Abstract

Owing to the nonlinear nature of the problem, the transform orders in fractional Fourier-domain filtering configurations have usually not been optimized but chosen uniformly. We discuss the optimization of these orders for multi-channel-filtering configurations by first finding the optimal filter coefficients for a larger number of uniformly chosen orders, and then maintaining the most important ones. The method is illustrated with the problem of synthesizing desired mutual-intensity distributions. The method we propose allows those fractional Fourier domains, which add little benefit to the filtering process but increase the overall cost, to be pruned, so that comparable performance can be attained with less cost, or higher performance can be obtained with the same cost. The method we propose is more likely to be useful when confronted with low-cost rather than high-performance applications, because larger improvements are obtained when the use of a smaller number of filters is desired.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. 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]
  23. D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J Aust. Math. Soc. B 38, 209–219 (1996).
    [CrossRef]
  24. 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]
  25. H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
    [CrossRef]
  26. Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
    [CrossRef]
  27. 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]
  28. Z. Zalevsky, D. Mendlovic, “Fractional Wiener filter,” Appl. Opt. 35, 3930–3936 (1996).
    [CrossRef] [PubMed]
  29. 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]
  30. M. F. Erden, H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647–1657 (1998).
    [CrossRef]
  31. M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (1998).
    [CrossRef]
  32. M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.
  33. M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.
  34. M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
    [CrossRef]
  35. İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.
  36. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications” in P. W. Hawkes, ed., Advances in Imaging and Electron Physics, Vol. 106 (Academic Press, San Diego, Calif., 1999) Chap. 4, pp. 239–291.
  37. D. Mendlovic, H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I, J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
  38. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II.” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
    [CrossRef]
  39. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  40. B. Barshan, M. A. Kutay, H. M. Ozaktas, “Optimal filtering with linear canonical transformations,” Opt. Commun. 135, 32–36 (1997).
    [CrossRef]

2000 (3)

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. Soc. A 2, 83–87 (2000).
[CrossRef]

Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

1999 (2)

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. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

1998 (6)

1997 (6)

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

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

S. Granieri, R. Arizaga, E. E. Sicre, “Optical correlation based on the fractional Fourier transform,” Appl. Opt. 36, 6636–6645 (1997).
[CrossRef]

J. Hua, L. Liu, G. Li, “Scaled fractional Fourier transform and its optical implementation,” Appl. Opt. 36, 8490–8492 (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]

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

1996 (5)

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

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J Aust. Math. Soc. B 38, 209–219 (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]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

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

1995 (2)

1994 (7)

1993 (3)

Agullo-Lopez, F.

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

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.

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

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. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (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]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

Arizaga, R.

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]

Candan, Ç.

Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (1998).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

Chen, L.

Chen, N. X.

Cong, W. X.

Dorsch, R. G.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Dragoman, D.

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

Dragoman, M.

D. Dragoman, M. Dragoman, “Near and far field optical beam characterization using the fractional Fourier tansform,” 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]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647–1657 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

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

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

Ferreira, C.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

García, J.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

Granieri, S.

Gu, B. Y.

Gu, B.-Y.

Güleryüz, Ö

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (1998).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

Hua, J.

Kuo, C. J.

Kutay, M. A.

Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (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, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (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, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (John Wiley & Sons, New York, 2001).

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications” in P. W. Hawkes, ed., Advances in Imaging and Electron Physics, Vol. 106 (Academic Press, San Diego, Calif., 1999) Chap. 4, pp. 239–291.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

Li, C.

Li, G.

Liu, L.

Liu, S.

Lohmann, A. W.

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[CrossRef]

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

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

Lopez, V.

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

Luo, Y.

M. Ozaktas, H.

Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[CrossRef]

Medlovic, D.

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. Soc. A 2, 83–87 (2000).
[CrossRef]

Mendlovic, D.

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

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

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

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

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

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (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 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, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications” in P. W. Hawkes, ed., Advances in Imaging and Electron Physics, Vol. 106 (Academic Press, San Diego, Calif., 1999) Chap. 4, pp. 239–291.

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, (Cambridge University Press, Cambridge, 1998) Chap. 4, pp.89–125

Mustard, D.

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J Aust. Math. Soc. B 38, 209–219 (1996).
[CrossRef]

Onural, L.

Ozaktas, H. M.

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. Soc. A 2, 83–87 (2000).
[CrossRef]

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (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. F. Erden, H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647–1657 (1998).
[CrossRef]

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ö Güleryüz, Ç. Candan, “Space-bandwidth-efficient realizations of linear systems,” Opt. Lett. 23, 1069–1071 (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]

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

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[CrossRef]

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

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (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]

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

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications” in P. W. Hawkes, ed., Advances in Imaging and Electron Physics, Vol. 106 (Academic Press, San Diego, Calif., 1999) Chap. 4, pp. 239–291.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (John Wiley & Sons, New York, 2001).

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, (Cambridge University Press, Cambridge, 1998) Chap. 4, pp.89–125

Özaktas, H.

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.

Pellat-Finet, P.

Sicre, E. E.

Soares, O. D. D.

Soffer, B. H.

Xu, J.

Yetik, I. S.

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.

Zalevsky, Z.

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. Soc. A 2, 83–87 (2000).
[CrossRef]

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[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]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, (Cambridge University Press, Cambridge, 1998) Chap. 4, pp.89–125

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (John Wiley & Sons, New York, 2001).

Zhang, Y.

Appl. Opt. (6)

IEEE Signal Process. Lett. (1)

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. (1)

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

IEEE Trans. Signal Process. (3)

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]

Ç. Candan, M. A. Kutay, H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Process. 48, 1329–1337 (2000).
[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]

J Aust. Math. Soc. B (1)

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J Aust. Math. Soc. B 38, 209–219 (1996).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. A (1)

Z. Zalevsky, D. Medlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. Soc. A 2, 83–87 (2000).
[CrossRef]

J. Opt. Soc. Am. A (10)

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

M. F. Erden, H. M. Ozaktas, “Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains,” J. Opt. Soc. Am. A 15, 1647–1657 (1998).
[CrossRef]

İ. Ş. Yetik, H. M. Ozaktas, B. Barshan, L. Onural, “Perspective projections in the space-frequency plane and fractional Fourier transforms,” J. Opt. Soc. Am. A 17, 2382–2390 (2000).
[CrossRef]

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

M. F. Erden, H. M. Ozaktas, D. Mendlovic, “Propagation of mutual intensity expressed in terms of the fractional Fourier transform,” J. Opt. Soc. Am. A 13, 1068–1071 (1996).
[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, “Fractional Fourier transforms and their optical implementation: II.” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
[CrossRef]

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

Opt. Commun. (4)

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

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

J. García, R. G. Dorsch, A. W. Lohmann, C. Ferreira, Z. Zalevsky, “Flexible optical implementation of fractional Fourier transform processors. Applications to correlation and filtering,” Opt. Commun. 133, 393–400 (1997).
[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. (3)

Opt. Rev. (1)

H. M. Ozaktas, B. Barshan, D. Mendlovic, “Convolution and filtering in fractional Fourier domains,” Opt. Rev. 1, 15–16 (1994).
[CrossRef]

Optics Commun. (1)

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

Signal Process. (1)

M. A. Kutay, H. Özaktaş, H. M. Ozaktas, O. Arikan, “The fractional Fourier domain decomposition,” Signal Process. 77, 105–109 (1999).
[CrossRef]

Other (6)

İ. Ş. Yetik, M. A. Kutay, H. Özaktaş, H. M. Ozaktas, “Continuous and discrete fractional Fourier domain decomposition, ”in Proceedings of the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing, (IEEE, Piscataway, N. J., 2000) Vol. I, pp. 93–96.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to the fractional Fourier transform and its applications” in P. W. Hawkes, ed., Advances in Imaging and Electron Physics, Vol. 106 (Academic Press, San Diego, Calif., 1999) Chap. 4, pp. 239–291.

M. A. Kutay, M. F. Erden, H. M. Ozaktas, O. Arikan, Ç. Candan, Ö Güleryüz, “Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations,” in Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Process. (IEEE, Piscataway, N. J., 1998) pp. 3433–3436.

M. A. Kutay, H. Özaktaş, M. F. Erden, H. M. Ozaktas, O. Arikan, “Solution and cost analysis of general multi-channel and multi-stage filtering circuits,” in Proceedings of the 1998 IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis (IEEE, Piscataway, N. J., 1998) pp. 481–484.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing, (John Wiley & Sons, New York, 2001).

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transform to optical pattern recognition,” in Optical Pattern Recognition, (Cambridge University Press, Cambridge, 1998) Chap. 4, pp.89–125

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Figures (4)

Fig. 1
Fig. 1

(a) Fourier-domain filtering. (b) Space-domain filtering. (c) ath-order fractional Fourier domain filtering. (d) Fractional Fourier domains. (e) Multistage filtering. (f) Multichannel filtering.

Fig. 2
Fig. 2

(a) The multichannel configuration consists of k = 1, 2, … , M parallel channels, each consisting of a fractional Fourier-transform stage F a k followed by a spatial filter h k followed by another fractional Fourier-transform stage of order -a k . (b), (c) and (d) show three alternative optical implementations of the fractional Fourier-transform stages appearing in (a) (b) Canonical implementation type I. (c) Canonical implementation type II. (d) Quadratic graded-index (GRIN) medium implementation. While the focal lengths of the lenses and their separations are shown in (b) and (c) the radial-refractive-index distribution is given by n 2(r) = n 0 2[1 - (r/χ)2].

Fig. 3
Fig. 3

(a) Desired Gaussian–Schell-model mutual intensity profile. (b) Normalized error versus P for different values of M. We show M = 2, P = 2, 4, 8, 12 by crosses; M = 4, P = 4, 8, 12 by asterisks M = 8, P = 8, 12 by open circles, and M = 12, P = 12 by dots. (c) Synthesized profile using uniform orders (M = 2). (d) Synthesized profile using optimized orders (M = 2, P = 8).

Fig. 4
Fig. 4

(a) Closest positive semi-definite approximation to the desired rectangular mutual intensity profile. (b) Normalized error versus P for different values of M. We show M = 2, P = 2, 4, 8, 12 by pluses; M = 4, P = 4, 8, 12 by asterisks M = 8, P = 8, 12 by open circles, and M = 12, P = 12 by dots. (c) Synthesized profile using uniform orders (M = 2). (d) Synthesized profile using optimized orders (M = 2, P = 8).

Equations (18)

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

fau=1-i cotaπ21/2  expiπcotaπ2u2-2 cscaπ2uu+cotaπ2u2 fudu.
Tms=F-aMΛM  Fa2 - a1Λ1Fa1,
Tmc=k=1MF-akΛkFak,
Fig. 2b: dI=s2λtanaπ/4, fI=s2λcscaπ2,
Fig. 2c: dII=s2λsinaπ/2, fII=s2λcotaπ4,
Fig. 2d: dGRIN=s2λaπ/2, χGRIN=s2λ,
Rgu1, u2= Rfu1, u2Hu1, u1×H*u2, u2du1du2,
Rg=HRfH,
R=UDU,
R=UD1/2UUD1/2U.
Rg=R˜gR˜g=R˜gR˜g=R˜g2, Rf=R˜fR˜f=R˜fR˜f=R˜f2,
R˜g=R˜g=UD1/2U,R˜f=R˜f=UD1/2U.
R˜gR˜g=HR˜fR˜fH.
R˜g=HR˜f,
H=R˜gR˜f-1.
Rfu1, u2=δu1-u2rectu12r0.
Rgu1, u2=exp-u1-u222r12exp-u12+u224r22.
Rgu1, u2=rect|u1-u2|2r1rectu12r2rectu22r2,

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