## Nonseparable Two-Dimensional Fractional Fourier Transform

Applied Optics, Vol. 37, Issue 23, pp. 5444-5453 (1998)

http://dx.doi.org/10.1364/AO.37.005444

Acrobat PDF (590 KB)

### 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

**OCIS Codes**

(070.2590) Fourier optics and signal processing : ABCD transforms

(100.0100) Image processing : Image processing

(100.3020) Image processing : Image reconstruction-restoration

**Citation**

Aysegul Sahin, M. Alper Kutay, and Haldun M. Ozaktas, "Nonseparable Two-Dimensional Fractional Fourier Transform," Appl. Opt. **37**, 5444-5453 (1998)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-37-23-5444

Sort: Year | Journal | Reset

### References

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

## 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.