## Image rotation, Wigner rotation, and the fractional Fourier transform

JOSA A, Vol. 10, Issue 10, pp. 2181-2186 (1993)

http://dx.doi.org/10.1364/JOSAA.10.002181

Acrobat PDF (562 KB)

### Abstract

In this study the degree *p* = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree *P* = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [
“
Fourier transforms of fractional order and their optical implementation,”
Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform.

© 1993 Optical Society of America

## 1. SURVEY AND MOTIVATION

- 1. Why are there actually
*three*titles? - 2. What is a fractional Fourier transform?
- 3. What does this transform have to do with optics?

*L*that is required for performing a Fourier transform of the coherent input image. Now imagine that the graded-index fiber is cut into pieces. A piece of length

*PL*(

*P*< 1) will do to the input image what Mendlovic and Ozaktas [1] call a fractional Fourier transform. Putting two pieces of lengths

*P*

_{1}

*L*and

*P*

_{2}

*L*together corresponds apparently to a fractional transform of degree

*P*

_{1}+

*P*

_{2}=

*P*.

*P*

_{1}+

*P*

_{2}=

*P*

_{2}+

*P*

_{1}, the fractional Fourier transform is also commutative if the definition is based on this gedankenexperiment.

*u*(

*x*,

*y*) or signal

*u*(

*t*) can be described indirectly and uniquely by a Wigner distribution function, which will be reintroduced briefly later in the paper. The Wigner distribution function (WDF) undergoes certain changes if something happens to the signal [from now on called

*u*(

*x*)]. For example, propagation in free space means a horizontal shearing of the WDF, and passage through a lens corresponds to a vertical shearing of the WDF. A Fraunhofer diffraction (i.e., an ordinary Fourier transform) lets the WDF rotate by 90°. Hence it is plausible to define a fractional Fourier transform as what happens to the signal

*u*(

*x*) while the WDF is rotated by an angle of

*ϕ*=

*Pπ*/2. The

*P*is the fractional degree. Notice that two consecutive rotations obey

*ϕ*

_{1}+

*ϕ*

_{2}=

*ϕ*

_{TOTAL}and

*ϕ*

_{1}+

*ϕ*

_{2}=

*ϕ*

_{2}+

*ϕ*

_{1}. Hence our definition is inherently additive and commutative.

- 4. If a rotation of the WDF is accepted as the primary (but indirect) definition of a fractional Fourier transform, how then is the corresponding operator for
*u*(*x*) itself defined? - 5. How can the WDF rotation be subdivided into other well-known WDF operations?
- 6. How can one implement experimentally the fractional Fourier transform?

*F*

^{(1)}a transform of degree

*P*= 1. A degree of 2 means that the Fourier transform is applied twice in a row:

*F*

^{(2)}=

*F*

^{(1)}

*F*

^{(1)}. So far, I have introduced only a slightly different nomenclature for a well-known procedure. However, new territory is entered if the degree

*P*is no longer an integer but, for example, 1/2, 1/3, and so on. How can we define

*F*

^{(}

^{P}^{)}?

*u*(

*x*) corresponds to a 90° rotation of the Wigner distribution. Naturally, we want

*F*

^{(1/2)}to correspond to a 45° rotation and

*F*

^{(2/3)}to a 60° rotation of the WDF, and so on. This raises the question of what kind of experiment upon

*u*(

*x*) would rotate the associated WDF accordingly. A preliminary answer will be found, but it is not quite satisfactory. Again, an isomorphic transition helps us. A special way of rotating a two-dimensional image consists of three steps, for which the three corresponding operations in Wigner space are known. Now we have gotten everything together for defining and implementing a fractional Fourier transform. As a by-product, we found a new method for rotating an image.

## 2. ABOUT THE WIGNER DISTRIBUTION FUNCTION

2. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. **25**, 26–30 (1978); [CrossRef]

2. “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. **69**, 1710–1716 (1979).

4. K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. **42**, 310–314 (1982) [CrossRef] .

*u*(

*x*):

*u*(0) happens to be very small or even zero we search for the maximum of |

*u*(

*x*)|

^{2}at

*x*=

*x*

*and modify Eq. (2.7) accordingly:*

_{M}*ν*, the so-called spatial frequency, has the dimension (1/length). Instead, we introduce the coordinate of the Fourier plane of a typical optical setup:

*λ*is the wavelength and

*f*

_{1}an arbitrary focal length, which we consider to be fixed from now on. Constant factors such as

*λf*

_{1}are dropped into the following definitions.

*x*,

*ξ*) plane. The associated physical process is a Fraunhofer diffraction.

*f*may be different from

*f*

_{1}.

*ξ*direction.

*Z*. This process can be described best in the spatial-frequency domain of the signal:

*Z*to the standard focal length

*f*

_{1}, and we rewrite relation (2.18) with the new variable

*ξ*:

*x*shearing of the WDF

## 3. DEFINITION OF THE FRACTIONAL FOURIER TRANSFORM

*P*, which may be a noninteger.

*P*= 1 we know that the associated WDF is rotated by 90° [relation (2.14)]:

*P*we specify the coordinate transformation (2.22) of the WDF as

4. K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. **42**, 310–314 (1982) [CrossRef] .

*u*(

*x*):

*u*

_{0}(

*x*). Nevertheless, neither the algorithm nor the suggested experiment is very elegant. But we did arrive at a working definition for the fractional Fourier transform. Two simple experiments for implementing our definition will evolve in Section 6. However, two preliminary steps are needed: image rotation (Section 4) and Wigner rotation (Section 5).

## 4. IMAGE ROTATION BASED ON IMAGE SHEARING

*β*−

*α*) has been accomplished. This method is commonly implemented with Dove prisms. It would be the preferred method if the WDF existed in two dimensions. Here I propose another way for rotating an image. This alternative method can be extended to four dimensions. It consists of three successive shearing operations (as illustrated in Fig. 1):

*x*

_{0},

*y*

_{0}) to (

*x*

_{3},

*y*

_{3}) if the coefficients satisfy

*x*shear,

*y*shear, and again

*x*shear. An equivalent result can be achieved by the following three steps:

*y*shear,

*x*shear, and

*y*shear.

5. A. W. Lohmann and N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. **22**, 780–783 (1983) [CrossRef] [PubMed] .

## 5. ROTATION OF THE WIGNER DISTRIBUTION FUNCTION

*u*(

*x*,

*y*), whose WDFs are four dimensional.

*u*(

*x*,

*y*) in free space in the

*Z*direction is described by multiplication of

*ũ*(

*ξ*,

*η*), with

*u*(

*x*,

*y*) is multiplied by

*x*,

*ξ*,

*y*,

*η*), is

*x*and

*y*:

*Q*

*≠*

_{x}*Q*

*. And a birefringent medium would be characterized by*

_{y}*R*

*≠*

_{x}*R*

*. The inversion from*

_{y}*W*(

*x*,

*y*,

*ξ*,

*η*) to

*u*(

*x*,

*y*) is straightforward:

## 6. TWO OPTICAL SYSTEMS FOR PERFORMING THE FRACTIONAL FOURIER TRANSFORM

*Z*=

*Rf*

_{1}and passage through a lens with a focal length

*f*=

*f*

_{1}/

*Q*. System I (RQR) is shown in Fig. 2 and is defined in Eq. (5.6), with the parameters

*Q*and

*R*established in Eq. (5.7).

*F*

^{(}

^{P}^{)}by analyzing the wave propagation through system II (Fig. 3). The complex amplitude

*u*

_{0}(

*x*

_{0}) is the input of the fractional Fourier transform:

*Z*coordinates refer to before the lens and behind the lens, respectively. Equations (6.2) and (6.3) are connected by a Fourier transform:

*ϕ*and the fractional degree

*P*are connected by

## 7. FINAL COMMENTS

*P*= 4/3. That result [6]

6. A. W. Lohmann and D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. **93**, 25–26 (1992) [CrossRef] .

7. A. W. Lohmann and D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A **9**, 2009–2112 (1992) [CrossRef] .

8. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. **25**, 241–265 (1980) [CrossRef] .

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

10. B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. **ASSP-30**, 25–31 (1982) [CrossRef] .

*P*as the key.

## ACKNOWLEDGMENTS

## REFERENCES

1. | H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published). |

2. | M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. |

3. | H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. |

4. | K.-H. Brenner and A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. |

5. | A. W. Lohmann and N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. |

6. | A. W. Lohmann and D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. |

7. | A. W. Lohmann and D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A |

8. | V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. |

9. | A. C. McBride and F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. |

10. | B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. |

**History**

Original Manuscript: November 9, 1992

Revised Manuscript: April 15, 1993

Manuscript Accepted: April 20, 1993

Published: October 1, 1993

**Citation**

Adolf W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A **10**, 2181-2186 (1993)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-10-10-2181

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### References

- H. M. Ozaktas, D. Mendlovic, “Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published).
- M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978); “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
- H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980). [CrossRef]
- K.-H. Brenner, A. W. Lohmann, “Wigner distribution function display of complex 1D-signals,” Opt. Commun. 42, 310–314 (1982). [CrossRef]
- A. W. Lohmann, N. Streibl, “Map transformations by optical anamorphic processing,” Appl. Opt. 22, 780–783 (1983). [CrossRef] [PubMed]
- A. W. Lohmann, D. Mendlovic, “An optical transform with odd cycles,” Opt. Commun. 93, 25–26 (1992). [CrossRef]
- A. W. Lohmann, D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9, 2009–2112 (1992). [CrossRef]
- V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,”J. Inst. Math. Appl. 25, 241–265 (1980). [CrossRef]
- A. C. McBride, F. H. Kerr, “On Namias’ fractional Fourier transforms,” IMA J. Appl. Math. 39, 159–175 (1987). [CrossRef]
- B. W. Dickinson, K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,”IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 25–31 (1982). [CrossRef]

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