期刊
IEEE TRANSACTIONS ON SIGNAL PROCESSING
卷 61, 期 13, 页码 3357-3367出版社
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TSP.2013.2259825
关键词
Higher-order singular value decomposition (HOSVD); multislice graph; spectral graph theory; tensor decompositions; wavelet transform
资金
- Swiss National Science Foundation [PP00P2-123438, PP00P2-146318]
- Societe Academique de Geneve
- FOREMANE foundation
- Center for Biomedical Imaging (CIBM) of the Geneva
- Lausanne Universities and EPFL
- Leenaards foundation
- Louis-Jeantet foundation
- Swiss National Science Foundation (SNF) [PP00P2_146318] Funding Source: Swiss National Science Foundation (SNF)
We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, Wavelets on Graphs via Spectral Graph Theory, Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of eigennetworks and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.
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