Bilkent University, Ankara, Turkey, 1996).

Shapiro, J. H.

Smithey, D. T.

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Soares, O. D. D.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

Soffer, B. H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.

Wetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.

Wood, J.

J. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166–3177 (1994).
[CrossRef]

Zhang, J.

J. Zhang, “The mean field theory in EM procedures for blind Markov random field image restoration,” IEEE Trans. Image Process. 2, 27–40 (1993).
[CrossRef] [PubMed]

IEEE Signal Process. Lett. (1)

H. M. Ozaktas, N. Erkaya, M. A. Kutay, “Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class,” IEEE Signal Process. Lett. 3(2), 40–41 (1996).
[CrossRef]

IEEE Trans. Image Process. (2)

J. Zhang, “The mean field theory in EM procedures for blind Markov random field image restoration,” IEEE Trans. Image Process. 2, 27–40 (1993).
[CrossRef] [PubMed]

M. R. Banham, A. K. Katsaggelos, “Spatially adaptive wavelet-based multiscale image restoration,” IEEE Trans. Image Process. 5, 619–634 (1996).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (5)

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).
[CrossRef]

J. Wood, D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166–3177 (1994).
[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. Arikan, M. A. Kutay, G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

L. M. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–245 (1980).
[CrossRef]

J. Mod. Opt. (1)

T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (6)

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
[CrossRef]

P. Pellat-Finet, G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
[CrossRef]

T. Alieva, F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

D. T. Smithey, M. Beck, M. G. Raymer, A. Faridanil, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244–1247 (1993).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Signal Process. (1)

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

Other (7)

A. Sahin, “Two-dimensional fractional Fourier transformation and its optical implementation,” Master’s thesis (Bilkent University, Ankara, Turkey, 1996).

J. R. Fonollosa, C. L. Nikias, “A new positive time-frequency distribution,” in Proceedings of the IEEE International Conference on Acoustic Speech and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. IV-301–IV-304.

B. D. O. Anderson, J. B. Moore, Optimal Filtering (Prentice-Hall, New York, 1979).

A. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

F. L. Lewis, Optimal Estimation (Wiley, New York, 1986).

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Figures (5)

Fig. 1
Fig. 1

(a) Original (desired) plane image; (b) corrupted image (SNR1), (c) estimated image obtained by filtering in optimum fractional Fourier domain (ax=0.4, ay=-0.6), (d) image restored by filtering in ordinary Fourier domain.

Fig. 2
Fig. 2

(a) MSE versus ay for ax=0.4, (b) MSE versus ax for ay=-0.6.

Fig. 3
Fig. 3

(a) Original (desired) plane image, (b) corrupted image (SNR0.1), (c) estimated image obtained by filtering in optimum fractional Fourier domain (ax=0.4, ay=-0.6), (d) image restored by filtering in ordinary Fourier domain.

Fig. 4
Fig. 4

(a) Original (desired) plane image, (b) degraded image, (c) estimated image obtained by filtering in optimum fractional Fourier domain (ax=0.7, ay=0.8), (d) image restored by filtering in ordinary Fourier domain.

Fig. 5
Fig. 5

(a) Original (desired) plane image, (b) degraded image, (c) estimated image obtained by filtering in optimum fractional Fourier domain (ax=0.4, ay=0.7), (d) image restored by filtering in ordinary Fourier domain.

Equations (23)

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o=H(f)+n,
fˆ=G(o).
[Fa(f)](x)=fa(x)=Ba(x, x)f(x)dx,
Ba(x, x)=Aϕ exp[iπ(x2 cot ϕ-2xx csc ϕ+x2 cot ϕ)],
Aϕ=(|sin ϕ|)-1/2 exp[iπ sgn(sin ϕ)/4-iϕ/2],
fax,ay(x, y)={Fax,ayf}(x, y)=Bax,ay(x, y; x, y)f(x, y)dxdy,
Bax,ay(x, y; x, y)=Bax(x, x)Bay(y, y).
o(x, y)=h(x, y; x, y)f(x, y)dxdy+n(x, y),
fˆ(x, y)=g(x, y; x, y)f(x, y)dxdy.
σe2=E(f-fˆ2),
f2=|f(x, y)|2dxdy.
Rfo(x, y; x, y)=gopt(x, y; x, y)Roo(x, y; x, y)dxdy
fˆ(x, y)=F-ax,-ay{m(x, y)Fax,ay[o(x, y)]},
fˆ(x, y)=F-ax,-ay{m(x, y)Fax,ay[o(x, y)]},=B-ax,-ay(x, y; x, y)m(x, y)×Bax,ay(x, y; x, y)o(x, y)×dxdydxdy,
σe2=E(f-fˆ2)=E(f(x, y)-F-ax,-ay{m(x, y)Fax,ay[o(x, y)]}2).
σe2=E(fax,ay-fˆax,ay2)=E[fax,ay(x, y)-m(x, y)oax,ay(x, y)2].
E{[fax,ay(x, y)-fˆax,ay(x, y)]oax,ay*(x, y)}=0,
mopt(x, y)=Rfax,ay,oax,ay(x, y; x, y)Roax,ay,oax,ay(x, y; x, y),
Rfax,ay,oax,ay(x, y; x, y),Roax,ay,oax,ay(x, y; x, y)
mopt(x, y)=Bax,ay(x, y; x, y)B-ax,-ay(x, y; x, y)Rf,o(x, y; x, y)dxdydxdyBax,ay(x, y; x, y)B-ax,-ay(x, y; x, y)Ro,o(x, y; x, y)dxdydxdy.
σe,o2=E[fax,ay(x, y)-fˆax,ay(x, y)]×[fax,ay(x, y)-fˆax,ay(x, y)]*dxdy={Rfax,ay,fax,ay(x, y; x, y)-2 Re[mopt*(x, y)×Rfax,ay,oax,ay(x, y; x, y)]+|mopt(x, y)|2Roax,ay,oax,ay(x, y; x, y)}dxdy,
h(x, y; x, y)=1αx+α0rectx-xαx+α0-12δ(y-y),
h(x, y; x, y)=exp{-πα2(x, y)×[(x-x)2+(y-y)2]},

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