Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new appro...

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Veröffentlicht in:IEEE transactions on signal processing Jg. 67; H. 18; S. 4855 - 4869
Hauptverfasser: Lu, Keng-Shih, Ortega, Antonio
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 15.09.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
bipartite graph
Clustering
Computation
Discrete Fourier transforms
Eigenvalues and eigenfunctions
Empirical analysis
fast algorithm
Fourier transforms
Graph Fourier transform
graph signal processing
Graph theory
Graphs
Image processing
Laplace equations
Signal processing
Signal processing algorithms
symmetric graph
Symmetry
Topology
Video compression
ISSN:1053-587X, 1941-0476
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Zusammenfassung:The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle <inline-formula><tex-math notation="LaTeX">\pi /4</tex-math></inline-formula>) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2019.2932882