OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 15, Iss. 4 — Apr. 1, 1998
  • pp: 825–833

Optimal image restoration with the fractional Fourier transform

M. Alper Kutay and Haldun M. Ozaktas  »View Author Affiliations


JOSA A, Vol. 15, Issue 4, pp. 825-833 (1998)
http://dx.doi.org/10.1364/JOSAA.15.000825


View Full Text Article

Acrobat PDF (1001 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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(N<sup>2</sup>) 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

OCIS Codes
(070.2590) Fourier optics and signal processing : ABCD transforms
(100.3020) Image processing : Image reconstruction-restoration

Citation
M. Alper Kutay and Haldun M. Ozaktas, "Optimal image restoration with the fractional Fourier transform," J. Opt. Soc. Am. A 15, 825-833 (1998)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-4-825


Sort:  Author  |  Year  |  Journal  |  Reset

References

  1. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).
  2. F. L. Lewis, Optimal Estimation (Wiley, New York, 1986).
  3. H. M. Ozaktas, B. Barshan, D. Mendlovic, and 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).
  4. M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45, 1129–1143 (1997).
  5. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdaǧi, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44, 2141–2150 (1996).
  6. H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101, 163–169 (1993).
  7. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation: part I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
  8. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transformations and their optical implementation: part II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993).
  9. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
  10. A. Sahin, H. M. Ozaktas, and D. Mendlovic, “Optical implementation of the two-dimensional fractional Fourier transform with different orders in two dimensions,” Opt. Commun. 120, 134–138 (1995).
  11. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
  12. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25, 241–245 (1980).
  13. A. C. McBride and F. H. Kerr, “On Namias’s fractional Fourier transform,” IMA J. Appl. Math. 39, 159–175 (1987).
  14. L. M. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
  15. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
  16. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
  17. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
  18. P. Pellat-Finet and G. Bonnet, “Fractional-order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
  19. L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Opt. Commun. 110, 517–522 (1994).
  20. T. Alieva, V. Lopez, F. Agullo-Lopez, and L. B. Almeida, “The angular Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
  21. T. Alieva and F. Agullo-Lopez, “Optical wave propagation of fractal fields,” Opt. Commun. 125, 267–274 (1996).
  22. D. T. Smithey, M. Beck, M. G. Raymer, and 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).
  23. J. R. Fonollosa and 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.
  24. J. Wood and D. T. Barry, “Radon transformation of time-frequency distributions for analysis of multicomponent signals,” IEEE Trans. Signal Process. 42, 3166–3177 (1994).
  25. A. W. Lohmann and B. H. Soffer, “Relationships between the Radon-Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
  26. H. M. Ozaktas, N. Erkaya, and 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).
  27. H. M. Ozaktas and O. Aytür, “Fractional Fourier domains,” Signal Process. 46, 119–124 (1995).
  28. A. Sahin, “Two-dimensional fractional Fourier transformation and its optical implementation,” Master’s thesis (Bilkent University, Ankara, Turkey, 1996).
  29. M. F. Erden, H. M. Ozaktas, and A. Sahin, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
  30. B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, New York, 1979).
  31. A. Jazwinski, Stochastic Processes and Filtering Theory (Academic, New York, 1970).
  32. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Wetterling, Numerical Recipes in Pascal (Cambridge U. Press, Cambridge, 1989), pp. 574–579.
  33. J. H. Shapiro, “Diffraction-limited atmospheric imaging of extended objects,” J. Opt. Soc. Am. 66, 469–477 (1976).
  34. J. Zhang, “The mean field theory in EM procedures for blind Markov random field image restoration,” IEEE Trans. Image Process. 2, 27–40 (1993).
  35. M. R. Banham and A. K. Katsaggelos, “Spatially adaptive wavelet-based multiscale image restoration,” IEEE Trans. Image Process. 5, 619–634 (1996).
  36. M. R. Banham and A. K. Katsaggelos, “Digital image restoration,” IEEE Trans. Signal Process. 14(2), 24–41 (1997).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited