SVD-Based Graph Fourier Transforms on Directed Product Graphs

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 in an effective way. The GFT on undirected graphs has been well studied and several approaches have been proposed to define GFTs on directed graphs. In this article, 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 graphs with strong correlation efficiently, and in the undirected graph setting they are essentially the joint GFT in the literature. In this article, we also consider the bandlimiting procedure in frequency domains of the proposed GFTs, and demonstrate their performances on denoising the hourly temperature data sets collected at 32 weather stations in the region of Brest (France) and at 218 locations in the United States.