Graph Fourier transform based on directed Laplacian

Singh, Rahul; Chakraborty, Abhishek; Manoj, B. S.

doi:10.1109/spcom.2016.7746675

Cited by 43 publications

(38 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we present some numerical results to assess the effectiveness of the proposed strategy for building the GFT basis. First, we illustrate some examples of application and then we compare the proposed approach with alternative definitions of GFT basis, as given in [5], [4], [14]. In all our experiments, the parameters of SOC and PAMAL methods are set as (unless stated otherwise): β = 100, τ = 0.5, γ = 1.5, ρ 1 = 50, ǫ k = (0.9) k , ∀k ∈ N, Λ min = −1000 · I Λ max = 1000 · I, Λ 1 = 0, c = c k,n i =c = 0.5, ∀i, k, n. Examples of bases for directed graphs.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Furthermore, the computation of the Jordan decomposition incurs into serious and intractable numerical instabilities when the graph size exceeds even moderate values [19] and more stable matrix decomposition methods have to be adopted to tackle its instability issues [20]. To overcome some of these criticalities, very recently the authors of [14] proposed a shift operator based on the directed Laplacian of a graph. Using the Jordan decomposition, the graph Laplacian is decomposed as…”

Section: Graph Fourier Basis and Directedmentioning

confidence: 99%

See 1 more Smart Citation

On the Graph Fourier Transform for Directed Graphs

Sardellitti

Barbarossa

Lorenzo

2017

IEEE J. Sel. Top. Signal Process.

110

104

View full text Add to dashboard Cite

The analysis of signals defined over a graph is relevant in many applications, such as social and economic networks, big data or biological networks, and so on. A key tool for analyzing these signals is the so-called graph Fourier transform (GFT). Alternative definitions of GFT have been suggested in the literature, based on the eigen-decomposition of either the graph Laplacian or adjacency matrix. In this paper, we address the general case of directed graphs and we propose an alternative approach that builds the graph Fourier basis as the set of orthonormal vectors that minimize a continuous extension of the graph cut size, known as the Lovász extension. To cope with the nonconvexity of the problem, we propose two alternative iterative optimization methods, properly devised for handling orthogonality constraints. Finally, we extend the method to minimize a continuous relaxation of the balanced cut size. The formulated problem is again nonconvex, and we propose an efficient solution method based on an explicit-implicit gradient algorithm

show abstract

Section: Numerical Resultsmentioning

confidence: 99%

Section: Graph Fourier Basis and Directedmentioning

confidence: 99%

On the Graph Fourier Transform for Directed Graphs

Sardellitti

Barbarossa

Lorenzo

2017

IEEE J. Sel. Top. Signal Process.

110

104

View full text Add to dashboard Cite

show abstract

“…It is possible to readily transpose the previous notions to the directed case, choosing either D out or D in to replace D in the different formulations: e.g. L = D in − A as in [10], L rw = I − D out −1 A as in [11], [1] (we recall that L d and R = A are equivalent for they share the same eigenvectors and thus, define the same GFT). These matrices are no longer strictly speaking Laplacians as they are no longer SDP, but one may nonetheless consider them as reference operators defining possible GFTs.…”

Section: Graph Fourier Transformmentioning

confidence: 99%

Design of Graph Filters and Filterbanks

Tremblay

Gonçalvès

Borgnat

2018

Cooperative and Graph Signal Processing

View full text Add to dashboard Cite

Basic operations in graph signal processing consist in processing signals indexed on graphs either by filtering them, to extract specific part out of them, or by changing their domain of representation, using some transformation or dictionary more adapted to represent the information contained in them. The aim of this chapter is to review general concepts for the introduction of filters and representations of graph signals. We first begin by recalling the general framework to achieve that, which put the emphasis on introducing some spectral domain that is relevant for graph signals to define a Graph Fourier Transform. We show how to introduce a notion of frequency analysis for graph signals by looking at their variations. Then, we move to the introduction of graph filters, that are defined like the classical equivalent for 1D signals or 2D images, as linear systems which operate on each frequency of a signal. Some examples of filters and of their implementations are given. Finally, as alternate representations of graph signals, we focus on multiscale transforms that are defined from filters. Continuous multiscale transforms such as spectral wavelets on graphs are reviewed, as well as the versatile approaches of filterbanks on graphs. Several variants of graph filterbanks are discussed, for structured as well as arbitrary graphs, with a focus on the central point of the choice of the decimation or aggregation operators. * N. Tremblay is with Univ. Grenoble Alpes, CNRS, GIPSA-lab,

show abstract

“…The eigenvalues of L G act as the graph frequencies and corresponding eigenvectors act as the graph harmonics [19], [20], [21]. Small λ's carry information about low frequency components of the signal, while high frequencies (details) are carried by large λ's.…”

Section: E Self Parameter Tuningmentioning

confidence: 99%