Multi-dimensional spectral graph wavelet transform

In this paper, we introduce the two-dimensional spectral graph wavelet transform (SGWT) of discrete functions defined on weighted Cartesian product graphs. The graphs we consider are undirected, non-planar, and can be cyclic with no multiple loops and no multiple edges. We build this transform with the help of spectral decomposition of N1N2×N1N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_1N_2 \times N_1N_2$$\end{document} Laplacian matrix L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document} of the Cartesian product graph G=G1□G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=G_1\square G_2$$\end{document}, where N1N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_1N_2$$\end{document} is the number of vertices of the Cartesian product graph. We have established the inversion formula for SGWT for continuous scale parameters s1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1$$\end{document} and s2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_2$$\end{document}. By sampling the scale parameters at discrete values, we obtain discrete SGWT coefficients at different scales and vertices. We have obtained the frame bounds of the frame formed by the set of scaling and wavelet coefficients. Further, we have proved the localization result which holds for both normalized and non-normalized form of Laplacian. Towards the end, we have given the implementation details of SGWT through an example to show how to calculate the graph wavelet coefficients.


Introduction
The scientific and engineering problems include analyzing data. Data can be given by a real valued function defined on some domain. Many functions on Cartesian product graphs are present in the real world, such as digital photographs, sensor observation time series, and movie ratings on Netflix. These functions are one, two or three dimensional and have directional characteristics along each factor graph. However, the one dimensional spectral wavelet transform does not distinguish these directions and assigns one dimensional spectra to signals on product graphs. If we have graph signals defined on individual factor graphs then we can separately use one dimensional spectral graph wavelet transform on each graph signal. Similarly, we can model discrete time-series data as a real valued function with domain a subset of integers.
Wavelet transforms have been shown to be extremely useful for a wide range of signal processing and analysis chal- Since their inception in the 1980's [1,2,4,5,8,12], wavelet transforms have seen a lot of success in a variety of signal and image processing applications, including compression [16,19,20], denoising [10,12] and inverse problems such as deconvolution [9,13,14].
Graph signal analysis has gained a lot of interest recently, leading to graph-based transforms like the graph Fourier transform (GFT) [5,17] and graph filters [8,15].
Motivated and inspired by the above work and need to process the data on non-Euclidean spaces, we have developed the two-dimensional multiscale wavelets and scaling functions on weighted cartesian product graphs in analogy with the wavelets on continuous field. We have proved the inversion formula for continuous scale parameters s 1 and s 2 . Upon discretizing the scale parameters, we obtain a finite number of coefficients arranged in different wavelet subbands. Also we have proved a localization result for Laplacian.
The paper is structured as follows. In Sect. 2, we have given the definitions of classical continuous wavelet transform in terms of operator T s using convolution theorem, two-dimensional discrete wavelet transform, Cartesian product graph and two-dimensional graph Fourier transform. In Sect. 3, we have defined the multi-dimensional spectral graph wavelet transform. In last section, we have proved inversion formula for multi-dimensional SGWT without discretizing the scale parameters s 1 , s 2 and further we obtain a finite number of wavelet and scaling function coefficients arranged into distinct wavelet subbands by sampling the scale parameters at discrete set of values. These coefficients form overcomplete frame and we describe how to calculate the corresponding frame bounds. Towards the end, we have given an example and the localization result which holds for both normalized and non-normalized form of Laplacian.

Preliminary
In this section, we define the classical continuous wavelet transform (CWT) in terms of operator T s using convolution theorem, two-dimensional discrete wavelet transform, Cartesian product graph and two-dimensional graph Fourier transform.

Classical continous wavelet transform
Wavelets were first introduced by Morlet as a family of functions created from translations and dilations of a single mother wavelet function ψ and are defined by where s > 0 is called a scaling parameter which estimates the degree of compression or scale, ψ is a function of real variable t and a ∈ R is a translation parameter, which specifies the time location of the wavelet. It is difficult to directly create the analogue of equation (1) then we can see the operator T s act on any function by convolution with ψ s as Taking the Fourier transform on both sides and using scaling property of Fourier transform, we have Applying inverse Fourier transform to (4), we obtain critically, s is mentioned above only in the argument of ψ, which is defined in the Fourier domain rather than the original signal domain. We see that the operator T s mapping f (t) to the collections of wavelet coefficients at scale s acts on f (t) by multiplying its Fourier transform by a bandpass filter function ψ(sω), which is scaled in the Fourier domain by s. The SGWT will be defined later on the basis of equation (5). Individual wavelets can be expressed using the operator T s on a translated delta impulse function δ a (t) = δ(t −a). From equation (3), it follows that which reduces to (T s δ a )(t) = ψ a,s (t) for even and realvalued ψ.

Two-dimensional discrete wavelet transform
The extension of a one-dimensional discrete wavelet transform to two dimensions is simple when two-dimensional scaling and wavelet functions are separable. The scaled and translated basis functions are initially defined as follows: Then the discrete scaling and wavelet transform of function f (x, y) of size M × N is given by , f (x, y)ψ j,m,n (x, y).

Cartesian product graph
Let R and C be the set of real and complex numbers respectively. A Kronecker product of two matrices A ∈ C m×n and B ∈ C r ×s is given by The graphs G 1 and G 2 are called factor graphs of G 1 G 2 . Suppose the factor graph G n for n = 1, 2 has vertex set V n = {0, 1, . . . , N n − 1} with adjacency matrix A n , the diagonal degree matrix D n and the Laplacian matrix L n . Then, ordering the vertices in V 1 × V 2 lexicographically, i.e., like (0, 0), (0, 1), (0, 2), . . . , (N 1 − 1, N 2 − 1), the adjacency matrix, the diagonal degree matrix and the Laplacian matrix of G 1 G 2 are given by A 1 ⊕ A 2 , D 1 ⊕ D 2 and L 1 ⊕ L 2 respectively. Here the operator ⊕ is the Kronecker sum defined by P ⊕ Q = P ⊗ I n + I m ⊗ Q for matrices P ∈ R m×m and Q ∈ R n×n , where I m is the identity matrix of order m. Suppose {λ (n) k } k=0,1,2,...,N n −1 be the non negative eigenvalues and {u (n) k } k=0,1,2,...,N n −1 be the orthonormal eigenfunctions of Laplacian matrix L n for n = 1, 2. The Kronecker sum L 1 ⊕ L 2 has an eigenvalue {λ (1) k 1 + λ (2) k 2 } and the corresponding eigenfunction χ (1) for any k 1 = 0, 1, . . . , N 1 − 1 and k 2 = 0, 1, . . . , N 2 − 1.

Two-dimensional graph Fourier transform
Consider the cartesian product graph G 1 G 2 and suppose the Laplacian matrix L n of its factor graphs have non negative eigenvalues in ascending order, the smallest eigen value is λ 0 = 0 as all the row-wise sums of L n are zero and the number of connected components of the weighted graph G 1 G 2 is equal to the multiplicity of the 0 eigen value [4]. Assuming the cartesian product graph is connected, the eigenvalues of G 1 G 2 are given by 0 = λ (n) 2 ≤ · · · ≤ λ (n) N n −1 and corresponding orthonormal eigen functions are given by {χ Once the vertices of G 1 G 2 have been assigned a particular numbering, any function f : for k 1 = 0, 1, . . . , N 1 − 1 and k 2 = 0, 1, . . . , N 2 − 1 and its inverse is The orthonormality of the eigenvectors χ (1) k 1 χ (2) k 2 makes the validity of the inverse Fourier transform a foregone conclusion. It can also be demonstrated that the graph Fourier coefficients meet Parseval's relation, i.e., for any two signals

Two-dimensional spectral graph wavelets
The spectral graph wavelet transform is based on examining the Fourier domain of classical continuous wavelet transform. We can now define the two-dimensional Spectral Graph Wavelet Transform as we have defined the graph Fourier transform. As previously stated, specifications of the SGWT needs fixing a non-negative real-valued kernel function g, which acts as a band-pass filter and is similar to the Fourier transform of the mother wavelet ψ. From equation (5) we will need that g(0) = 0 and that lim λ (1) k 2 ) = 0. Particular choices for the kernel g will be discussed later.
The wavelet operators producing the SGWT coefficients at each scale are obtained as rescaled kernel functions of the graph Laplacian operator. One may define a function of a self adjoint operator by using the continuous functional calculus [6] based on the spectral representation of the operator. For the finite dimensional graph Laplacian, this is affordable by the eigenvectors and eigenvalues of the Laplacian matrix L 1 ⊕ L 2 . Specifically, we set the wavelet operator by and T g f gives the wavelet coefficients for the signal f at unit scale along both the components of the graph. The operator is defined by its action on the eigenvectors χ (1) This means that the operator T g works on any graph signal f by modulating each of its graph Fourier coefficients as follows: The inverse Fourier transform is then used to illustrate We can compare this relation with equation (5) which describes the mapping from signal to wavelet coefficients for the CWT. Next we define the wavelet operator at scale s 1 and s 2 as T s 1 , . As the domain of kernel function g(λ (1) k 2 ) is continuous so, we can give the proper definition of T s 1 ,s 2 g , for any positive s 1 and s 2 . The individual wavelets are obtained by localizing these operators by applying them to δ n 1 ,n 2 , where δ n 1 ,n 2 ∈ R N 1 N 2 is the signal with value 1 on vertex (n 1 , n 2 ) and zeroes on all other vertices. So, the two dimensional spectral graph wavelets ψ s 1 ,s 2 ,n 1 ,n 2 with scale s centered at (n 1 , n 2 ) is defined as We now observe that δ n 1 ,n 2 (λ (1) Using equation (9) and (11) in equation (10), we get then the wavelet coefficient W f (s 1 , s 2 , n 1 , n 2 ) are given as inner product of f with wavelet ψ s 1 ,s 2 ,n 1 ,n 2 i.e. via Similarly, we have two-dimensional spectral graph wavelet transform W f (s 1 , s 2 , n 1 , n 2 ) given as the value of (T s 1 ,s 2 g f ) (n 1 , n 2 ).
As the wavelet kernel g satisfies g(0+0) = 0, the wavelets ψ s 1 ,s 2 ,n 1 ,n 2 are all orthogonal to the eigenvector χ (1) 0 χ (2) 0 and are close to orthogonal to χ (1) k 1 χ (2) k 2 for eigenvectors where λ (1) k 1 + λ (2) k 2 is close to zero. It is beneficial to introduce a set of spectral graph scaling functions in order to represent the lower frequency content of signals in a stable manner. These are defined similarly to scaling functions for the traditional wavelet transform, which are required to represent low frequency content of signals when the scale parameter is not allowed to become arbitrarily large. We define the spectral graph scaling functions similarly to the wavelets, using a non-negative real valued scaling kernel function h(λ (1) k 1 + λ (2) k 2 ) which may be regarded as a low-pass filter. The scaling kernel function h satisfies h(0 + 0) > 0 and lim λ (1) (2) k 2 ) = 0, from which the scaling function operator T h = h(L 1 ⊕ L 2 ) is defined. The scaling functions centered at vertex (n 1 , n 2 ) are given by φ n 1 ,n 2 = T h δ n 1 ,n 2 and the scaling function coefficients are defined by S f (n 1 , n 2 ) = φ n 1 ,n 2 , f .

Properties of the two-dimensional SGWT
The inverse of the continuous transform, localization qualities in the small-scale limit and frame bounds for the scale discretized transform are among the many properties of the two-dimensional spectral graph wavelet transform described next.

Inversion of continuous SGWT
The signal transform is useful for signal processing or signal analysis only if the transform can be inverted i.e., if the signal can be reconstructed corresponding to a given set of transform coefficients. When the scale parameters s 1 , s 2 are not discretized in that case the continuous SGWT admits an inverse expression that has an analogous form to the inverse formula for the continuous wavelet transform. Each wavelet coefficient W f (s 1 , s 2 , n 1 , n 2 ) can be considered as computing the "amount" of the wavelet ψ s 1 ,s 2 ,n 1 ,n 2 present in the original function f . As all the wavelets are orthogonal to the eigenvector χ (1) 0 χ (2) 0 , the subspace χ (1) 0 χ (2) 0 must be treated separately.
Theorem 1 Let f ∈ R N 1 N 2 be a signal, and let f # be the projection of f onto the orthogonal complement of the span of χ (1) 0 χ (2) 0 . Let g be a kernel function satisfying g(0 + 0) = 0, and admissibility condition Then the continuous reconstruction formula holds: The complete reconstruction of f is then given by Proof We will expand left side of equation (16) and use equation (12) and (14) to write ψ s 1 ,s 2 ,n 1 ,n 2 and W f (s 1 , s 2 , n 1 , n 2 ) in terms of the Laplacian eigenvectors χ (1) We have Applying this and summing over k 1 and k 2 , we have Using the substitution u 1 = s 1 λ (1) k 1 and u 2 = s 2 λ (2) k 2 , provided that λ (1) k 1 , λ (2) k 2 are non-zero, reduces the integral appearing in equation (17) to which is finite and equals C g by the admissibility condition for g. If λ (1) k 2 = 0 which holds only for k 1 = k 2 = 0, then the integral is 0 as g(0 + 0) = 0. As a result, equation (17) is the exact inverse Fourier transform at vertex (m 1 , m 2 ), with the k 1 = k 2 = 0 omitted from the sum. As the omitted k 1 = k 2 = 0 term is exactly 0 . Hence the theorem is proved.

Frame bounds for two-dimensional SGWT
As indicated before, practical computations of the SGWT must involve discretizing the scale parameter s to a finite set of values. We fix our notation by making I the number of scales selected for parameter s . Let {s 1 , s 2 , s 3 , . . . , s I } denote the specific scale values. At each scale, the SGWT generates a set of N 1 N 2 coefficients (commonly referred to as a "subband") W f (s i , s j , n 1 , n 2 ) for 1 ≤ n 1 ≤ N 1 , 1 ≤ n 2 ≤ N 2 . Together with the N 1 N 2 scaling function coefficients, the full transform with I scales may be deemed as a mapping from Consider the frame generated by the complete set of wavelets and scaling functions to get some insight into how stable the coefficients for the entire set of N 1 N 2 I 2 wavelets and N 1 N 2 scaling functions are for expressing signals. In short, a collection of vectors n ∈ H is said to be a frame with frame limits A and B for a Hilbert space H if for all f ∈ H it is true that where the coefficients c n are given by c n = , f . The numerical stability of retrieving the original signal f from the coefficients c n is described by the constants A and B. If A = B, the set n is referred to as a tight frame and the signal may be retrieved simply by Although A and B will not always be equal, the guiding principle that the frame is easier to invert if B A is near to 1 still applies. These frame bounds provide a exact approximation for the speed of convergence of the conjugate-gradients algorithm for inverting the discrete SGWT. For more information of the fundamentals of the theory of frames, see [3].
where h and g are the scaling functions and wavelet kernels. Then the set is frame with frame bounds A, B given by (2) ), G(λ (1) + λ (2) ).
Proof For a fixed function f using equation (14), we can write as where we have used the orthonormality of the χ (1) k 1 χ (2) k 2 . For the scaling function coefficients. Similarly, we have By giving a fixed numbering to the elements of , we have = N 1 N 2 (I 2 +1) l=1 γ l . Depending on l, γ l , f can be either a scaling function coefficient or wavelet coefficient. Equation (21) and (22) show that By the definition of A and B, we have The Parseval's relation f 2 = k 1 k 2 | f (λ (1) k 1 + λ (2) k 2 )| 2 then implies that A and B are frame bounds for the frame .

Limit of small scales
As classical wavelets may be engineered to be confined in both the spatial and frequency domains, they can be extremely useful for signal processing. If g is used as a bandpass filter, the multi-dimensional spectral graph wavelet can be built to be localized in the frequency domain. However, we have yet to show that the spectral graph wavelets can be localized in the spatial (i.e. vertex) domain.
The spatial localization features of classical wavelets formed from a single wavelet by dilation and translation can be inferred directly from the mother wavelet ψ(t). The resulting wavelet ψ s,a (t) will be well localized on [a − ls, a + ls] if ψ(t) is well localized on the interval [−l, l]. In the limit of small scales as s → 0, this means that ψ s,a (t) → 0 for all t = a, as long as the original mother wavelet ψ(t) decays to zero as t → ∞.
This distance measure treats the graph G 1 G 2 as a binary graph i.e., the particular values of the non-zero edge weights are not used, we get the following localization result for integer powers of L 1 ⊕ L 2 .
Theorem 4 Let G = G 1 G 2 be a weighted product graph and L = L 1 ⊕ L 2 the graph Laplacian of G. Fix an integer l > 0, and pick vertices m 1 m 2 and n 1 n 2 . Then (L l ) m 1 m 2 , n 1 n 2 = 0 whenever d G 1 G 2 (m 1 m 2 , n 1 n 2 ) > l.
Proof By the construction of L, we have that L p 1 p 2 ,q 1 q 2 = 0 for any vertices p 1 p 2 = q 1 q 2 that are not connected (i.e. where a p 1 p 2 ,q 1 q 2 = 0). Writing out repeated matrix multiplication, we see Suppose for the sake of contradiction that (L l ) m 1 m 2 ,n 1 n 2 = 0. This implies that atleast one of the terms in the above sum is non-zero, which shows the existence of a sequence of vertices k 1 , k 2 , . . . k l−1 with L m 1 m 2 ,k 1 = 0, L k 1 ,k 2 = 0, ..., L k l−1 ,n 1 n 2 = 0. However, this is precisely a path of length l from vertex m 1 m 2 to vertex n 1 n 2 with possible repeats of vertices. Removing these possible repeated vertices gives a path of length p ≤ l from the vertex m 1 m 2 to n 1 n 2 , which implies d G (m 1 m 2 , n 1 n 2 ) ≤ l, which is a contradiction.
The Laplacian matrix for G 1 and G 2 is given by L n = D n − W n for n = 1 and 2 as follows: then the Laplacian matrix of product graph is given by The orthonormal eigenvectors corresponding to the eigenvalues 0, 0, 0, 4 of L are the minimum and maximum scales are adapted to the spectrum of L as. Given an upper bound λ max = 4 and a value K = 2(say) that is considered design parameter of the transform, we set λ min = λ max K = 4 2 = 2. The discrete scale values are chosen by first specifying the maximum scale s 1 and the minimum scale s 4 . . Now, let us define wavelet function ψ at vertex (v 1 , u 1 ) and at scale s 3 , s 3 along both components of the graph and for the eigenvalue λ = λ 1 + λ 2 = 2 + 2 = 4, where 2 is the eigenvalue of L n ψ s 3 ,s 3 ,v 1 ,u 1 = g(s 3 λ 1 + s 3 λ 2 )δ v 1 ,u 1 Consider the function f ∈ R 4 on V 1 × V 2 giving real values [2,1,1,4], then the wavelet coefficient W f (s 3 , s 3 , v 1 , u 2 ) of f is given by Similarly, we can calculate the other wavelet coefficients at different vertices and scales. Such examples can be constructed on graphs with higher number of vertices involving matrices of higher order and we can perform calculations for higher order matrices by using matlab. Multi-dimensional spectral graph wavelet transform is applied to irregular domains [18] and can have potential applications in areas such as transportation network, road network and neutral network, where network topology plays an important role.

Author contributions
The authors confirm contribution to the paper as follows: The first author has conceived, designed and perform the analysis of the paper and have wrote the paper. The corresponding author has drafted the article or revised it critically for important intellectual content and approved the version to be published.
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