The Fractional Fourier Transform on Graphs

Home , Graph Fourier Transform

Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia The Fractional Fourier Transform on Graphs

Yi-qian Wang∗, Bing-zhao Li† and Qi-yuan Cheng‡ ∗ Xuteli School, Beijing Institute of Technology, Beijing, China E-mail: wangyiqian [email protected] † School of Mathematics and statistics, Beijing Institute of Technology, Beijing, China E-mail: li [email protected] Tel/Fax: +86-10-68913829 ‡ Xuteli School, Beijing Institute of Technology, Beijing, China E-mail: [email protected] Tel/Fax: +86-10-81383312

Abstract—The emerging field of signal processing on graphs The second one is called discrete signal processing on merges algebraic or spectral graph theory with discrete signal graphs (DSPG) [2], [3], which originates from the algebraic processing techniques to process signals on graphs. In this signal processing (ASP) and uses adjacency matrix (the graph paper, a definition of the fractional Fourier transform on graphs (GFRFT) is proposed and consolidated, which extends the dis- shift operator) as its basic building block. Particularly, in this crete fractional Fourier transform (DFRFT) in the same sense framework, the graph Fourier transform expands the signal the graph Fourier transform (GFT) extends the discrete Fourier onto eigenvectors of the adjacency matrix, and defines the transform (DFT). The definition is based on the eigenvalue spectrum by the corresponding eigenvalues. As the graph decomposition method of defining DFRFT, for it satisfies all the shift operator, or the adjacency matrix, is not restricted to be agreeable properties expected of the discrete fractional Fourier transform. Properties of the GFRFT are discussed, and examples symmetric, the second approach can be applied to arbitrary of GFRFT of some graph signals are given to illustrate the graphs. transform. An important technique to investigate time-series or graph I.INTRODUCTION signals is to adopt appropriate transforms. For conventional signal processing, Fourier analysis is one of the most basic and the most commonly used. Beside the Fourier transform per Data with high-dimensional structures residing on the ver- se, joint time-frequency analysis tools, including the Wigner tices of weighted graphs are being generated at an increasing distribution [15], the short-time Fourier transform (STFT) rate in a wide range of applications, such as social, economic, [15], [16] and the continuous wavelet transform (CWT) [17] transportation, energy, sensor, biological networks and so on. are frequently used in speech processing, radar, or quantum This large set of data, compared to ordinary time-series or physics. images, has complex structures and calls for new types of The fractional Fourier transform (FRFT) was proposed processing techniques, which ultimately leads to the emerging in 1980 [18], and later introduced to the signal processing field of signal processing on graphs [1], [2]. community in 1994 [19]. It is a set of linear transformations Signal processing on graphs extends the conventional dis- that generalizes the Fourier transform, and can convert a signal crete signal processing (DSP), by modelling the underlying into any intermediate domain between time and frequency. The structure of the signal by a graph, and indexing the signal FRFT can also be interpreted as expanding a signal onto a by vertices. Related work covers graph-based transforms [3], basis of linear frequency modulation signals. Its applications [4], [5], [6], filtering [3], [7], [8], sampling [9], [10], [11], cover a wide range, from filter design to pattern recognition. reconstruction and recovery [12], [13], and so on. Before introducing the fractional point of view into the There are two fundamental frameworks of signal processing field of DSP , it needs to be discretized for DSP. Since the on graphs, based on spectral and algebraic approaches respec- G discretization of the fractional Fourier transform cannot be ob- tively. Both of the frameworks define fundamental notions and tained by directly sampling in time domain, the discretization concepts for the theory, yet their disparate bases contribute to of the FRFT has been widely studied. There are three main- different definition and techniques for analysing graph signals. The first one derives from spectral graph theory, which stream types of method to defining and calculating the discrete uses graph Laplacian matrix as its basic building block [1]. fractional Fourier transform (DFRFT): linear combination- It expands the graph signal into the eigenfunctions of the type [20], sampling-type [21] and eigendecomposition-type Laplace operator to define the graph Fourier transform, and the [22], [23], [24]. The different methods have been studied and corresponding spectrum is represented by the eigenvalues. This compared in details in [25]. approach, however, is limited to undirected graphs only, as the In this paper, a definition of the fractional Fourier transform standard graph Laplacian matrix is required to be symmetric on graphs (GFRFT) is proposed and consolidated by the and positive semi-definite. following steps. In Section II, the theories on which the study is based are briefly reviewed. Section III presents the definition This work is sponsored by the National Natural Science Foundation of of GFRFT and several properties. In Section IV, examples China (No. 61671063). of graph signals under GFRFT are given to illustrate the

978-1-5386-1542-3@2017 APSIPA APSIPA ASC 2017 Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia transform. Finally, Section V concludes the paper.

II.PRELIMINARIES A. Graph Fourier transform

In this study, the DSPG (or ASP) framework is adopted, and Fig. 1. Time-series with the length of N represented by a cycle graph. the following reviews some of the most fundamental notions and concepts. A detailed introduction to the theory can be This matrix is diagonalizable and the eigendecomposition found in [2], [3]. is For a dataset where data elements are connected to each  −j 2π 0  other according to a certain relation, the relation is denoted e N 1 −1 . by a graph G = (V, A), where V = v0, . . . , vN−1 is the set of C = DFT  .  DFT (6) N N  .  N vertices, and A is the weighted adjacency matrix of the graph. −j 2π 0 e N Each data element is indexed by a node vn, and each weighted where DFT denotes the discrete Fourier transform matrix. edge An,m ∈ C from vm to vn denotes the relation of the mth N data element to the nth one. If the graph is undirected, that In this case, as intuitively, the GFT matrix F = DFTN . is the relation is bidirectional, we have An,m = Am,n, which It should be noted that the definition of GFT and its inverse means A is symmetrical. depends on the choice of eigenvectors, which is unnecessarily Based on this graph, the dataset is referred to as graph unique. How to choose the eigenvectors, however, is beyond signal, which is defined as a map the scope of this paper. Rather, it is assumed they are fixed. Nevertheless, the desirable choice of the eigenvectors to opti- s :V → C, (1) mize the analysis remains an open issue. v 7→ s , (2) n n B. Continuous fractional Fourier transform and could be written as a complex-valued vector The continuous fractional Fourier transform [18], [19] can T s = s s . . . s ∈ N . be viewed as a linear operator, which rotates the time axis 0 1 N−1 C to the frequency axis by an angle of φ (rather than π/2 Parallel to the classical Fourier transform which expands as a Fourier operator) in the time-frequency plane, and the the signal onto a basis of exponent signals, the graph Fourier transformed signal would be a representation of the signal basis is given by the Jordan eigenvectors of the adjacency along the rotated axis u making an angle of φ with the time matrix A, and the distinct eigenvalues λ0, λ1, ˙,λM−1 are axis. the graph frequencies that form the spectrum of the graph. The ath-order FRFT is defined for 0 < |a| < 2 through its Jordan eigenvectors corresponding to a frequency λm are integral kernel (Mehler kernel) as called frequency components. Z ∞ a As Jordan eigenvectors form the matrix V that reduces A F f(u) := Ka(u, t)f(t)dt −∞ (7) to its Jordan normal form 2 2 K (u, t) := K ejπ(u cot φ−2ut csc φ+t cot φ) −1 a φ A = VJAV , where the graph Fourier transform of a graph signal can be defined φ := aπ/2, as K := exp[−j(πsgn(φ)/4 − φ/2)]/| sin φ|0.5. ˆs = Fs := V−1s, (3) φ The kernel K (u, t) is defined separately for a = 0 and a = where F = V−1 is the graph Fourier transform matrix. The a ±2 as K (u, t) := δ(u − t) and K (u, t) := δ(u + t). The nth component sˆ represents the frequency component of the 0 ±2 n definition is extended outside the interval [−2, 2] by noting signal s. that F 4l+af(u) = F af(u) for any integer l. The inverse graph Fourier transform is defined as The Mehler kernel is known to have the spectral expansion −1 s = F ˆs := Vˆs, (4) ∞ X −j π ka Ka(u, t) = ψk(u)e 2 ψk(t) (8) The DSPG framework is the extension of the classical DSP k=0 theory. Periodic time-series can be presented by a cycle graph shown in Fig. 1 [14]. where ψk(t) denotes the kth Hermite-Gaussian function, The directed edges represent the flow of time from past to which is also the eigenfunction of Fourier transform. future. The adjacency matrix of a cycle graph is The properties of FRFT include: a −1 −a a † †   1) Unitarity: (F ) = F = (F ) , where (·) denotes 1 Hermitian conjugation; 1  F a ◦ F b = F b ◦ F a = F a+b A = C =   . (5) 2) Index additivity: ;  ..  0  .  3) Zero rotation: F = Id; 1 4) Reduction to Fourier transform when a = 1: F 1 = F.

978-1-5386-1542-3@2017 APSIPA APSIPA ASC 2017 Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia

C. Discrete fractional Fourier transform of the DFRFT matrix to the ath order, the GFRFT matrix can be given by For a discrete fractional Fourier transform in the strict sense, N−1 it is supposed to have the following properties [23]: X Fa[m, n] = χ [m]λaχ [n]. (12) 1) Unitarity; k k k 2) Index additivity; k=0 3) Reduction to Fourier transform when a = 1; Then we can generalize the definition into a broader domain 4) Resemblance to the continuous FRFT. using Jordan decomposition. As mentioned previously, when It can be easily checked that all three mainstream types of the adjacency matrix is not diagonalizable, the GFT matrix can A = VJ V−1 DFRFT reduce to Fourier transform when a = 1, and their be defined by the Jordan decomposition of A . other properties are as listed in Table 1 [25]. Clearly, the eigen- Further, the GFT matrix is decomposed into its Jordan eigenvectors decomposition method satisfies all the desirable properties, so −1 −1 in the current study, a similar method is used to generalize the F = V = PJP , (13) transform onto graphs. and the definition of the GFRFT matrix would be Using eigendecomposition method, the DFRFT is defined as Fa := PJaP−1. (14) N−1 a X j π ka Note that the matrix function of the Jordan normal form F [m, n] = uk[m]e 2 uk[n], (9) k=0 can be computed by applying the function to each Jordan block, so that the elements of the kth super-diagonal of the (k) denoting the discrete Hermite-Gaussian series as uk[n]. The f (λ) resulting block are k! . Specifically, for the ath-order definition of the discrete Hermite-Gaussian series is given by a a−k power function, the kth super-diagonal is k λ , where the the common eigenvectors of two commutative matrix – those k binomial coefficients are defined as a = Q a+1−i . For of the discrete Fourier transform and the discrete Laplacian k i=1 i example, operator, so that the DFRFT would resemble its continuous j π ka  a version. The e 2 in the middle can be interpreted as the λ1 1 0 0 0 eigenvalue of DFT to the ath order.  0 λ1 1 0 0     0 0 λ1 0 0  =   III.THEFRACTIONAL FOURIERTRANSFORMONGRAPHS  0 0 0 λ2 1  0 0 0 0 λ2 A. Definition  a a−1 a a−2  λ1 1 λ1 2 λ1 0 0 The definition generalizes the discrete fractional Fourier a a a−1  0 λ1 1 λ1 0 0   a  transform in the same sense the graph Fourier transform  0 0 λ1 0 0  . generalizes the discrete Fourier transform.  a a a−1  0 0 0 λ2 1 λ2  a We first assume that the GFRFT can be expressed in the 0 0 0 0 λ2 matrix form as a a It can be verified that this definition of matrix power function ˆs = F s := F s. (10) holds the index additivity JaJb = Ja+b [26]. Once we have defined the GFRFT matrix, the GFRFT to To begin with a simple circumstance, consider a sig- the ath order would then be nal s on graph G with a diagonalizable adjacency matrix −1 a a A = VΛAV . Assume that the GFT matrix F can be ˆs = F s := F s. (15) orthogonally diagonalized, i.e. Its inverse (F a)−1 would be F −a, for N−1 −1 T X T F −a ◦ F a = F−aFa = PJ−aP−1PJaP−1 F = V = χΛχ = χkλkχk , (11) −a a −1 −1 (16) k=0 = PJ J P = PP = I. where λk and χk denote the kth eigenvalue of F and its B. Properties corresponding eigenvector respectively. Parallel to definition 1) Zero rotation F0 = PJ0P−1 = PIP−1 = PP−1 = I, (17) TABLE I so the 0th-order GFRFT of a signal is the signal itself. COMPARINGTHREEMAINTYPESOF DFRFTS 2) Reduction to GFT when a = 1 Properties linear combination√ sampling √ eigendecomposition √ unitary √ √ F1 = PJ1P−1 = PJP−1 = F, (18) index additive √× √ resemble FRFT × so the 1th-order GFRFT of a signal is the GFT signal.

978-1-5386-1542-3@2017 APSIPA APSIPA ASC 2017 Proceedings of APSIPA Annual Summit and Conference 2017 12 - 15 December 2017, Malaysia

3) Index additivity As mentioned earlier, the matrix power V. CONCLUSIONS function is index additive, so we have To sum up, a generalization of the graph Fourier transform is presented in the paper, which is designated fractional Fourier a b a −1 b −1 a b −1 F F = PJ P PJ P = PJ J P transform on graphs, or abbreviated as GFRFT. The transform, −1 (19) = PJa+bP = Fa+b. depending on the order a, can be interpreted as a rotation by an angle φ = aπ/2 in the vertex-frequency plane. In particular, 4) Reduction to DFRFT on cycle graphs it holds the following properties: It has been discussed that when A = C, the GFT 1) When a = 0, the GFRFT is the identity operator. matrix and the DFT matrix are identical, so F can be 2) When a = 1, the GFRFT reduce to the GFT. diagonalized by the Hermitian-Gaussian vector. Then 3) When the graph’s A = C, the GFRFT reduce to the the GFRFT matrix will be the same as the predeﬁned conventional FRFT. DFRFT matrix. 4) Two successive GFRFTs to the ath and bth order respec- 5) Convolution theorem tively, are equivalent to a single GFRFT to the a + bth The convolution in the fractional domain on the graph order. can be given by Further, the current study could possibly open new areas in

N−1 the ﬁeld of signal processing on graphs, and lead to further X (f ∗ g)[n] := fˆ[m]ˆg[m]F −a[n, m], research. Hopefully, useful processing techniques and speciﬁc m=0 (20) applications will be developed with the fractional insight. −a or f ∗ g := F (ˆf ◦ ˆg). ACKNOWLEDGMENT This work is supported by the National Natural Science Here, the ◦ denotes the Hadamard product of matrices. Foundation of China (No. 61671063). Hence, in this way, the convolution theorem holds The author also wishes to acknowledge valuable suggestions from Jie-dong Jiang and Shi-ji Lyu. They are both with School fd∗ g = ˆf ◦ ˆg. (21) of Mathematical Sciences, Peking University.

6) Translation property REFERENCES T The translation operator i can be given by convolution [1] D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, with a δi vector, whose components are all 0 except “The emerging field of signal processing on graphs. Extending high- δi[i] = 1. dimensional data analysis to networks and other irregular domains,” IEEE Signal Process. Mag., vol. 30, pp. 83-98, May 2013. √ √ −a ˆ ˆ [2] A. Sandryhaila and J. M. F. Moura, “Big data processing with signal Tif := Nf ∗ δi = N F (f ◦ δi) processing on graphs,” IEEE Signal Process. Mag., vol. 31, no. 5, pp. √ (22) −a ˆ a 80-90, 2014. = N F (f ◦ Fi ), [3] A. Sandryhaila and J. M. F. Moura, “Discrete signal processing on graphs,” IEEE Trans. Signal Process., vol. 61, no. 7, pp. 1644-1656, a a where Fi denotes the ith column of the matrix F . Then April 2013. its GFRFT is calculated to be [4] D. K. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on graphs via spectral graph theory,” Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011. √ [5] D. I. Shuman, B. Ricaud, P. Vandergheynst, “A windowed graph Fourier a a −a ˆ a F (Tif) = N F F (f ◦ Fi ) transform,” Stat. Signal Proc. Workshop, Aug. 2012. √ (23) [6] S. Chen, R. Varma, A. Sandryhaila, and J. Kovaceviˇ c,´ “Vertex-frequency ˆ a = N f ◦ Fi , analysis on graphs,” Appl. Comput. Harmon. Anal., vol. 40, no. 2, pp. 260-291, Mar. 2016. which corresponds to the time-delay property. [7] S. K. Narang and A. Ortega, “Perfect reconstruction two-channel wavelet filter banks for graph structured data,” IEEE Trans. Signal Process., vol. 60, pp. 2786-2799, June 2012. IV. EXAMPLES [8] S. K. Narang and A. Ortega, “Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs” IEEE Trans. Signal Process., vol. 61, no. 19, pp. 4671-4685, Dec. 2015. Examples of bipolar rectangular signal on the cycle graph [9] S. Chen, R. Varma, A. Sandryhaila, and J. Kovaceviˇ c,´ “Discrete signal [14] and the Minnesota road graph [29] are given in Fig. 2 processing on graphs: sampling theory,” IEEE Trans. Signal Process., vol. and Fig. 3 respectively to illustrate the GFRFT. We can have 63, no. 24, pp. 6510-6523, Dec. 2015. [10] A. Anis, and A. Ortega, “Towards a sampling theorem for signals on a sense for the rotation of domain from vertex to frequency arbitrary graphs,” Proc. IEEE Int. Conf. Acoust., Speech Signal Process., by observing the variation of GFRFT with different order a. May. 2014, pp. 3864-3868. Specifically, the first example shows that the GFRFT reduce [11] P. Hu , W. C. Lau, “A survey and taxonomy of graph sampling,” Computer Science, Aug. 2013. to the conventional DFRFT on the cycle graph. [12] S. Chen, A. Sandryhaila, J. M. F. Moura, and J. Kovaceviˇ c,´ “Signal re- The calculation is based on the Graph Signal Processing covery on graphs: Variation minimization,” IEEE Trans. Signal Process., vol. 63, no. 17, Sept. 2015. toolbox (GSPBox) [30] on MATLAB, with minor modifica- [13] S. K. Narang and A. Ortega, “Signal recovery on graphs: fundamental tions for the DSPG framework. The eigenvalues are calculated limits of sampling strategies,” IEEE Trans. Signal and Inf. Proc. over by the function eig() and sorted in the descending order. Networks, vol. 2, no. 4, pp. 539 - 554, Dec. 2016.

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(a) a = 0 (b) a = 0.2 (c) a = 0.4

(d) a = 0.6 (e) a = 0.8 (f) a = 1

Fig. 2. The GFRFT of bipolar rectangular signal on the cycle graph with different a.

(a) a = 0 (b) a = 0.2 (c) a = 0.4

(d) a = 0.6 (e) a = 0.8 (f) a = 1

Fig. 3. The GFRFT of bipolar rectangular signal on the Minnesota road graph with different a.

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