Data Analytics on Graphs Part II: Signals on Graphs

Stanković, Ljubiša; Mandic, Danilo P.; Daković, Miloš; Brajović, Miloš; Scalzo, Bruno; Li, Shengxi; Constantinides, A.G.

doi:10.1561/2200000078-2

Cited by 26 publications

(23 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, N}, which are connected with edges. The weight of edges are W mn [24][25][26]. For the vertices m and n which are not connected, by definition W mn = 0.…”

Section: Basic Graph Definitionsmentioning

confidence: 99%

“…is called a convolution of signals on a graph [25,29]. However, this procedure could be computationally unacceptable for very large graphs.…”

Section: Undirected Circular Graphmentioning

confidence: 99%

“…. , h M is direct in the classical DFT case only, while it is more complex in the general graph case [25]. For large M, the solution to the system of equations in (12), for the unknown parameters h 0 , h 1 , .…”

Section: Undirected Circular Graphmentioning

confidence: 99%

“…Of course, the sum of all parts of signal, filtered by H k (L), produces the reconstruction relation, ∑ K k=0 H k (L)x = x, what is obvious from the identity in (25), that is, from (I − L/λ max ) + L/λ max = I. Example 2.…”

Section: Binomial Decompositionmentioning

confidence: 99%

“…, K − 1, in the form given by ( 28) or ( 33) should be realized using their approximations by polynomials whose order is (M − 1). Although the approximation based on the Chebyshev polynomial is most commonly used for this purpose [25,31], we will revisit alternative approaches [38] as well.…”

Section: Polynomial Lgft Approximationmentioning

confidence: 99%

See 4 more Smart Citations

From Time–Frequency to Vertex–Frequency and Back

et al. 2021

Self Cite

View full text Add to dashboard Cite

The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

show abstract

“…, N}, which are connected with edges. The weight of edges are W mn [24][25][26]. For the vertices m and n which are not connected, by definition W mn = 0.…”

Section: Basic Graph Definitionsmentioning

confidence: 99%

“…is called a convolution of signals on a graph [25,29]. However, this procedure could be computationally unacceptable for very large graphs.…”

Section: Undirected Circular Graphmentioning

confidence: 99%

Section: Undirected Circular Graphmentioning

confidence: 99%

Section: Binomial Decompositionmentioning

confidence: 99%

Section: Polynomial Lgft Approximationmentioning

confidence: 99%

See 3 more Smart Citations

From Time–Frequency to Vertex–Frequency and Back

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

Diffusion optical tomography reconstruction based on convex–nonconvex graph total variation regularization

Xie

Liu

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

Graph total variation (GTV) is a powerful regularization tool for diffuse optical tomography (DOT) reconstruction since it combines the powerful representation ability of graph and the edge-preserving ability of total variation (TV) regularization. However, as everyone knows, the classical TV regularization trend underestimates the large edge values. In this paper, we propose a convex-nonconvex graph total variation (CNC-GTV) regularization for DOT reconstruction. In particular, we construct a nonconvex regularization by subtracting the generalized Huber function from the GTV regularization. We show that the global convexity of the objective function can be guaranteed by adjusting the nonconvex control parameters. Moreover, we present an alternating direction multiplier method (ADMM) to solve the proposed DOT reconstruction model. Numerical experiments show that the proposed model outperforms existing models in visual and numerical results.

show abstract