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Spectral design of signal-adapted tight frames on graphs

Behjat, Hamid LU and Van De Ville, Dimitri (2019) In Signals and Communication Technology p.177-206
Abstract

Analysis of signals defined on complex topologies modeled by graphs is a topic of increasing interest. Signal decomposition plays a crucial role in the representation and processing of such information, in particular, to process graph signals based on notions of scale (e.g., coarse to fine). The graph spectrum is more irregular than for conventional domains; i.e., it is influenced by graph topology, and, therefore, assumptions about spectral representations of graph signals are not easy to make. Here, we propose a tight frame design that is adapted to the graph Laplacian spectral content of a given class of graph signals. The design exploits the ensemble energy spectral density, a notion of spectral content of the given signal set that... (More)

Analysis of signals defined on complex topologies modeled by graphs is a topic of increasing interest. Signal decomposition plays a crucial role in the representation and processing of such information, in particular, to process graph signals based on notions of scale (e.g., coarse to fine). The graph spectrum is more irregular than for conventional domains; i.e., it is influenced by graph topology, and, therefore, assumptions about spectral representations of graph signals are not easy to make. Here, we propose a tight frame design that is adapted to the graph Laplacian spectral content of a given class of graph signals. The design exploits the ensemble energy spectral density, a notion of spectral content of the given signal set that we determine either directly using the graph Fourier transform or indirectly through approximation using a decomposition scheme. The approximation scheme has the benefit that (i) it does not require diagonalization of the Laplacian matrix, and (ii) it leads to a smooth estimate of the spectral content. A prototype system of spectral kernels each capturing an equal amount of energy is defined. The prototype design is then warped using the signal set’s ensemble energy spectral density such that the resulting subbands each capture an equal amount of ensemble energy. This approach accounts at the same time for graph topology and signal features, and it provides a meaningful interpretation of subbands in terms of coarse-to-fine representations.

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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
Signals and Communication Technology
series title
Signals and Communication Technology
pages
30 pages
publisher
Springer
external identifiers
  • scopus:85063228248
ISSN
1860-4870
1860-4862
ISBN
978-3-030-03573-0
978-3-030-03574-7
DOI
10.1007/978-3-030-03574-7_4
language
English
LU publication?
yes
id
12050474-47c6-44aa-91fc-bbc8de3d9158
date added to LUP
2019-04-05 15:12:59
date last changed
2019-04-23 04:48:52
@inbook{12050474-47c6-44aa-91fc-bbc8de3d9158,
  abstract     = {<p>Analysis of signals defined on complex topologies modeled by graphs is a topic of increasing interest. Signal decomposition plays a crucial role in the representation and processing of such information, in particular, to process graph signals based on notions of scale (e.g., coarse to fine). The graph spectrum is more irregular than for conventional domains; i.e., it is influenced by graph topology, and, therefore, assumptions about spectral representations of graph signals are not easy to make. Here, we propose a tight frame design that is adapted to the graph Laplacian spectral content of a given class of graph signals. The design exploits the ensemble energy spectral density, a notion of spectral content of the given signal set that we determine either directly using the graph Fourier transform or indirectly through approximation using a decomposition scheme. The approximation scheme has the benefit that (i) it does not require diagonalization of the Laplacian matrix, and (ii) it leads to a smooth estimate of the spectral content. A prototype system of spectral kernels each capturing an equal amount of energy is defined. The prototype design is then warped using the signal set’s ensemble energy spectral density such that the resulting subbands each capture an equal amount of ensemble energy. This approach accounts at the same time for graph topology and signal features, and it provides a meaningful interpretation of subbands in terms of coarse-to-fine representations.</p>},
  author       = {Behjat, Hamid and Van De Ville, Dimitri},
  isbn         = {978-3-030-03573-0},
  issn         = {1860-4870},
  language     = {eng},
  pages        = {177--206},
  publisher    = {Springer},
  series       = {Signals and Communication Technology},
  title        = {Spectral design of signal-adapted tight frames on graphs},
  url          = {http://dx.doi.org/10.1007/978-3-030-03574-7_4},
  year         = {2019},
}