Sampling, Filtering and Sparse Approximations on Combinatorial Graphs

Pesenson, Isaac Z.; Pesenson, Meyer Z.

doi:10.1007/s00041-009-9116-7

Cited by 67 publications

(50 citation statements)

References 24 publications

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“…The next theorem can be considered as a form of the Paley-Wiener theorem and it essentially follows from our more general results in [12][13][14][15][16][17][18]. One can easily to verify that an immersion f = ( f 1 , . .…”

Section: Definitionmentioning

confidence: 96%

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

Pesenson

2010

Journal of Mathematical Analysis and Applications

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We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.

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Section: Definitionmentioning

confidence: 96%

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

Pesenson

2010

Journal of Mathematical Analysis and Applications

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“…Suppose that the DS condition (8) is satisfied for the signal and sampling subspaces. Following the expression in (9), the signal recovery is given asx…”

Section: B Unconstrained Casementioning

confidence: 99%

Generalized Sampling on Graphs With Subspace and Smoothness Priors

Tanaka

Eldar

2020

IEEE Trans. Signal Process.

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We consider a framework for generalized sampling of graph signals that parallels sampling in shift-invariant (SI) subspaces. This framework allows for arbitrary input signals, which are not constrained to be bandlimited. Furthermore, the sampling and reconstruction filters can be different. We present design methods of the correction filter that compensates for these differences and can be obtained in closed form in the graph frequency domain. This paper considers two priors on graph signals: The first is a subspace prior, where the signal is assumed to lie in a periodic graph spectrum (PGS) subspace. The PGS subspace is proposed as a counterpart of the SI subspace used in standard sampling theory. The second is a smoothness prior that imposes a smoothness requirement on the graph signal. We suggest recovery methods both for the case when the recovery filter can be optimized and in the setting in which a predefined filter must be used. Sampling is performed in the graph frequency domain, which is a counterpart of "sampling by modulation" in SI subspaces. We compare our approach with existing sampling methods in graph signal processing. The effectiveness of the proposed generalized sampling is validated numerically through several experiments.

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“…Graph sampling has been extensively studied, we will comment on a few of these papers and refer to them for a more complete review of the literature and list of references. References [23]- [25] consider the space of bandlimited graph signals (Paley-Wiener spaces) and establish that low-pass graph signals can be perfectly reconstructed from their values on some subsets of vertices (sampling sets). Sampling has received considerable attention in the GSP literature [26]- [53].…”

Section: Introductionmentioning

confidence: 99%

Topics in Graph Signal Processing: Convolution and Modulation

Shi

Moura

2019

2019 53rd Asilomar Conference on Signals, Systems, and Computers

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To analyze data supported by arbitrary graphs G, DSP has been extended to Graph Signal Processing (GSP) by redefining traditional DSP concepts like shift, filtering, and Fourier transform among others. This paper revisits modulation, convolution, and sampling of graph signals as appropriate natural extensions of the corresponding DSP concepts. To define these for both the vertex and the graph frequency domains, we associate with generic data graph G and its graph shift A a graph spectral shift M and a spectral graph G s . This leads to a spectral GSP theory that parallels in the graph frequency domain the existing GSP theory in the vertex domain. The paper applies this to design and recovery sampling techniques for data supported by arbitrary directed graphs.

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Sampling, Filtering and Sparse Approximations on Combinatorial Graphs

Cited by 67 publications

References 24 publications

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

Generalized Sampling on Graphs With Subspace and Smoothness Priors

Topics in Graph Signal Processing: Convolution and Modulation

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