Spectral Folding And Two-Channel Filter-Banks On Arbitrary Graphs

Pavéz, Eduardo; Girault, Benjamin; Ortega, Antonio; Chou, Philip A.

doi:10.1109/icassp39728.2021.9414066

Cited by 4 publications

(2 citation statements)

References 31 publications

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“…They implement the discrete Fourier transform (DFT) formula in efficient manner. Nowadays, implementations benefit from parallel computing, which has allowed reaching a milestone in terms of speed [8,9]. From this class of procedures, the discrete cosine transform (DCT) algorithm is integrated in many standards of signal compression and coding [10].…”

Section: Introductionmentioning

confidence: 99%

John von Neumann’s Time-Frequency Orthogonal Transforms

Ștefănoiu

Culita

2023

Mathematics

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John von Neumann (JvN) was one of the greatest scientists and minds of the 20th century. His research encompassed a large variety of topics (especially from mathematics), and the results he obtained essentially contributed to the progress of science and technology. Within this article, one function that JvN defined long time ago, namely the cardinal sinus (sinc), was employed to define transforms to be applied on 1D signals, either in continuous or discrete time. The main characteristics of JvN Transforms (JvNTs) are founded on a theory described at length in the article. Two properties are of particular interest: orthogonality and invertibility. Both are important in the context of data compression. After building the theoretical foundation of JvNTs, the corresponding numerical algorithms were designed, implemented and tested on artificial and real signals. The last part of the article is devoted to simulations with such algorithms by using 1D signals. An extensive analysis on JvNTs effectiveness is performed as well, based on simulation results. In conclusion, JvNTs prove to be useful tools in signal processing.

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Section: Introductionmentioning

confidence: 99%

John von Neumann’s Time-Frequency Orthogonal Transforms

Ștefănoiu

Culita

2023

Mathematics

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“…In our preliminary version of this work [39], we first introduced the spectral folding property and two channel filter banks on arbitrary graphs (Sections IV-B and IV-E), while an application of GFBs to image compression has been recently published [27]. In this paper, we further develop the GFB theory, providing all proofs not given in [39], discussing the results in more depth and introducing several novel contributions, including:…”

Section: Introductionmentioning

confidence: 99%

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Pavéz¹,

Girault²,

Ortega³

et al. 2022

Preprint

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We study the design of filter banks for signals defined on the nodes of graphs. We propose novel two channel filter banks, that can be applied to arbitrary graphs, given a positive semi definite variation operator, while using downsampling operators on arbitrary vertex partitions. The proposed filter banks also satisfy several desirable properties, including perfect reconstruction, and critical sampling, while having efficient implementations. Our results generalize previous approaches only valid for the normalized Laplacian of bipartite graphs. We consider graph Fourier transforms (GFTs) given by the generalized eigenvectors of the variation operator. This GFT basis is orthogonal in an alternative inner product space, which depends on the choices of downsampling sets and variation operators. We show that the spectral folding property of the normalized Laplacian of bipartite graphs, at the core of bipartite filter bank theory, can be generalized for the proposed GFT if the inner product matrix is chosen properly. We give a probabilistic interpretation to the proposed filter banks using Gaussian graphical models. We also study orthogonality properties of tree structured filter banks, and propose a vertex partition algorithms for downsampling. We show that the proposed filter banks can be implemented efficiently on 3D point clouds, with hundreds of thousands of points (nodes), while also improving the color signal representation quality over competing state of the art approaches.

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Spectral Mappings for Graph Wavelets

Tay

2022

IEEE Trans. Signal Process.

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Spectral Folding And Two-Channel Filter-Banks On Arbitrary Graphs

Cited by 4 publications

References 31 publications

John von Neumann’s Time-Frequency Orthogonal Transforms

John von Neumann’s Time-Frequency Orthogonal Transforms

Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators

Spectral Mappings for Graph Wavelets

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