TwoChannel CriticallySampled Graph Wavelets
With Spectral Domain Sampling
Abstract
We propose twochannel criticallysampled wavelet transforms for signals on undirected graphs that utilize spectral domain sampling. Unlike conventional approaches based on vertex domain sampling, our transforms have the following desirable properties: 1) perfect reconstruction regardless of the characteristics of the underlying graphs and graph variation operators and 2) a symmetric structure; i.e., both analysis and synthesis filter banks are built using similar building blocks. The relationship between the proposed wavelets and those using vertex domain sampling are also described. The effectiveness of our approach is evaluated by comparing its performance in nonlinear approximation and denoising with that of conventional graph wavelets and filter banks.
I Introduction
Ia Motivation
Graph signal processing focuses on graph signals, discrete signals defined on the vertices of a graph [Shuman2013, Sandry2013, Ortega2018]. Graph signals can represent a broad range of irregularly structured data, such as signals on brain, sensor, social, and traffic networks, point cloud attributes, and images/videos. Developing sparse representations for these signals by using appropriate bases or frames is important, because these signals are often highdimensional. Promising applications for such sparse representations of graph signals include feature extraction [Leonar2013, Miller2015, Shahid2016], denoising [Onuki2016, Pang2017, Yamamo2016, Cheung2018], compression [Shuman2016, Hu2015, Liu2017, Cheung2018], and others in many different areas [Zhang2014, Ono2015, Higash2016, Bronst2017, Segarr2016, Rustam2013].
As is the case with classical signal processing, multiscale transforms or dictionaries are important tools for achieving sparse representations of graph signals. Sampling strategies are crucial for controlling the level of redundancy in the graph signal in a multiscale signal representation. Undecimated transforms [Hammon2011, Sakiya2016a, Shuman2015] require a significant storage overhead for the transformed coefficients. Other approaches, which can achieve different tradeoffs in terms of redundancy, performance, computation cost and storage, are oversampled [Tay2015, Sakiya2014a, Tanaka2014a, Sakiya2016a], critically sampled (CS) [Narang2012, Narang2013, Tay2017, Tay2017a, Trembl2016, Jin2017, Sakiya2016a, Ekamba2015, Teke2016b] and undersampled transforms [Sakiya2016b].
For time domain signals, downsampling by a factor of two followed by upsampling by a factor of two corresponds to replacing every other sample by zero. In the frequency domain, the resulting signal has two components, the original frequency content of the signal and an added aliasing term (a modulated version of the original spectrum) [Vaidya1993, Oppenh2009, Vetter2014]. In contrast, in the case of graph signals, downsampling/upsampling in the vertex domain (i.e., replacing some of the values on the graph vertices by zero) does not preserve the shape of the original graph signal spectrum [Tanaka2017], with the sole exception of bipartite graphs.
To date, most graph transforms that make use of sampling apply it in the vertex domain. These approaches are limited to specific graph types and variation operators. For example, some designs require bipartite graphs and symmetric normalized graph Laplacians are needed for perfect reconstruction [Narang2012, Narang2013, Sakiya2016a, Tay2017a]. Some graph transforms are not restricted to specific graph types or variation operators but have other limitations. For example, in [Ekamba2015, Narang2009, Trembl2016] the transform operates on a graph different from the original one, while in [Jin2017, Chen2015] the original graph is used, but even though the decomposition can use efficient polynomial graph filters, the reconstruction requires significantly more complex global interpolation. The choices of sampling leading to perfect reconstruction are not unique in [Jin2017, Chen2015], and for any given sampling choice the interpolator is obtained via matrix inversion, which can lead to numerical issues.
In this paper, we propose novel CS graph wavelet transforms (GWTs), using the recently introduced spectral domain sampling [Tanaka2017], that overcome the problems described above. These GWTs have the following properties:

Perfect reconstruction is guaranteed for any graph and for any variation operator as long as the operator is diagonalizable and has real eigenvalues.

The (frequency domain) sampling that leads to perfect reconstruction is unique, while the analysis and synthesis operations have the same complexity and a matrix inversion is not required to compute the reconstruction operator.
Moreover, we show that the GWTs obtained in the vertex domain and those obtained using spectral domain sampling are identical in some special cases. We also assess their performance through experiments on denoising and nonlinear approximation. This paper significantly extends our preliminary study [Watana2018] by adding rigorous proofs to the theoretical results and providing much more comprehensive experiments.
The rest of the paper is organized as follows. We review related work in Section II. Sampling methods in the vertex and spectral domains are introduced in Section LABEL:sec:II. Section LABEL:sec:III reviews the conventional CS GWTs. The proposed CS GWTs are presented in Section LABEL:sec:IV along with the octaveband structure and polyphase representation. The relationship between the vertex and spectral domain sampling approaches is studied in Section LABEL:sec:V. Section LABEL:sec:VI presents a few potential applications of the proposed CS GWTs, together with comparisons with the conventional methods. Finally, Section LABEL:sec:VII is the conclusion.
IB Notation
A graph consists of a set of edges and vertices , where the number of vertices is . We consider undirected graphs without selfloops and nonnegative edge weights. A graph signal is a function , and it can be represented in vector form , whose th sample is regarded as a signal value on the th vertex of the graph.
is an adjacency matrix of the graph whose thelement represents the weight of the edge between the th and th vertices. is a diagonal degree matrix whose elements are defined as . The combinatorial and symmetric normalized graph Laplacians are defined as and , respectively. Since a graph Laplacian is a real symmetric matrix, the eigendecomposition of (or ) can always be represented as , where is an eigenvector matrix, is an eigenvalue matrix having eigenvalues () of as diagonal elements, and represents the transpose of a matrix.
For a symmetric normalized graph Laplacian, its eigenvalues are bounded in . In addition, the maximum eigenvalue becomes and the eigenvalues are distributed symmetrically with respect to only for the bipartite case.
The graph Fourier transform (GFT) is defined as
(1) 
while the other definitions of the GFT, such as those in , can be used as long as the GFT matrix is nonsingular.
Ii Related Work
Several CS GWTs using vertex domain sampling have been proposed for signals on bipartite graphs. They can be used on nonbipartite graphs by dividing the original graph into several bipartite graphs and then using a “multidimensional” decomposition. Filter design methods for this class of CS GWT include: graphQMF [Narang2012], which utilizes quadrature mirror filters; graphBior [Narang2013] a biorthogonal and polynomial filter solution with spectral factorization; a frequency conversion method [Sakiya2016a] that transforms time domain filters into graph spectral filters; nearorthogonal polynomial filter design methods proposed in [Tay2017, Tay2017a]. Oversampled graph filter banks were introduced in [Tanaka2014a, Sakiya2014a] as an extension of CS GWTs for bipartite graphs.
The above methods are for designing filters in the graph frequency domain. There are also CS graph filter banks whose filters are designed in the vertex domain. For example, a liftingbased transform [Narang2009] divides the original graph into even and oddindexed vertices and performs vertex domain filtering. The subgraphbased biorthogonal filter bank [Trembl2016] decomposes the original graph into several partitions. Wavelets on a balanced tree [Gavish2010] provide CS perfect reconstruction transforms using vertex domain filtering. The splinebased graph wavelet is a CS perfect reconstruction for cyclic graphs [Ekamba2015]. There is a CS graph filter bank for a specific class of graphs, called structures [Teke2016b]^{1}^{1}1For more general graphs, we need to redesign bases for a new graph Fourier transform.. However, all of these methods require simplifying the graph, i.e., eliminating some of the edges in the original graph, in order to ensure critical sampling and invertibility.
CS graph filter banks can also be designed with careful vertex domain sampling. An channel CS graph filter bank [Jin2017] was designed that selects sampled vertices for each subband in order to satisfy the uniqueness set condition. In the context of sampling theory of graph signals, band limiting the input graph signal followed by vertex domain sampling has been proposed as a graph filter bank [Chen2015]. However, such approaches have several limitations. First, they have to select an appropriate sampling set for perfect reconstruction. In other words, arbitrarily selected sampling sets do not generally lead to a perfect reconstruction transform. Second, the sampling set is not unique; different sampling sets significantly affect the overall performance of the graph transforms in applications. Here, an efficient vertex selection was proposed in [Anis2017], where the filter bank system has a symmetric structure. However, the perfect reconstruction condition is only guaranteed for bipartite graphs [Anis2017]. Third, many approaches are perfect reconstruction only if ideal filters are used in the analysis transform. That means there is no flexibility in the design of the filter. However, nonideal filters are sometimes preferred when the eigenvalue distribution of the variation operator is irregular (described in Section LABEL:sec:idealvsnonideal). Fourth, they often need to calculate the reconstruction operator for the synthesis side [Jin2017, Chen2015]. This leads to a matrix inversion with a high computational cost.
The performance of CS GWTs with vertex domain sampling varies according to the graph reduction method used. Graph coloring [Narang2012, Aspval1984, Narang2013, Harary1977], Kron reduction [Dorfle2013, Shuman2016], maximum spanning trees [Nguyen2015], weighted maxcut [Narang2010], and graph coarsening using algebraic distance [Ron2011] are examples of the various graph reduction methods.
The properties of the existing and proposed GWTs are summarized in Table II. It should be emphasized that all of the existing approaches have limitations on their design, e.g., eligible graphs/variation operators, sampling set guaranteeing perfect reconstruction, or filter design. Our approach overcomes the limitations by employing a novel sampling in the graph frequency domain, and it is the only approach that has all of the following features: i) spectral domain filtering, ii) orthogonality, iii) perfect reconstruction, and iv) applicability to any graph.
GWTs / Properties  Analysis^{1}  Synthesis^{2}  Filter^{3}  Graphs^{4}  VO^{5}  Orth.^{6}  Comp.^{7}  PR^{8} 

GraphQMF [Narang2012]  Filt. VS  VS Filt.  S  Bipartite  SNL  O  ✓  
GraphBior [Narang2013]  Filt. VS  VS Filt.  S  Bipartite  SNL  B  ✓  ✓ 
Frequency conversion [Sakiya2016a]  Filt. VS  VS Filt.  S  Bipartite  SNL  O, B  ✓  
Nearorth [Tay2017a]  Filt. VS  VS Filt.  S  Bipartite  SNL  B  ✓  ✓ 
structure [Teke2016b]  Filt. VS  VS Filt.  S  structure^{9}  Adjacency  B  ✓  
Generalized Spline [Ekamba2015]  Filt. VS  Interpolation  S  Cyclic  NAD  N/A  ✓  ✓ 
Lifting [Narang2009]  Filt. VS  VS Filt.  V  Bipartite  Any  B  ✓  ✓ 
Wavelets on balanced tree [Gavish2010]  Filt. VS  VS Filt.  V  Tree  Any  B  ✓  ✓ 
Subgraph [Trembl2016]  Filt. VS  VS Filt.  V  Any  LoS  B  ✓  ✓ 
Qualified sampling [Chen2015]  Filt. VS  Interpolation  S  Any  Any  N/A  ✓  
Uniqueness set [Jin2017]  Filt. VS  Interpolation  S  Any  Any  N/A  ✓  
Sampling set selection [Anis2017]  Filt. VS  VS Filt.  S  Any  SNL  O, B  ✓  *^{10} 
Proposed  Filt. SS  SS Filt.  S  Any  Any  O, B  ✓ 