Optimal image restoration with the fractional Fourier transform

Journal of the Optical Society of America A: Optics and Image Science, and Vision

Volumn 15, Issue 4, 1998, Pages 825-833

  Alper Kutay, M ^a   Ozaktas, Haldun M ^a  

^a BILKENT UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; DIFFRACTION; ESTIMATION; FOURIER OPTICS; FOURIER TRANSFORMS; HOLOGRAPHY; LIGHT PROPAGATION; MATHEMATICAL MODELS; OPTICAL SYSTEMS; SIGNAL PROCESSING; STATISTICAL OPTICS; WHITE NOISE;

FRACTIONAL FOURIER TRANSFORM; MEAN SQUARE ERROR; WIENER FILTERING;

IMAGE RECONSTRUCTION;

EID: 0032049778     PISSN: 10847529     EISSN: 15208532     Source Type: Journal    
DOI: 10.1364/JOSAA.15.000825     Document Type: Article

References (36)

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. Bozdagi, “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 timefrequency 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 timefrequency 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 timefrequency 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. Aytiir, “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).