有向积图上的图傅里叶变换
Graph Fourier transforms on directed product graphs
摘要：
图形傅立叶变换（graphFourier transform，GFT）是图形信号处理的基本工具之一，它能有效地将图形信号分解为不同的频率分量，用不同的变化模式来表示强相关的图形信号。无向图上的GFT已经得到了很好的研究，并且已经提出了几种方法来定义有向图上的GFT。本文基于图的Laplacian的奇异值分解，提出了两个关于两个有向图的笛卡尔积图的GFT。我们表明，建议的GFT可以表示时空数据集的有向网络上的强相关性有效，在无向图设置，他们基本上是联合GFT在文献中。在本文中，我们还考虑了所提出的GFT的谱域中的带限过程，并展示了其对2014年1月布雷斯特（法国）地区的温度数据集进行去噪的性能。
英文摘要：
Graph Fourier transform (GFT) is one of the fundamental tools in graph signal processing to decompose graph signals into different frequency components and to represent graph signals with strong correlation by different modes of variation effectively. The GFT on undirected graphs has been well studied and several approaches have been proposed to define GFTs on directed graphs. In this paper, based on the singular value decompositions of some graph Laplacians, we propose two GFTs on the Cartesian product graph of two directed graphs. We show that the proposed GFTs could represent spatial-temporal data sets on directed networks with strong correlation efficiently, and in the undirected graph setting they are essentially the joint GFT in the literature. In this paper, we also consider the bandlimiting procedure in the spectral domain of the proposed GFTs, and demonstrate its performance to denoise the temperature data set in the region of Brest (France) on January 2014.
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