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Vertex-Frequency Analysis of Graph Signals pp 223–254Cite as

Oversampled Transforms for Graph Signals

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Part of the book series: Signals and Communication Technology ((SCT))

Abstract

In this chapter, oversampled transforms for graph signals are introduced. Oversampling is done in two ways: One is oversampled graph Laplacian and the other is oversampled graph transforms. Both are described here. The advantage of the oversampled transforms is that we can take a good trade-off between performance (in context to sparsifying the graph signals) and storage/memory space for transformed coefficients. Furthermore, any graph can be converted into an oversampled bipartite graph by using the oversampled graph Laplacian. It leads to that well-known graph wavelet transforms/filter banks for bipartite graphs can be applied to the signals on any graphs with a slight sacrifice of redundancy. Actual performances are compared through several numerical experiments.

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Notes

  1. 1.

    There always exists \(\widetilde{q}(1-l)\) which satisfies the perfect reconstruction condition (45) [11]. Therefore, \(\widetilde{p}_{\frac{M}{2}-1}(1-l)\) also has a unique solution with real coefficients as long as all of the arbitrary design filters have real coefficients. Additionally, since the perfect reconstruction condition is only imposed on \(\widetilde{p}_{\frac{M}{2}-1}(1-l)\), \(J^{(h)}_k\) and \(J^{(g)}_k\) in (53) can be set arbitrarily regardless of K.

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Acknowledgements

This work was supported in part by JST PRESTO Grant Number JPMJPR1656 and JSPS KAKENHI Grant Number JP16H04362.

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Authors and Affiliations

  1. Tokyo University of Agriculture and Technology, Fuchu, Japan

    Yuichi Tanaka & Akie Sakiyama

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  1. Yuichi Tanaka
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  2. Akie Sakiyama
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Correspondence to Yuichi Tanaka .

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Editors and Affiliations

  1. Faculty of Electrical Engineering, University of Montenegro, Podgorica, Montenegro

    Ljubiša Stanković

  2. Department of Electrical and Computer Engineering, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Ervin Sejdić

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Tanaka, Y., Sakiyama, A. (2019). Oversampled Transforms for Graph Signals. In: Stanković, L., Sejdić, E. (eds) Vertex-Frequency Analysis of Graph Signals. Signals and Communication Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-03574-7_6

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  • DOI: https://doi.org/10.1007/978-3-030-03574-7_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-03573-0

  • Online ISBN: 978-3-030-03574-7

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