Abstract

Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example.

© 1998 Optical Society of America

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  1. A. C. McBride, F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  2. 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]
  3. H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
    [CrossRef]
  4. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional order Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  5. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  6. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  7. 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]
  8. L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  9. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  10. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [CrossRef]
  11. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
    [CrossRef]
  12. M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
    [CrossRef]
  13. D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming–optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
    [CrossRef] [PubMed]
  14. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions,” Opt. Commun. 120, 134–138 (1995).
    [CrossRef]
  15. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  16. L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
    [CrossRef]
  17. T. Alieva, V. Lopez, F. Agullo-Lopez, L. B. Almedia, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
  18. D. Mendlovic, Z. Zalevsky, N. Konforti, R. G. Dorsch, A. W. Lohmann, “Incoherent fractional Fourier transform and its optical implementation,” Appl. Opt. 34, 7615–7620 (1995).
    [CrossRef] [PubMed]
  19. R. G. Dorsch, “Fractional Fourier transformer of variable order based on a modular lens system,” Appl. Opt. 34, 6016–6020 (1995).
    [CrossRef] [PubMed]
  20. A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
    [CrossRef]
  21. C. C. Shih, “Optical interpretation of a complex-order Fourier transform,” Opt. Lett. 20, 1178–1180 (1995).
    [CrossRef] [PubMed]
  22. R. G. Dorsh, A. W. Lohmann, Y. Bitran, D. Mendlovic, H. M. Ozaktas, “Chirp filtering in the fractional Fourier domain,” Appl. Opt. 33, 7599–7602 (1994).
    [CrossRef]
  23. Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
    [CrossRef] [PubMed]
  24. A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).
  25. A. Sahin, H. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  26. R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
    [CrossRef]
  27. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  28. G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, Princeton, N.J., 1989)
  29. H. M. Ozaktas, O. Arikan, M. A. Kutay, G. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
    [CrossRef]
  30. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990.)
  31. Tanju Erdem. Bilkent University, Department of Electrical Engineering, 06533, Bilkent, Ankara, Turkey. (Personal communication.)
  32. 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]
  33. M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  34. A. W. Lohmann, Z. Zalevsky, D. Mendlovic, “Synthesis of pattern recognition filters for fractional Fourier processing,” Opt. Commun. 128, 199–204 (1996).
    [CrossRef]
  35. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
  36. B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
    [CrossRef]
  37. H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
    [CrossRef]
  38. H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
    [CrossRef] [PubMed]
  39. M. A. Kutay, H. M. Ozaktas, “Optimal image restoration with the fractional Fourier transform,” J. Opt. Soc. Am. A 15, 825–833 (1998).
    [CrossRef]
  40. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [CrossRef]

1998 (2)

1997 (3)

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (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]

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

1996 (6)

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

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

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

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

1995 (7)

1994 (10)

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]

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]

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (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]

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

Y. B. Karasik, “Expression of the kernel of a fractional Fourier transform in elementary functions,” Opt. Lett. 19, 769–770 (1994).
[CrossRef] [PubMed]

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

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

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

1993 (5)

1987 (2)

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

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

1979 (1)

Agullo-Lopez, F.

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

Alieva, T.

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

Almedia, L. B.

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

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (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]

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

Barshan, B.

Bastiaans, M. J.

Bernardo, L. M.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

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

Bitran, Y.

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. Bozdagi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
[CrossRef]

Champ, K. Y.

R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
[CrossRef]

Dorsch, R. G.

Dorsh, R. G.

Erdem, Tanju

Tanju Erdem. Bilkent University, Department of Electrical Engineering, 06533, Bilkent, Ankara, Turkey. (Personal communication.)

Erden, M. F.

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

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Ferreira, C.

Folland, G. B.

G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, Princeton, N.J., 1989)

Garcia, J.

Jha, A. K.

R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
[CrossRef]

Karasik, Y. B.

Kerr, F. H.

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

Konforti, N.

Kutay, M. A.

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

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (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]

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

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. Almedia, “The fractional 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.

A. Sahin, H. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

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

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

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

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

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

D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming–optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, N. Konforti, R. G. Dorsch, A. W. Lohmann, “Incoherent fractional Fourier transform and its optical implementation,” Appl. Opt. 34, 7615–7620 (1995).
[CrossRef] [PubMed]

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 the two dimensions,” Opt. Commun. 120, 134–138 (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]

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

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

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

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

Onural, L.

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, 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]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Ozaktas, H.

Ozaktas, H. M.

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

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable fractional Fourier transformer,” Opt. Commun. 136, 52–60 (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]

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

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

H. M. Ozaktas, D. Mendlovic, “Every Fourier optical system is equivalent to consecutive fractional Fourier domain filtering,” Appl. Opt. 35, 3167–3170 (1996).
[CrossRef] [PubMed]

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

D. Mendlovic, Y. Bitran, C. Ferreira, J. Garcia, H. M. Ozaktas, “Anamorphic fractional Fourier transforming–optical implementation and applications,” Appl. Opt. 34, 7451–7456 (1995).
[CrossRef] [PubMed]

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

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

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

D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (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, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[CrossRef]

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

Sahin, A.

A. Sahin, H. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable 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 the two dimensions,” Opt. Commun. 120, 134–138 (1995).
[CrossRef]

A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).

Scott, P. D.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Shih, C. C.

Soares, O. D. D.

L. M. Bernardo, O. D. D. Soares, “Fractional Fourier transforms and imaging,” J. Opt. Soc. Am. A 11, 2622–2626 (1994).
[CrossRef]

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

Soffer, B. H.

Wang, Y. H.

R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
[CrossRef]

Zalevsky, Z.

Appl. Opt. (7)

Electron. Lett. (1)

R. N. Bracewell, K. Y. Champ, A. K. Jha, Y. H. Wang, “Affine theorem for two dimensional Fourier transform,” Electron. Lett. 29, 304–309 (1993).
[CrossRef]

IEEE Trans. Signal Process. (3)

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

L. B. 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. Mod. Opt. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (9)

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]

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

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

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

A. W. Lohmann, “A fake zoom lens for fractional Fourier experiments,” Opt. Commun. 115, 437–443 (1995).
[CrossRef]

B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[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. Eng. (1)

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[CrossRef]

Opt. Lett. (3)

Signal Process. (1)

H. M. Ozaktas, “Repeated fractional Fourier domain filtering is equivalent to repeated time and frequency domain filtering,” Signal Process. 54, 81–84 (1996).
[CrossRef]

Other (4)

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

Tanju Erdem. Bilkent University, Department of Electrical Engineering, 06533, Bilkent, Ankara, Turkey. (Personal communication.)

A. Sahin, “Two-dimensional fractional Fourier transform and its optical implementation,” M.S. thesis (Bilkent University, Ankara, Turkey, 1996).

G. B. Folland, Harmonic Analysis in Phase Space (Princeton U. Press, Princeton, N.J., 1989)

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Figures (9)

Fig. 1
Fig. 1

Transform orders and directions for (a) the separable transform and (b) the nonseparable transform.

Fig. 2
Fig. 2

Optical setup for simulating anamorphic sections of free space.

Fig. 3
Fig. 3

Optical setup for realizing the nonseparable fractional Fourier transform.

Fig. 4
Fig. 4

(a) Original image. (b) Noisy image with a value of SNR = 1.

Fig. 5
Fig. 5

Original image. (b) Noisy image with a value of SNR = 0.1.

Fig. 6
Fig. 6

Images with a value of SNR = 1: (a) Image filtered by the separable transform. (b) Image filtered by the nonseparable transform.

Fig. 7
Fig. 7

Images with a value of SNR = 0.1: (a) Image filtered by the separable transform. (b) Image filtered by the nonseparable transform.

Fig. 8
Fig. 8

Normalized MSE as a function of θ1 for a value of SNR = 1.

Fig. 9
Fig. 9

Normalized MSE as a function of θ2 for a value of SNR = 1.

Equations (48)

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a 1 f x x = -   B a 1 x ,   x f x d x ,
B a 1 x ,   x = A ϕ 1 exp i π x 2 cot   ϕ 1 - 2 xx   csc   ϕ 1 + x 2 cot   ϕ 1 ,
a 1 , a 2 f x ,   y x ,   y = - -   B a 1 , a 2 x ,   y ;   x ,   y f x ,   y d x d y ,
B a 1 ,   a 2 x ,   y ;   x ,   y = B a 1 x ,   x B a 2 y ,   y .
G x ,   y = 1 Δ   F dx - cy Δ ,   - bx + ay Δ ,
θ 1 , θ 2 a 1 , a 2 f r r = - -   B a 1 , a 2 , θ 1 , θ 2 r ,   r f r d r ,
B a 1 , a 2 , θ 1 , θ 2 r ,   r = A ϕ 1 , ϕ 2 exp i π r T Ar + 2 r T Br + r T Cr , A ϕ 1 , ϕ 2 = A ϕ 1 A ϕ 2 ,     r = x   y T ,     r = x   y T ,
A = cot   ϕ 1 0 0 cot   ϕ 2 ,
B = - cos   θ 2 csc   ϕ 1 cos θ 1 - θ 2 sin   θ 1 csc   ϕ 1 cos θ 1 - θ 2 - sin   θ 2 csc   ϕ 2 cos θ 1 - θ 2 - cos   θ 1 csc   ϕ 2 cos θ 1 - θ 2 ,
C = cos 2   θ 2 cos 2 θ 1 - θ 2 cot   ϕ 1 + sin 2   θ 2 cos 2 θ 1 - θ 2 cot   ϕ 2 - sin   θ 1 cos   θ 2 cos 2 θ 1 - θ 2 cot   ϕ 1 + sin   θ 2 cos   θ 1 cos 2 θ 1 - θ 2 cot   ϕ 2 - sin   θ 1 cos   θ 2 cos 2 θ 1 - θ 2 cot   ϕ 1 + sin   θ 2 cos   θ 1 cos 2 θ 1 - θ 2 cot   ϕ 2 cos 2   θ 1 cos 2 θ 1 - θ 2 cot   ϕ 2 + sin 2   θ 1 cos 2 θ 1 - θ 2 cot   ϕ 1 .
θ 1 , θ 2 a 1 , a 2 f ax + by ,   cx + dy x ,   y = k θ 1 , θ 2 a 1 , a 2 f x ,   y a x + b y ,   c x + d y , k = exp C ϕ 1 x 2 + C ϕ 2 y 2 + C ϕ 1 , ϕ 2 xy , ϕ 1 = ϕ 1 cot - 1 D θ 1 θ 2 a   cos   θ 1 + b   sin   θ 2 2 - c   cos   θ 1 + d   sin   θ 2 2 , ϕ 2 = ϕ 2 cot - 1 D θ 1 θ 2 d   cos   θ 2 + c   sin   θ 1 2 - b   cos   θ 2 + c   sin   θ 1 2 , θ 1 = cos - 1 a   cos   θ 1 + b   sin   θ 2 2 d   cos   θ 2 + c   sin   θ 1 2 - b   cos   θ 2 + a   sin   θ 1 2 D θ 1 θ 2 1 / 2 , θ 2 = cos - 1 d   cos   θ 2 + c   sin   θ 1 2 a   cos   θ 1 + b   sin   θ 2 2 - c   cos   θ 1 + d   sin   θ 2 2 D θ 1 θ 2 1 / 2 , a = csc   ϕ 1 d   cos   θ 2 + c   sin   θ 1 cos   θ 1 + b   cos   θ 2 + a   sin   θ 2 sin   θ 2 csc   ϕ 1 cos θ 1 - θ 2 , b = csc   ϕ 2 c   cos   θ 1 + d   sin   θ 2 cos   θ 1 + a   cos   θ 1 + b   sin   θ 2 sin   θ 2 csc   ϕ 1 cos θ 1 - θ 2 , c = csc   ϕ 1 d   cos   θ 2 + c   sin   θ 1 sin   θ 1 - b   cos   θ 2 + a   sin   θ 1 cos   θ 2 csc   ϕ 2 cos θ 1 - θ 2 , d = csc   ϕ 2 a   cos   θ 1 + b   sin   θ 2 cos   θ 2 - c   cos   θ 1 + d   sin   θ 2 sin   θ 1 csc   ϕ 2 cos θ 1 - θ 2 ,
D θ 1 θ 2 = a   cos   θ 1 + b   sin   θ 2 2 d   cos   θ 2 + c   sin   θ 1 2 - b   cos   θ 2 + a   sin   θ 1 2 c   cos   θ 1 + d   sin   θ 2 , C ϕ 1 = cot   ϕ 1 a   cos   θ 1 + b   sin   θ 2 2 - c   cos   θ 1 + d   sin   θ 2 2 - D θ 1 θ 2 a   cos   θ 1 + b   sin   θ 2 2 - c   cos   θ 1 + d   sin   θ 2 2 - cot 2   ϕ 1 D θ 1 θ 2 , C ϕ 2 = cot   ϕ 2 d   cos   θ 2 + c   sin   θ 1 2 - b   cos   θ 2 + c   sin   θ 1 2 - D θ 1 θ 2 d   cos   θ 2 + c   sin   θ 1 2 - b   cos   θ 2 + a   sin   θ 1 2 - cot 2   ϕ 2 D θ 1 θ 2 , C ϕ 1 , ϕ 2 = a b   cot   ϕ 1 + c d   cot   ϕ 2 .
a 1 , a 2 f ax + by ,   cx + dy x ,   y = k θ 1 , θ 2 a 1 , a 2 f x ,   y a x + b y ,   c x + d y , Δ = ad - bc ,     k = 1 Δ exp C ϕ 1 x 2 + C ϕ 2 y 2 + C ϕ 1 , ϕ 2 xy , ϕ 1 = ϕ 1 cot - 1 a 2 d 2 - b 2 c 2 Δ 2 a 2 - c 2 ,     ϕ 2 = ϕ 2 cot - 1 a 2 d 2 - b 2 c 2 Δ 2 d 2 - b 2 , θ 1 = cos - 1 a 2 d 2 - b 2 a 2 d 2 - b 2 c 2 1 / 2 ,     θ 2 = cos - 1 d 2 a 2 - c 2 a 2 d 2 - b 2 c 2 1 / 2 , a = csc   ϕ 1 d   cos   θ 1 + b   sin   θ 2 Δ   csc   ϕ 1 cos θ 1 - θ 2 ,     b = csc   ϕ 2 c   cos   θ 1 + a   sin   θ 2 Δ   csc   ϕ 2 cos θ 1 - θ 2 , c = csc   ϕ 1 d   sin   θ 1 - b   cos   θ 2 Δ   csc   ϕ 2 cos θ 1 - θ 2 ,     d = csc   ϕ 2 a   cos   θ 2 - c   sin   θ 1 Δ   csc   ϕ 2 cos θ 1 - θ 2 ,
k = 1 Δ exp C ϕ 1 x 2 + C ϕ 2 y 2 + C ϕ 1 , ϕ 2 xy , Δ = ad - bc , C ϕ 1 = cot   ϕ 1 Δ 4 a 2 - c 2 - a 2 d 2 - b 2 c 2 2 Δ 4 a 2 - c 2 - cot 2   ϕ 1 a 2 d 2 - b 2 c 2 2 , C ϕ 2 = cot   ϕ 2 Δ 4 d 2 - b 2 - a 2 d 2 - b 2 c 2 2 Δ 4 d 2 - b 2 - cot 2   ϕ 2 a 2 d 2 - b 2 c 2 2 , C ϕ 1 , ϕ 2 = a b   cot   ϕ 1 + c d   cot   ϕ 2 .
B a 1 , a 2 , θ 1 , θ 2 - 1 r ,   r = A - ϕ 1 , - ϕ 2 exp - i π r T Cr + 2 r T B T r + r T Ar ,
θ 1 , θ 2 a 1 , a 2 f x ,   y = a 1 , a 2 f cos   θ 1 x + sin   θ 1 y / cos θ 1 - θ 2 , - sin   θ 2 x + cos   θ 2 y / cos θ 1 - θ 2 .
B a 1 , a 2 , θ 1 , θ 2 * x ,   y ;   x ,   y = B a 1 , a 2 , θ 1 , θ 2 - 1 x ,   y ;   x ,   y .
W g r ,   μ = W f Ar + B μ ,   Cr + D μ ,
r = x   y T ,     μ = μ x   μ y T ,
A = 1 cos θ 1 - θ 2 cos   ϕ 1 cos   θ 1 cos   ϕ 2 sin   θ 1 - cos   ϕ 1 sin   θ 2 cos   ϕ 2 cos   θ 2 ,
B = 1 cos θ 1 - θ 2 - sin   ϕ 1 cos   θ 1 - sin   ϕ 2 sin   θ 1 sin   ϕ 1 sin   θ 2 - sin   ϕ 2 cos   θ 2 ,
C = 1 cos θ 1 - θ 2 sin   ϕ 1 cos   θ 2 sin   ϕ 2 sin   θ 2 - sin   ϕ 1 sin   θ 1 sin   ϕ 2 cos   θ 1 ,
D = 1 cos θ 1 - θ 2 cos   ϕ 1 cos   θ 2 cos   ϕ 2 sin   θ 2 - cos   ϕ 1 sin   θ 1 cos   ϕ 2 cos   θ 1 .
W r ,   μ = -   f r + r 2 f * r - r 2 exp i 2 π μ T r d r ,
r = x   y T ,     r = x   y T ,     μ = μ x   μ y T .
exp - i π x 2 λ f x + y 2 λ f y + xy λ f xy ,
exp i π x 2 λ d x + y 2 λ d y + xy λ d xy ,
d x = s 4 f xy 2 - f x f y λ 2 f x f xy 2 ,     d y = s 4 f xy 2 - f x f y λ 2 f y f xy 2 ,
d xy = s 4 f x f y - f xy 2 2 λ 2 f x f y f xy ,
d x = s x in s x out   cos θ 1 - θ 2 2 λ   cos   θ 2   csc   ϕ 1 ,
d y = s y in s y out   cos θ 1 - θ 2 2 λ   cos   θ 1   csc   ϕ 2 ,
d xy = s x in s y out   cos θ 1 - θ 2 2 λ   sin   θ 2   csc   ϕ 2 = - s y in s x out   cos θ 1 - θ 2 2 λ   sin   θ 1   csc   ϕ 1 ,
1 λ f x 1 = 2   cos   θ 2   csc   ϕ 1 s x in s x out   cos θ 1 - θ 2 - cos 2   θ 2 cos   ϕ 1 + sin 2   θ 2 cot   ϕ 2 s x in 2 cos 2 θ 1 - θ 2 ,
1 λ f y 1 = 2   cos   θ 1   csc   ϕ 2 s y in s y out   cos θ 1 - θ 2 - cos 2   θ 1 cos   ϕ 2 + sin 2   θ 1 cot   ϕ 1 s y in 2 cos 2 θ 1 - θ 2 ,
1 λ f xy 1 = 2   sin   θ 2   csc   ϕ 2 s x in s y out   cos θ 1 - θ 2 + 2   sin   θ 1 cos   θ 2 cot   ϕ 1 - 2   cos   θ 1 sin   θ 2 cot   ϕ 2 s x in s y in   cos 2 θ 1 - θ 2 ,
1 λ f x 2 = 2   cos   θ 2 csc   ϕ 1 s x in s x out   cos θ 1 - θ 2 - cot   ϕ 1 s x out 2 ,
1 λ f y 2 = 2   cos   θ 1 csc   ϕ 2 s y in s y out   cos θ 1 - θ 2 - cot   ϕ 2 s y out 2 ,
1 λ f xy 2 = 2   sin   θ 2   csc   ϕ 2 s x in s y out   cos θ 1 - θ 2 .
s x out s y out = - s x in   sin   θ 1 csc   ϕ 1 s y in   sin   θ 2 csc   ϕ 2 .
s x in s y in = - s x out   sin   θ 2   csc   ϕ 2 s y out   sin   θ 1   csc   ϕ 1 .
o = f + n ,
f ˆ = θ 1 , θ 2 a 1 , a 2 - 1 g · θ 1 , θ 2 a 1 , a 2 o ,
σ e 2 = E | f - f ˆ | 2 .
g opt x ,   y = R f ˜ , õ x ,   y ;   x ,   y R õ , õ x ,   y ;   x ,   y .
R f ˜ ,   õ x ,   y ;   x ,   y = E f ˜ x ,   y õ x ,   y ,
R õ , õ x ,   y ;   x ,   y = E õ x ,   y õ x ,   y .
f x ,   y + C exp 1.6 i π x - 7.3 2 + exp 1.4 i π y + 7.3 2 ,
SNR =   | f x ,   y | 2 d x d y   | n x ,   y | 2 d x d y .

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