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The Dual Graph Shift Operator: Identifying the Support of the Frequency Domain
Journal of Fourier Analysis and Applications ( IF 1.2 ) Pub Date : 2021-05-13 , DOI: 10.1007/s00041-021-09850-1
Geert Leus , Santiago Segarra , Alejandro Ribeiro , Antonio G. Marques

Contemporary data is often supported by an irregular structure, which can be conveniently captured by a graph. Accounting for this graph support is crucial to analyze the data, leading to an area known as graph signal processing (GSP). The two most important tools in GSP are the graph shift operator (GSO), which is a sparse matrix accounting for the topology of the graph, and the graph Fourier transform (GFT), which maps graph signals into a frequency domain spanned by a number of graph-related Fourier-like basis vectors. This alternative representation of a graph signal is denominated the graph frequency signal. Several attempts have been undertaken in order to interpret the support of this graph frequency signal, but they all resulted in a one-dimensional interpretation. However, if the support of the original signal is captured by a graph, why would the graph frequency signal have a simple one-dimensional support? Departing from existing work, we propose an irregular support for the graph frequency signal, which we coin dual graph. A dual GSO leads to a better interpretation of the graph frequency signal and its domain, helps to understand how the different graph frequencies are related and clustered, enables the development of better graph filters and filter banks, and facilitates the generalization of classical SP results to the graph domain.



中文翻译:

对偶图移位运算符:确定对频域的支持

当代数据通常由不规则结构支持,可以通过图形方便地捕获。考虑到这种图形支持对于分析数据至关重要,这导致了一个称为图形信号处理(GSP)的领域。GSP中最重要的两个工具是图移位运算符(GSO)和图傅立叶变换(GFT),图稀疏矩阵是考虑图拓扑的稀疏矩阵,图傅里叶变换将图信号映射到一个跨数的频域中图相关的傅立叶样基向量的集合。图形信号的这种替代表示被称为图形频率信号。为了解释该图形频率信号的支持,已经进行了几次尝试,但是它们都导致了一维解释。但是,如果原始信号的支持是通过图形捕获的,为什么图形频率信号具有简单的一维支持?与现有工作不同的是,我们提出了对图频率信号的不规则支持,我们称其为对偶图。双重GSO可以更好地解释图形频率信号及其域,有助于理解不同图形频率之间的相关性和聚类关系,可以开发更好的图形滤波器和滤波器组,并有助于将经典SP结果推广到图域。

更新日期:2021-05-14
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