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Bandlimited graph signal reconstruction by diffusion operator
EURASIP Journal on Advances in Signal Processing volume 2016, Article number: 120 (2016)
 The Erratum to this article has been published in EURASIP Journal on Advances in Signal Processing 2017 2017:12
Abstract
Signal processing on graphs extends signal processing concepts and methodologies from the classical signal processing theory to data indexed by general graphs. For a bandlimited graph signal, the unknown data associated with unsampled vertices can be reconstructed from the sampled data by exploiting the spatial relationship of graph signal. In this paper, we propose a generalized analytical framework of unsampled graph signal and introduce a concept of diffusion operator which consists of localmean and globalbias diffusion operator. Then, a diffusion operatorbased iterative algorithm is proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices for accelerating the convergence. We then prove that the proposed reconstruction strategy converges to the original graph signal. The simulation results demonstrate the effectiveness of the proposed reconstruction strategy with various downsampling patterns, fluctuation of graph cutoff frequency, robustness on the classic graph structures, and noisy scenarios.
Introduction
Recent years have witnessed an enormous growth of interest in efficient paradigms and techniques for representation, analysis, and processing of largescale datasets emerging in various fields and applications, such as sensor and transportation networks, social networks and economic networks, and energy networks [1, 2]. The irregular structure is the most important characteristic of those largescale datasets, which limits the applicability of many approaches used for smallscale datasets. This big data problem motivates the emerging field of signal processing on graphs.
Signal processing on graphs extends the classical signal processing techniques and paradigms to the irregular domain [3–5]. Graphs are useful representation tools for representing largescale datasets with geometric structures. The relational structure of largescale dataset is represented with graph, in which data elements correspond to the vertices, the relationship between data elements is represented by the edge, and the strength of relationship is reflected in the edge weight. The graph signal can be regarded as a vector signal which contains the spatial relationship of the vertices. Due to the complicated relationship and large data volume, it is necessary to transform the original graph signal to smallscale modality. The downsampling can be treated as any decrease in dimension via an operator, and conversely, the interpolation can be treated as any increase in dimension via an operator. The main purpose of downsampling method is that the original graph signal may be reconstructed through its entries on only a subset of the vertices by exploiting the character of smoothness. Pesenson in [6] established a PaleyWiener function based sampling theory on combinatorial graphs. He proposed a concept of uniqueness set for downsampling and gave a sufficient condition that the downsampling set needs to satisfy for unique reconstruction. For the reconstruction of sampled graph signal, the main methodology of current algorithms is to extend the PapoulisGerchberg algorithm [7, 8] from the classical regular domain to the graph irregular domain. S. K. Narang in [9] proposed an iterative least square reconstruction (ILSR) algorithm for reconstructing bandlimited graph signal from partially observed samples. ILSR adopts the method of projection onto convex sets (POCS) to iteratively project the sampled data onto the downsampling subspace and the lowpass filtering subspace. In [10, 11], X. Wang proposed a concept of localset and two localset based iterative reconstruction algorithms (IWR and IPR) for recovering the sampled bandlimited graph signal. The localset is formed by partitioning the graph into several disjoint subgraphs. iterative propagating reconstruction (IPR) is also an iterative reconstruction algorithm based on the philosophy of POCS. Compared with ILSR, IPR propagates the reconstructed residual associated with the sampled vertex to the unsampled vertices in the respective localsets. Given the benefit which is obtained from the propagating of reconstructed residual, IPR has faster convergence than ILSR. However, the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph, which should not be limited in the localset. Besides, since the differences among the reconstruction residuals associated with the unsampled vertices cannot be ignored, the even propagation may not be able to achieve the assignment.
Related works on downsampling and reconstruction of graph signals include the methods proposed in [12, 13]. In [12], the authors proposed a sampling theory and reconstruction method on graphs for bandlimited graph signals. The sampling theory proposed in [12] focuses on the graph adjacency matrix and noniterative reconstruction method. In [13], the authors proposed a sampling aggregation method for the graph signals, where the observations are aggregated to one vertex. Different from them, we focus on the iterative reconstruction method for bandlimited graphs in this paper.
The main contribution of this paper is that we present a generalized analytical framework of graph signals associated with the unsampled vertices to further improve the convergence rate of bandlimited graph signal reconstruction. We decompose the graph signals associated with the unsampled vertices into three components, i.e., the extrapolated component, the localmean diffusion component, and the globalbias diffusion component. Based on this scheme, we propose an iterative diffusion operatorbased reconstruction algorithm. The correspondence between the proposed algorithm and the current reconstruction algorithms (ILSR and IPR) is also analyzed, which will be helpful to future works on the reconstruction of bandlimited graph signal. Besides, the theoretical analysis for the proposed iterative reconstruction algorithm is also presented. Then, we demonstrate the performance of the proposed algorithm and the current algorithms with various downsampling patterns, fluctuation of graph cutoff frequency, robustness on the classic graph structures, and noisy scenarios. Finally, we adopt the temperature data of the USA and the electricity consumption data of Shandong province of China as the examples of realworld data to test the performance of reconstruction algorithms. The simulation results show that a better performance of the proposed algorithm can be achieved.
The rest of this paper is organized as follows. In Section 2, the previous works of downsampling and reconstruction for bandlimited graph signal are briefly reviewed. In Section 3, we propose the concept of diffusion operator and its corresponding iterative reconstruction algorithm. In Section 4, we analyze and prove the convergence of the proposed algorithm. In Section 5, we demonstrate the proposed algorithm by using the synthetic and realworld data on various graphs. In Section 6, conclusions are drawn.
The previous work for downsampling and reconstruction
A simple, connected, and undirected graph G=(V,E) is a collection of vertices V and edges E, with V={1,2,…,N} representing the set of vertices of the graph and E={w _{ i,j },∀i,j∈V} representing the set of edges connecting vertex i and j with weight w _{ ij }, where w _{ ii }=0. The adjacency matrix of the graph is defined as A(i,j)=w _{ ij }. The degree d _{ i } of a vertex i is defined as the sum of the weights of the edges connected to the vertex i. The degree matrix of the graph is a diagonal matrix defined as D=diag{d _{1},d _{2},…,d _{ N }}. The Laplacian matrix of the graph is defined as L=D−A. The normalized Laplacian matrix \(\mathcal {L}\) is a symmetric positive semidefinite matrix and can be decomposed as
where U ^{T} denotes the transpose of Laplacian eigenvector U, Λ=diag{λ _{1},λ _{2},…,λ _{ N }} is a diagonal matrix of real Laplacian eigenvalues ordering as λ _{1}≤λ _{2}≤⋯≤λ _{ N }, and its corresponding orthogonal set of Laplacian eigenvectors denoted as U={u _{1},u _{2},…,u _{ N }}, with u _{ i } is the ith column vector of Laplacian eigenvector matrix.
For example, the Minnesota path graph is shown in Fig. 1, which contain 2642 vertices and 6606 edges. A graph signal f is represented as a vector mapping f:V→R ^{N}, such that f(i) is the value of the signal on vertex i. \({\hat f_{i}}=<{f},{u}_{i}>\) is the graph Fourier transform (GFT) of f. Similar with the Fourier transform in classical signal processing, graph Fourier transform performs the expansion of a graph signal into a Laplacian eigenvector basis of signals [14]. The eigenvectors and eigenvalues of the Laplacian matrix provide a spectral interpretation of the graph signal. For more concise comparison, the eigenvalues of the Laplacian matrix can be regarded as the graph frequency and form the spectrum of graph, and the Laplacian eigenvectors that correspond to a frequency λ _{ m } are called the graph frequency component corresponding to the mth frequency.
Downsampling of a bandlimited graph signal
Downsampling on graphs can efficiently extract valuable information by exploiting the spatial relationship of graph structure. If the GFT of a graph signal has support only in frequency [0,w), the graph signal is regarded as bandlimited in the range of [0,w), with the w is called the graph cutoff frequency [6, 15]. The space of bandlimited graph signal is often called PaleyWiener space (PWS) and is denoted as
where λ denotes the Laplacian eigenvalue and \(\hat {f}(\lambda)\) denotes the graph frequency component corresponding to λ. We denote S as a downsampling set of the vertices of the graph, and S ^{c}=V∖S denotes its complementary set. The purpose of downsampling operation is to select valuable vertices to form the downsampling set S. The concept of uniqueness set is defined in [6], which provides a sufficient condition for exact reconstruction from the sampled graph signal. A subset of vertices S⊂V is a uniqueness set in PWS, if for any two signals g,h, the fact that they coincide on S implies they coincide on V : g(S)=h(S)⇒ g=h.
Currently, there are two solutions for finding an appropriate downsampling set. In [16], the author formulates a greedy heuristic algorithm to obtain an estimation of the optimal downsampling set. In [11], the author proposes a localset based downsampling forming algorithm. The graph is divided into disjoint subgraphs and each subgraph selects one vertex as the downsampling vertex, which is called onehop sampling method. The onehop sampling method is a rather economical and efficient choice of downsampling set forming method when there is no restriction on the number of vertices in the downsampling set or no locationlimited of downsampling vertices. In Fig. 1, we adopt the onehop sampling method to form the downsampling set, where the sampled vertices are denoted as the redstarvertices and the unsampled vertices are denoted as the blueroundnessvertices.
Reconstruction of a bandlimited graph signal
The main methodology of the reconstruction algorithm is to establish the relationship between the sampled vertices and the unsampled vertices according to the spatial structure of graph. In classical signal processing, the authors in [7, 8] propose an iterative extrapolation strategy (PapoulisGerchberg Algorithm) for reconstructing the original signal. The basic idea of PapoulisGerchberg algorithm lies in alternatingly imposing the initially known values in the time domain and the finite support constraint in the frequency domain, until convergence is reached. The iterative process of PapoulisGerchberg algorithm can be written as follows
where f ^{c} denotes the classical continuous signal, \({{P_{T}^{c}}}\) denotes the time domain downsampling operator, I denotes the identity operator, \({{P_{w}^{c}}}\) denotes the frequency domain cutoff operator, and \({{\mathcal {F}}_{c}}\) and \({\mathcal {F}}_{c}^{ 1}\) represent the classical Fourier transform and classical Fourier inverse transform, respectively. In each iteration, the PapoulisGerchberg algorithm replaces the downsampling part of the estimated reconstruction signal \({{f_{k}^{c}}}\) by the actual known segment and then combines the extrapolation segment to form the next iterative signal. In other words, at kth iteration, the solution \({f_{k}^{c}}\) is obtained from \(f_{k1}^{c}\) and satisfies the two constraints of time domain downsampling and frequency domain bandlimited.
According to the principle of PapoulisGerchberg Algorithm, an iterative least square reconstruction (ILSR) algorithm is proposed in [9, 17] for the signal processing on graphs. At each iteration, ILSR resets the signal samples on the downsampling set S to the actual given samples and then projects the graph signal onto the lowpass filtering subspace. Denote P _{ T } as the vertex domain downsampling operator and f _{ d } as the sampled graph signal, then the downsampling process can be represented as follows
Besides, the vertex domain downsampling operator P _{ T } is a diagonal matrix
where 1_{ S } is the set indicator vector, whose ith entry is equal to one, if i∈S, or zero otherwise. The iterative process of ILSR can be written as follows
where \({f_{k}^{L}}\) denotes the kth iterative reconstructed residual of ILSR, P _{ w } denotes the graph frequency cutoff operator, w denotes the graph frequency domain cutoff frequency, and \({\mathcal {F}}\) and \({{\mathcal {F}}^{ 1}}\) denote the graph Fourier transform and inverse transform, respectively. At the first iteration, the initial reconstructed graph signal f _{0} is obtained by projecting the sampled graph signal f _{ d } onto the lowpass filtering subspace. In this paper, we define the difference between the graph signal f and the sampled reconstructed graph signal f _{ k } as the reconstructed residual \({f_{k}^{s}}\), i.e., \({f_{k}^{s}} = {f}  {f_{k}}\). Moreover, the reconstructed graph signal f _{ k } is denoted by \({f_{k}^{L}}\) for ILSR and \({f_{k}^{P}}\) for IPR. In [10, 11], the author proposes a localset based IPR algorithm. In each iteration, IPR adopts a local propagative operator to locally and evenly propagate the reconstructed residual. The iterative process of IPR is shown as follows
where \({f_{k}^{P}}\) denotes the kth iterative reconstructed graph signal of IPR, \({{\mathcal {Q}}_{w}}\left (\cdot \right)\) is a graph frequency domain cutoff operator, δ _{ N(v)}=(δ _{ N(v)}(1),δ _{ N(v)}(2),…,δ _{ N(v)}(N))^{T} with δ _{ N(v)}(m)=1 only when m∈N(v). According to Eq. (7), IPR first propagates the reconstructed residual locally and evenly to the localset that each sampled vertex belongs to and then projects the new signal onto the lowpass filtering subspace. Since IPR propagates the reconstructed residual in the localset at each iteration, IPR converges faster than ILSR. However, IPR only focus on the propagating of reconstructed residual within the localsets. In the next section, we propose a diffusion operator based iterative reconstruction strategy, which extends the reconstructed residual to more generalized diffusion.
Different from the POCS method, the sampling theory proposed in [12] recovers the sampled graph signal by employing an interpolation operator Φ=U _{ w }(P _{ T } U _{ w })^{−1}, where w denotes the bandwidth of bandlimited graph signal, U denotes the eigenvector matrix of graph adjacency matrix A, U _{ w } denotes the first w columns of U, and P _{ T } denotes the sampling operator. In this paper, we follow the methodology of the POCS method.
The diffusion operatorbased reconstruction strategy
In this section, we establish a generalized analysis framework of the graph signals associated with the unsampled vertices. The concept of localmean and globalbias diffusion operator is firstly defined. Then, we propose an iterative diffusion operatorbased reconstruction algorithm. Discussions on the current reconstruction algorithms are also included in this section.
The generalized analytical framework of unsampled graph signal
The essence of the reconstruction algorithm is to establish the relationship between the sampled vertices and the unsampled vertices according to the spatial correlation. However, the current research does not pay much attention to the component analysis of graph signals associated with the unsampled vertices. In this paper, we analyze the graph signals associated with the unsampled vertices from the perspective of mean and bias and then establish a generalized analytical framework of unsampled graph signal. The bandlimited graph signal is smooth, where the graph signals associated with the vertices vary slowly in comparison to the neighboring vertices [3]. In Section 2, the reconstruction residual is defined as the difference between the actual graph signal and the reconstructed graph signal. Due to the fact that the actual graph signal and the reconstructed graph signal are both bandlimited, the reconstructed residual is also bandlimited. Thus, it can be seen that the reconstructed residuals associated with vertices vary slowly in comparison to the neighboring vertices. This important property may allow the diffusion of reconstructed residuals associated with the sampled vertices to the unsampled vertices, since we only know the reconstructed residuals associated with sampled vertices at each iteration of POCS method. Thus, we decompose the actual graph signals associated with the unsampled vertices as
where f _{acu} denote the actual graph signals associated with the unsampled vertices, f _{rec} denote the extrapolated graph signal obtained by projecting onto the lowpass filtering subspace, and f _{res} denote the diffused reconstructed residuals obtained from the sampled vertices. Since ILSR directly projects the sampled signal onto the lowpass filtering subspace, it can be seen that f _{res} are set to zeros and f _{acu} is only obtained from f _{rec}. In IPR, the reconstructed residuals associated with sampled vertices are copied and assigned to the vertices in the corresponding localset, and then the projection procedure is conducted. It can be seen that the vertices within the localset have the same value of reconstructed residuals. It may not fit into the property of actual reconstructed residuals associated with the unsampled vertices. Besides, practically the graph signal associated with every vertex is either directly or indirectly influenced by all the vertices in a graph, which should not be limited in the localset. Thus, for the purpose of accurate analysis, we decompose the diffused reconstructed residuals f _{res} into two components, and Eq. (8) can be rewritten as
where \(f_{\text {res}}^{\text {mean}}\) denote the localmean component of the diffused reconstructed residuals, and \(f_{\text {res}}^{\text {bias}}\) denote the globalbias component of the diffused constructed residuals. The motivation of the diffused reconstructed residual decomposition is that we expect to establish threelayer analytical framework for the unsampled graph signals. The first layer of unsampled graph signal is obtained by projecting the sampled graph signal onto the lowpass filtering subspace, which is f _{rec}. The second layer of unsampled graph signal is obtained from the reconstructed residual of adjacent sampled vertices, which is \(f_{\text {res}}^{\text {mean}}\). For the localmean component, the reconstructed residuals associated with the sampled vertices are regarded as the mean of local region around the sampled vertices and are diffused from sampled vertices to their adjacent unsampled vertices. The third layer of unsampled graph signal is obtained by the reconstructed residual of all the sampled vertices, which is \(f_{\text {res}}^{\text {bias}}\). The main purpose of the globalbias component is that the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph. The globalbias component is expected to further exploit the reconstructed residuals associated with the sampled vertices, which is beneficial to accelerate the convergence of reconstruction. Besides, the globalbias component is also used to form the differences among the graph signals associated with the unsampled vertices, since the localmean component assigns the same value of reconstruction residuals from the sampled vertices to their adjacent vertices. The collaboration of the localmean component and globalbias component is expected to approximate the actual reconstructed residuals associated with the unsampled vertices by further exploiting the smoothness of the bandlimited graph signals.
Besides, Eq. (9) can be regarded as the composition model of unsampled graph signal of ILSR, when \(f_{\text {res}}^{\text {mean}}\) and \(f_{\text {res}}^{\text {bias}}\) are set to zeros. Similarly, Eq. (9) can be regarded as the composition model of unsampled graph signal of IPR, when \(f_{\text {res}}^{\text {bias}}\) is set to zero. The f _{rec} of IPR and ILSR are both obtained by directly projecting the sampled graph signal onto the lowpass filtering subspace. Based on the decomposition, in the following parts of this section, we will propose two diffusion operators (the localmean diffusion operator and the globalbias diffusion operator) to achieve the localmean diffusion and globalbias diffusion of reconstruction residuals associated with the sampled vertices.
The localmean diffusion operator
In this part, the concept of localmean diffusion operator is introduced. Then, discussions on the localmean diffusion operator and local propagation are also included in this part.
According to the decomposition model of (9), the second layer of unsampled graph signal is the localmean diffusion component. In this part, we proposed a localmean diffusion operator, which is used to assign the reconstructed residual from the sampled vertices to their adjacent unsampled vertices, to achieve the localmean diffusion operation. Denote by ρ(v,S) the fewest distance of an unsampled vertex v from the downsampling set S, i.e., the fewest number of edges in a shortest path from v to a vertex in the downsampling set S. Besides, we assume that the fewest distance of a sampled vertex from the downsampling set is zero.
Definition 1
For a given reconstructed residual \({f^{s}} = {\left [ {{f_{1}^{s}},{f_{2}^{s}},\ldots,{f_{N}^{s}}} \right ]^{T}}\), we define the localmean diffusion operator P _{ m } as
where \(f_{\text {res}}^{\text {mean}}\) is an Nby1 vector and denotes the localmean component of reconstructed residual, B _{ i }=C A ^{i}, C is a diagonal matrix with the diagonal element as the reciprocal of the degree of the corresponding vertex, A denotes the adjacency matrix of graph, i denotes the fewest number of edges in a shortest path from an unsampled vertex to a vertex in the downsampling set S, r denotes the maximal number of hop between sampled vertex and unsampled vertex on the graph, and δ _{ i } denotes δfunction of ρ with entries
In other words, δ _{ i } is a diagonal matrix with the diagonal element as the ρ(u,S) of corresponding vertex u.
Indeed, an underlying idea behind the localmean diffusion operator is to incorporate the structure of the adjacency matrix into the localmean diffusion operation of reconstructed residual. In definition 1, we employ the powers of the adjacency matrix A to sequentially diffuse the reconstructed residuals associated with the sampled vertices to the adjacency unsampled vertices. Besides, C is used to obtain the weighted value of diffused reconstructed residual.
In [11], the authors propose an operation G firstly to propagate the reconstructed residual in respective localsets and then projects the combinatorial signal onto the lowpass filtering subspace. The first step of G provides a solution to propagate the reconstructed residuals associated with the sampled vertices with the help of localsets. In this paper, we redefine the first propagating step of G as G _{ p }. It can be seen that G _{ p } can be regarded as a special case of localmean diffusion operation based on localset. Thus, it is easy to see that G _{ p } and P _{ m } achieve the same performance with the precondition of localset, but P _{ m } can diffuse the reconstructed residuals in the more comprehensive scenarios.
The globalbias diffusion operator
The main motivation of the globalbias component is that the graph signals associated with the unsampled vertices are influenced by all the sampled vertices in graph, which should not be limited in the local region. Besides, the globalbias component is used to provide the difference of reconstructed residual of local region. In this part, we propose a globalbias diffusion operator, which is used to establish the relationship between one sampled vertex and all unsampled vertices, to achieve the globalbias diffusion operation.
We firstly analyze the diffusion character of single element of reconstructed residuals and then extend to all the elements of the downsampling set. Assuming that all the values of reconstructed residual are zeros except at the first vertex, that is, only the first vertex is selected as the sampled vertex. Then, the reconstructed residual can be represented as f ^{s}=[f ^{s}(1),0,…,0]^{T}, where f ^{s}(1) denotes the reconstructed residual resided on the first vertex and \({\hat f^{s}} = {\left [ {{u_{11}}{f^{s}}\left (1 \right),{u_{12}}{f^{s}}\left (1 \right),\ldots,{u_{1N}}{f^{s}}\left (1 \right)} \right ]^{T}}\) denotes graph frequency component. Since the graph signal is bandlimited, we assume that there are m graph frequency components within the bandwidth w, that is, \({\hat f^{s}} = {\left [ {{u_{11}}{f^{s}}\left (1 \right),\ldots,{u_{1m}}{f^{s}}\left (1 \right),0,..,0} \right ]^{T}}\). Then, to transform \({\hat f^{s}}\) from the graph frequency domain to the vertex domain and denote \({\tilde f^{s}}\) as the new presentation, which can be written as follows
It can be seen that Eq. (10) establishes the relationship of reconstructed residual between the first vertex and all the unsampled vertices. Then, we can extend the same assumption to the every element of downsampling set S. Thus, from the perspective of the single unsampled vertex, its estimated reconstructed residual is the sum of diffused reconstructed residuals of all the sampled vertices. Denote \({\bar f^{s}}\left (i \right)\) as the estimated reconstructed residual of the ith unsampled vertex, which can be written as follows
The globalbias diffusion process can be understood from two aspects. From the perspective of sampled vertex, the reconstructed residual of the single sampled vertex is diffused to all the unsampled vertices. From the perspective of unsampled vertex, the estimated reconstructed residual of the single unsampled vertex is the sum of diffused reconstructed residuals of all the sampled vertices. Then, based on the discussion above, we define the globalbias diffusion operator to diffuse the reconstructed residual to all the unsampled vertices.
Definition 2
For a given reconstructed residual \({f^{s}} = {\left [ {{f_{1}^{s}},{f_{2}^{s}},\ldots,{f_{N}^{s}}} \right ]^{T}}\), we define the globalbias diffusion operator P _{ b } as
where \(f_{\text {res}}^{\text {bias}}\) is an Nby1 vector and denotes the globalbias diffusion component of estimated reconstructed residual, U ^{′} denotes the modified Laplacian eigenvector matrix which the rows corresponding to the sampled vertices are set to the zero sequence and the columns corresponding to out of the bandwidth w are set to the zero sequence, and U ^{″} denotes the modified Laplacian eigenvector matrix in which the rows corresponding to the unsampled vertices are set to the zero sequence and the columns corresponding to out of the bandwidth w are set to the zero sequence.
In each iteration, the globalbias diffusion operator P _{ b } diffuses the iterative reconstructed residual from the sampled vertices to all the unsampled vertices.
The diffusion operatorbased reconstruction algorithm
In this part, we propose a diffusion operator based iterative reconstruction (IGDR) algorithm.
Definition 3
For a given reconstructed residual \({f^{s}} = {\left [ {{f_{1}^{s}},{f_{2}^{s}},\ldots,{f_{N}^{s}}} \right ]^{T}}\), we define the diffusion operator P _{ d } as
where P _{ m } denotes the localmean diffusion operator, and P _{ b } denotes the globalbias diffusion operator.
In each iteration, the diffusion operator diffuses the reconstructed residuals from the sampled vertices to the unsampled vertices, where the localmean diffusion operator diffuses the reconstructed residual associated with the sampled vertices to their adjacent unsampled vertices and the globalbias diffusion operator diffuses the reconstructed residual associated with the sampled vertices to the all unsampled vertices. According to the discussion above, we propose a diffusion operator based reconstruction algorithm (IGDR), and its process can be written as follows
where \({f_{k}^{G}}\) denotes the kth iterative reconstructed signal of IGDR, \({\left ({{f_{d}}  {P_{T}}{f_{k}^{G}}} \right)}\) denote the reconstructed residual on the downsampling set, and \({\left ({I  {P_{T}}} \right){P_{d}}\left ({{f_{d}}  {P_{T}}{f_{k}^{G}}} \right)}\) denote the reconstructed residual is diffused from the downsampling set to the nondownsampling set by the diffused operator.
Discussion
ILSR, IPR, and IGDR followed the methodology of POCS, and the sampled data are iteratively projected onto the downsampling subspace and lowpass filtering subspace. The difference of ILSR, IPR, and IGDR lies in the way of their dealing with the reconstructed residual. For ILSR, the sampled signal is directly projected onto the lowpass filtering subspace. For IPR, the reconstructed residuals associated with the sampled vertices are firstly copied and propagated to the unsampled vertices in the corresponding localsets, and then the signal is projected onto the lowpass filtering subspace. For IGDR, the diffusion operation of reconstructed residuals consists of two components, which are localmean diffusion operation and globalbias diffusion operation. The localmean diffusion operation is that the reconstructed residuals associated with the sampled vertices are diffused to the their adjacent vertices. The globalbias diffusion operation establishes the diffusion relationship between one sampled vertex and all the unsampled vertices. The localmean diffusion operation is used to form the basic component of estimated reconstructed residuals associated with the unsampled vertices, due to one sampled vertex has stronger relationship with the adjacent vertices than others. The globalbias diffusion operation is used to form the differences of estimated reconstructed residuals associated with the unsampled vertices in the local region. The collaboration of the localmean diffusion operation and globalbias diffusion operation is used to accelerate the convergence of reconstruction. The illustration of the iterations of the three algorithms is shown in Fig. 2.
Besides, it can be seen that ILSR can be regarded as a special case of IGDR, when the localmean diffusion component and globalbias diffusion component are set to zeros. Similarly, IPR can be regarded as a special case of IGDR, when the localmean diffusion component is designed by the localset and the globalbias diffusion component is set to zero.
The analysis of iterative error and convergence
In this section, we present the theoretical analysis of convergence for the proposed iterative reconstruction algorithm.
Proposition 1
The bandlimited graph signal f can be reconstructed from its downsampling set S by IGDR, for a given graph cutoff frequency w, the reconstructed error bound of IGDR is
if satisfying
where η ^{k} denotes the kth iterative error, \({\mathcal {F}}\) and \({{\mathcal {F}}^{ 1}}\) denote the graph Fourier transform and the inverse graph Fourier transform, respectively, w denote the graph cutoff frequency, P _{ T } denotes the downsampling operator which chooses the sampled vertices and pads the unsampled vertices with zeros, P _{ w } denote the graph frequency cutoff operator, I denotes the identity operator, and P _{ d } is diffusion operator. The notation R _{max}, J _{max}, α _{1}, and α _{2} are the parameters of graph signal and the downsampling set, and the detailed explanation can be seen in proof.
Proof
: The proof is postponed to Appendix. □
The simulation results
In this section, we adopt the Minnesota path graph [18] as the graph structure to demonstrate the performance of proposed algorithm and current algorithms, which is evaluated from the convergence rate, sensitivity with graph cutoff frequency, the influence on the different downsampling set, and the robustness with additive noise. We use the synthetic data as the bandlimited graph signal on the Minnesota path graph, and the process is shown as follows:

1.
Generate a random Gaussian signal on graph.

2.
Transform the graph signal into graph spectral domain by graph Fourier transformation and remove the frequency components higher than the given graph cutoff frequency.

3.
Transform the graph signal from the graph spectral domain to the vertex domain.
The Minnesota path graph is shown in Fig. 1. The proposed algorithm and the current algorithms are compared on the three classic graph structures for demonstrating the robustness. We also use the realworld data, which are the temperature (2014.1.1) of 94 cities of the USA and the electricity consumption data (2015) of Shandong province of China, to test the performance of the proposed reconstruction algorithm. Moreover, since IPR needs the help of localset, we use onehop sampling method to form the downsampling set for fair comparison. Besides, we define the concept of relative error. Let x denote a vector and \(\tilde {x}\) denote the estimated value of x, then the relative error is defined by \(\varepsilon = \left \ {\tilde x  x} \right \ \left / \left \ x \right \\right.\).
The convergence performance
In this part, we compare the performance of iterative reconstruction for ILSR, IPR, and IGDR. We adopt the maximum degree division based onehop sampling method to form the downsampling set, and the number of vertices in the downsampling set is 873. The graph cutoff frequency is set to 0.45 (the range of normalized graph frequency is from 0 to 2). Moreover, the convergence of reconstruction algorithms on the random downsampling set is also considered. For the random downsampling set, 873 vertices are selected completely at random among all the vertices. Since IPR needs the help of localset, in this experiment, we only consider the performance of ILSR and IGDR. The convergence curves of ILSR, IPR, and IGDR are illustrated in Fig. 3. It is obvious that the convergence rate of the proposed IGDR is improved compared with the current iterative reconstruction algorithms. Besides, it can be seen that the convergence is faster by using the maximum degree division based onehop sampling set than the random downsampling set.
Graph cutoff frequency
Since the main methodology of current reconstruction is iteratively projecting on the downsampling subspace and lowpass filtering subspace, the graph cutoff frequency is a crucial quantity. In this simulation, the effect on the variation of the graph cutoff frequency is investigated. The downsampling set is formed by the onehop sampling method, and the number of vertices in the downsampling set is 873. The graph cutoff frequency varies from 0.4 to 0.5, which the step size is 0.005. Figure 4 shows the final relative error of 10 iterative reconstructions for ILSR, IPR, and IGDR. It can be seen that IGDR has much higher recovery accuracy than the current iterative reconstruction algorithms via the variation of graph cutoff frequency.
Robustness with additive noise
This simulation focuses on the robustness against the additive noise of IGDR and the current reconstruction algorithms. The independent and identically distributed Gaussian sequence is involved in the observation of sampled graph signal. We adopt the onehop sampling method to form the downsampling set, and the number of vertices in the downsampling set is 873. The signal to noise ratio (SNR) is considered with 20 and 40 dB. The graph cutoff frequency is set to 0.45. Moreover, the convergence of reconstruction algorithms on the random downsampling set is also considered. For the random downsampling set, 873 vertices are selected completely at random among all the vertices. Since IPR needs the help of localset, in this experiment, we only consider the performance of ILSR and IGDR. The performances are illustrated in Fig. 5. It can be seen that all the algorithms have almost the same reconstructed performance against the additive noise, but IGDR holds the fastest convergence. Besides, it can be seen that the convergence is faster by using the maximum degree division based onehop sampling set than the random downsampling set.
Robustness with different downsampling sets
The choice of downsampling set may affect the performance of convergence and robustness. We use three different downsampling sets to reconstruct the same bandlimited graph signal. The graph cutoff frequency is set to 0.3. The first downsampling set is the maximum degree division based onehop sampling set, which is formed by the algorithm in [11] and with 873 sampled vertices. The second downsampling set is followed by the minimum degree based greedy algorithm in [19], with 923 sampled vertices. The greedy algorithm is to iteratively remove connected vertices with the smallest degrees from the original graph into the new subset, until the cardinality of the new subset reaches the given maximal cardinality or there is no connected vertex. This greedy algorithm can be regarded as a solution of uniform sampling. For the third downsampling set, 923 vertices are selected completely at random among all the vertices. Since IPR need the help of localset and the second and third downsampling set do not contain localset, in this experiment, we only consider the performance of ILSR and IGDR. The convergence curves of the three downsampling sets using ILSR and IGDR are shown in Fig. 6. It can be seen that the convergence is faster by using the maximum and minimum degree divisionbased downsampling set than the randomly selected downsampling set. The convergence of IGDR is faster than ILSR by using all the three downsampling sets. We can find that the different downsampling set may influence on the convergence rate. Besides, all the three reconstruction algorithms follow the methodology of projection onto convex sets. The unsampled data are extrapolated by alternatively projecting the sampled data onto the downsampling subspace and lowpass filtering subspace. The cornerstone of this method is the close relationship between the sampled data and unsampled data. Based on the theory of signal processing on graphs, the edge denotes the relationship of vertices, which also denotes the relationship of graph signal. That is, the vertex has stronger relationship with its adjacent vertices than others. Thus, from the perspective of experience, if the sampled vertices and the unsampled vertices are uniformly distributed on the graph, the reconstruction algorithm may present more efficient performance than others. The results of this simulation may support this analysis.
The performance on three classic graph structures
We demonstrate the performance of IGDR on three classical graph structures: the ErdosRenyi graph, the smallworld graph, and the scalefree graph. The ErdosRenyi graph is a random graph with a certain connection probability for each edge. We generate a 40vertices ErdosRenyi graph, in which the probability of edge connecting is 0.1. The smallworld structure is a typical graph in which most vertices are not neighbors of one another, but most vertices can be reached from every other by a small number of hops or steps. We adopt the WattsStrogatz model [20] to generate a smallworld graph with 100 vertices. The scalefree graph is a graph whose degree distribution follows a power law. We generate a 100vertices scarefree network according to the BarabasiAlbert model [21]. The graph cutoff frequency is set to 0.5 for all three graph structures. The downsampling set is formed by the onehop sampling method for all the three graphs. For eliminating the random effect of all three graph structures, each simulation result is averaged over 100 random network topologies, and one of those graph structures is shown in Fig. 7 a–c. Figure 8 shows the relative error for ILSR, IPR, and IGDR on the three graph structures. It is obvious that IGDR has steady performance for different graph structures and has also lower relative error than other reconstruction algorithms.
Sensitivity with imprecise knowledge of graph cutoff frequency
For a bandlimited graph signal, the graph cutoff frequency is a crucial quantity during the reconstruction and is known as a priori knowledge. In the actual applications, the graph cutoff frequency may be an imprecise value rather than ground truth. In this simulation, the effect on the imprecise knowledge of graph cutoff frequency is investigated. The downsampling set is formed by the onehop sampling method, and the number of vertices in the downsampling set is 873. The actual graph cutoff frequency is set to 0.35. The imprecise values of graph cutoff frequency are set to 0.3 and 0.4, which are smaller and larger than the actual value. Figure 9 shows the reconstruction performances of ILSR, IPR, and IGDR. It can be seen that the relative error of smallervalue is larger than the larger value and actual value. The simulation results show that the current reconstruction algorithms and the proposed algorithm have the same performances on the imprecise knowledge of graph cutoff frequency, and the smaller value of the imprecise knowledge of graph cutoff frequency has more sensitive than the larger value.
Realworld data
In this simulation, realworld data is used to test the performance of the proposed reconstructed algorithm. As an example of realworld data, we adopt the daily temperature data (2014.1.1) which measured by the 94 weather stations across the USA. The data is collected by the National Climatic Data Center [22]. We represent these stations with an undirected twonearest neighbor graph, in which every weather station corresponds to a vertex and is connected to two closest weather stations by edges. The graph is shown in Fig. 7 d. We use onehop sampling method to form the downsampling set, and the number of vertices in the downsampling set is 31. Since temperature data varies slowly across the graph, most of the graph signal’s energy is concentrated in the low frequencies. The graph cutoff frequency can be estimated by the following two respects: projecting the historical or snapshot data onto the graph frequency domain and the convergence condition in Proposition 1. Thus, the graph cutoff frequency is set to 0.65. In Fig. 10, we can find that IGDR has much faster convergence than the current algorithms.
As another example of realworld data, the electricity consumption (2015) of Shandong province of China is selected. The data is provided by the Shandong statistical yearbook (2015) [23]. The graph structure consists of 17 vertices, which are 17 cities of Shandong providence. The edge of graph structure denotes the electric power transmission line between the two cities. The graph structure is shown in Fig. 11 a. We use onehop sampling method to form the downsampling set, and the number of vertices in the downsampling set is 7. The graph cutoff frequency is set to 0.55. In Fig. 11 b, it can be seen that IGDR has much faster convergence than current reconstruction algorithms.
Conclusions
In this paper, the problem of the bandlimited graph signal reconstruction was studied. We established a generalized analytical framework of graph signals associated with the unsampled vertices. We defined a concept of diffusion operator, which consists of localmean diffusion operator and globalbias diffusion operator. Employing the diffusion operator, we proposed an iterative algorithm to reconstruct the unknown data associated with the unsampled vertices from the observed samples. In each iteration, the reconstructed residuals associated with the sampled vertices are diffused to all the unsampled vertices. We also presented the analysis of iterative reconstructed error and convergence of the proposed algorithm. The simulation results are showed that the techniques presented in this paper perform beyond the current reconstruction algorithms. Moreover, the main purpose of this paper is to present a generalized model to accelerate the convergence rate of reconstruction algorithm. The design of the localmean diffusion operator and globalbias diffusion operator according to the character of network topology and graph signals can be investigated in the future work.
Appendix
Proof
: The key notation used in the proof is shown in Table 1.
The graph frequency cutoff operator is defined as the diagonal matrix P _{ w }=diag{1_{ w }}, where 1_{ w } is the set indicator vector, whose ith entry is equal to one, if i∈[0,w), or zero otherwise (the range of normalized graph frequency is from 0 to 2). Thus, it can be seen that P _{ w } remove the highfrequency components ([w,2]). Since only [0,w) of the energy of graph signal can be preserved, \({{\mathcal {F}}^{ 1}}{P_{w}}{\mathcal {F}}\) is a contraction mapping. The downsampling operator is defined as a diagonal matrix P _{ T }=diag{1_{ S }}, where 1_{ S } is the set indicator vector, whose ith entry is equal to one, if i∈S, or zero otherwise. Due to only the nondownsampling vertices are preserved, (I−P _{ T }) is a contraction mapping. Besides, let f denote a bandlimited graph signal and its graph cutoff frequency is w. Notice that \({{\mathcal {F}}^{ 1}}{P_{w}}{\mathcal {F}}f = f\) and f _{ d }=P _{ T } f, we have,
The diffusion operator P _{ d } consists of the localmean diffusion operator P _{ m } and the globalbias diffusion operator P _{ b }. We have
We first analyze the character of ∥I−P _{ m } P _{ T }∥. According to the analysis in Section 3.2, the localmean diffusion operator P _{ m } assign the signal from sampled vertices to their adjacent unsampled vertices. Let f denote a bandlimited graph signal, S denote the downsampling set, and K(u) denote the localdiffused unsampled vertices set of the sampled vertex u by the localmean diffusion operator. Then, we have
where d(v) denote the degree of the vertex v. It can be seen that the part a of Eq. (18) is the downsampling part of ∥f∥^{2}, then
where α _{1} denote the proportion of quadratic sum of graph signals associated with sampled vertices in the quadratic sum of graph signals associated with all the vertices, i.e.,
It can be seen that the range of α _{1} is from 0 to 1. The value of α _{1} approximates to \(\frac {M}{N}\), when the graph signal f is very smooth. For the part b of Eq. (18), we have
We first discuss the character of the part c of Eq. (20). There is always a shortest path within K(u) from any localdiffused unsampled vertex v∈K(u) to sampled vertex u, which is denoted as (v,v _{1},v _{2},…,v _{ r−1},u). Then, we have
where H _{ K(u)} denote the maximal distance from sampled vertex u to any localdiffused unsampled vertex within K(u). For each localdiffused unsampled vertex v∈K(u), the path to the sampled vertex u is not longer than H _{ K(u)}. Let X _{ K(u)} denote the cardinality of K(u). The maximal distance within K(u) is counted for no more than X _{ K(u)} times. Then, we have
Let R _{max} denote the maximal value of the product of X _{ K(u)} and H _{ K(u)} in graph, i.e.,
Thus, we have
where V denotes the set of vertices of the graph and E denotes the set of edges connecting vertices. Since f is bandlimited, the components of \({\hat f}\) associated with the frequencies higher than the graph cutoff frequency w are zero. Then, we analyze the character of the part d of Eq. (20). J(u) is denoted as
and let J _{max} denote the maximal value of J(u), i.e.,
Thus, considering Eq. (19), we have
Combining (18), (19), (22), and (23), we have
Then, we analyze the character of ∥P _{ b } P _{ T }∥. According to the explanation in Section 3.3, the globalbias diffusion operation first projects the reconstructed residual associated with the downsampling set onto the lowpass filtering subspace, then preserves the nondownsampling set of new signal. Thus, we have \({P_{b}}{P_{T}} = \left ({I  {P_{T}}} \right){{\mathcal {F}}^{ 1}}{P_{w}}{\mathcal {F}}{P_{T}}\). Due to (I−P _{ T }), \({{\mathcal {F}}^{ 1}}{P_{w}}{\mathcal {F}}\), and P _{ T } are contraction mapping, it can be seen that \(\left ({I  {P_{T}}} \right){{\mathcal {F}}^{ 1}}{P_{w}}{\mathcal {F}}{P_{T}}\) is contraction mapping, i.e.,
Let α _{2} denote the proportion of quadratic sum of P _{ b } P _{ T } f in the quadratic sum of graph signal f. Therefore,
It can be seen that the range of α _{2} is from 0 to 1. Since P _{ T } and P _{ w } are the known parameter, the value of α _{2} can be obtained according to graph signal and downsampling set. Besides, the value of α _{2} approximates to \(\left ({\frac {{\left ({N  M} \right)}}{N}\frac {Y}{N}\frac {M}{N}} \right)\), when the graph signal f is very smooth. Then, combining (17), (24) and (25), we have
Therefore, the operation (I−P _{ d } P _{ T }) is a contraction mapping, if
Thus, for a given graph cutoff frequency, if satisfying the condition of (27), we have
It can be seen that IGDR reduces the reconstructed error at each iteration step. Then, the reconstructed error bound of IGDR satisfies
and Proposition 1 is proved. □
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The authors would like to thank the editor and the reviewers for constructive comments and suggestions that led to improvements in the manuscript.
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Yang, L., You, K. & Guo, W. Bandlimited graph signal reconstruction by diffusion operator. EURASIP J. Adv. Signal Process. 2016, 120 (2016). https://doi.org/10.1186/s1363401604214
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Keywords
 Diffusion operator
 Downsampling and reconstruction
 Bandlimited graph signal processing