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The Fractional Fourier Transform and Its Applications
Presenter: Pao-Yen Lin Research Advisor: Jian-Jiun Ding , Ph. D. Assistant professor Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Outlines Introduction Fractional Fourier Transform (FrFT) Linear Canonical Transform (LCT) Relations to other Transformations Applications
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Introduction Generalization of the Fourier Transform Categories of Fourier Transform a) Continuous-time aperiodic signal b) Continuous-time periodic signal (FS) c) Discrete-time aperiodic signal (DTFT) d) Discrete-time periodic signal (DFT)
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Fractional Fourier Transform (FrFT)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Fractional Fourier Transform (FrFT) Notation is a transform of is a transform of
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Fractional Fourier Transform (FrFT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Fractional Fourier Transform (FrFT) (cont.) Constraints of FrFT Boundary condition Additive property
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Definition of FrFT Eigenvalues and Eigenfunctions of FT Hermite-Gauss Function
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Definition of FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Definition of FrFT (cont.) Eigenvalues and Eigenfunctions of FT
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Definition of FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Definition of FrFT (cont.) Eigenvalues and Eigenfunctions of FrFT Use the same eigenfunction but α order eigenvalues
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Definition of FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Definition of FrFT (cont.) Kernel of FrFT 把An 代入上頁的結果
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Definition of FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Definition of FrFT (cont.)
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Properties of FrFT Linear. The first-order transform corresponds to the conventional Fourier transform and the zeroth-order transform means doing no transform. Additive.
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Linear Canonical Transform (LCT)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Linear Canonical Transform (LCT) Definition where
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Linear Canonical Transform (LCT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Linear Canonical Transform (LCT) (cont.) Properties of LCT When , the LCT becomes FrFT. Additive property where
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Relation to other Transformations
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to other Transformations Wigner Distribution Chirp Transform Gabor Transform Gabor-Wigner Transform Wavelet Transform Random Process
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Relation to Wigner Distribution
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Wigner Distribution Definition Property
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Relation to Wigner Distribution
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Wigner Distribution
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Relation to Wigner Distribution (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Wigner Distribution (cont.) WD V.S. FrFT Rotated with angle
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Relation to Wigner Distribution (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Wigner Distribution (cont.) Examples slope=
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Relation to Chirp Transform
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Chirp Transform for Note that is the same as rotated by
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Relation to Chirp Transform (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Chirp Transform (cont.) Generally,
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Relation to Gabor Transform (GT)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor Transform (GT) Special case of the Short-Time Fourier Transform (STFT) Definition
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Relation to Gabor Transform (GT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor Transform (GT) (cont.) GT V.S. FrFT Rotated with angle
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Relation to Gabor Transform (GT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor Transform (GT) (cont.) Examples (a)GT of (b)GT of (c)GT of (d)WD of
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 GT V.S. WD GT has no cross term problem GT has less complexity WD has better resolution Solution: Gabor-Wigner Transform
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Relation to Gabor-Wigner Transform (GWT)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor-Wigner Transform (GWT) Combine GT and WD with arbitrary function
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Relation to Gabor-Wigner Transform (GWT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor-Wigner Transform (GWT) (cont.) Examples In (a) In (b)
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Relation to Gabor-Wigner Transform (GWT) (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Gabor-Wigner Transform (GWT) (cont.) Examples In (c) In (d)
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Relation to Wavelet Transform
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Wavelet Transform The kernels of Fractional Fourier Transform corresponding to different values of can be regarded as a wavelet family.
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Relation to Random Process
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process Classification Non-Stationary Random Process Stationary Random Process Autocorrelation function, PSD are invariant with time t
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) Auto-correlation function Power Spectral Density (PSD)
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) FrFT V.S. Stationary random process Nearly stationary
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) FrFT V.S. Stationary random process for
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) FrFT V.S. Stationary random process PSD:
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) FrFT V.S. Non-stationary random process Auto-correlation function PSD rotated with angle
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) Fractional Stationary Random Process If is a non-stationary random process but is stationary and the autocorrelation function of is independent of , then we call the -order fractional stationary random process.
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Relation to Random Process (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Relation to Random Process (cont.) Properties of fractional stationary random process After performing the fractional filter, a white noise becomes a fractional stationary random process. Any non-stationary random process can be expressed as a summation of several fractional stationary random process.
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Applications of FrFT Filter design Optical systems Convolution Multiplexing Generalization of sampling theorem
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Filter design using FrFT
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Filter design using FrFT Filtering a known noise Filtering in fractional domain signal noise
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Filter design using FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Filter design using FrFT (cont.) Random noise removal If is a white noise whose autocorrelation function and PSD are: After doing FrFT Remain unchanged after doing FrFT!
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Filter design using FrFT (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Filter design using FrFT (cont.) Random noise removal Area of WD ≡ Total energy signal signal
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Optical systems Using FrFT/LCT to Represent Optical Components Using FrFT/LCT to Represent the Optical Systems Implementing FrFT/LCT by Optical Systems
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Using FrFT/LCT to Represent Optical Components
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Using FrFT/LCT to Represent Optical Components Propagation through the cylinder lens with focus length Propagation through the free space (Fresnel Transform) with length
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Using FrFT/LCT to Represent the Optical Systems
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Using FrFT/LCT to Represent the Optical Systems input output
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Implementing FrFT/LCT by Optical Systems
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Implementing FrFT/LCT by Optical Systems All the Linear Canonical Transform can be decomposed as the combination of the chirp multiplication and chirp convolution and we can decompose the parameter matrix into the following form
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Implementing FrFT/LCT by Optical Systems (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Implementing FrFT/LCT by Optical Systems (cont.) input output The implementation of LCT with 2 cylinder lenses and 1 free space input output The implementation of LCT with 1 cylinder lens and 2 free spaces
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Convolution Convolution in domain Multiplication in domain
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Convolution (cont.)
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Multiplexing using FrFT
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Multiplexing using FrFT TDM FDM
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Multiplexing using FrFT
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Multiplexing using FrFT Inefficient multiplexing Efficient multiplexing
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Generalization of sampling theorem
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Generalization of sampling theorem If is band-limited in some transformed domain of LCT, i.e., then we can sample by the interval as
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Generalization of sampling theorem (cont.)
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Generalization of sampling theorem (cont.)
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Conclusion and future works
Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Conclusion and future works Other relations with other transformations Other applications
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 References [1] Haldun M. Ozaktas and M. Alper Kutay, “Introduction to the Fractional Fourier Transform and Its Applications,” Advances in Imaging and Electron Physics, vol. 106, pp. 239~286. [2] V. Namias, “The Fractional Order Fourier Transform and Its Application to Quantum Mechanics,” J. Inst. Math. Appl., vol. 25, pp , [3] Luis B. Almeida, “The Fractional Fourier Transform and Time-Frequency Representations,” IEEE Trans. on Signal Processing, vol. 42, no. 11, November, [4] H. M. Ozaktas and D. Mendlovic, “Fourier Transforms of Fractional Order and Their Optical Implementation,” J. Opt. Soc. Am. A 10, pp , [5] 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, vol. 11, no. 2, pp , Feb [6] A. W. Lohmann, “Image Rotation, Wigner Rotation, and the Fractional Fourier Transform,” J. Opt. Soc. Am. 10,pp , [7] S. C. Pei and J. J. Ding, “Relations between Gabor Transform and Fractional Fourier Transforms and Their Applications for Signal Processing,” IEEE Trans. on Signal Processing, vol. 55, no. 10, pp , Oct
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 References [8] Haldun M. Ozaktas, Zeev Zalevsky, M. Alper Kutay, The fractional Fourier transform with applications in optics and signal processing, John Wiley & Sons, [9] M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, “Optimal Filter in Fractional Fourier Domains,” IEEE Trans. Signal Processing, vol. 45, no. 5, pp , May [10] J. J. Ding, “Research of Fractional Fourier Transform and Linear Canonical Transform,” Doctoral Dissertation, National Taiwan University, [11] C. J. Lien, “Fractional Fourier transform and its applications,” National Taiwan University, June, [12] Alan V. Oppenheim, Ronald W. Schafer and John R. Buck, Discrete-Time Signal Processing, 2nd Edition, Prentice Hall, [13] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000.
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Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University 2018/12/31 Chenquieh!
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