Multi-dimensional Graph Fourier Transform
This provides a foundation for novel filtering and stationarity methods for signals on product graphs, applicable to various data types like images and sensor data, but it is incremental as it builds on existing GFT concepts.
The paper tackles the problem that existing graph Fourier transforms (GFT) do not distinguish directional characteristics in signals on Cartesian product graphs, such as images or time series, and often produce multi-valued spectra. The result is a multi-dimensional GFT that rearranges 1-D spectra into a multi-dimensional frequency domain, enabling directional frequency analysis and resolving multi-valuedness in some cases.
Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish these directions, and assigns 1-D spectra to signals on product graphs. Further, these spectra are often multi-valued at some frequencies. Our main result is a multi-dimensional graph Fourier transform that solves such problems associated with the conventional GFT. Using algebraic properties of Cartesian products, the proposed transform rearranges 1-D spectra obtained by the conventional GFT into the multi-dimensional frequency domain, of which each dimension represents a directional frequency along each factor graph. Thus, the multi-dimensional graph Fourier transform enables directional frequency analysis, in addition to frequency analysis with the conventional GFT. Moreover, this rearrangement resolves the multi-valuedness of spectra in some cases. The multi-dimensional graph Fourier transform is a foundation of novel filterings and stationarities that utilize dimensional information of graph signals, which are also discussed in this study. The proposed methods are applicable to a wide variety of data that can be regarded as signals on Cartesian product graphs. This study also notes that multivariate graph signals can be regarded as 2-D univariate graph signals. This correspondence provides natural definitions of the multivariate graph Fourier transform and the multivariate stationarity based on their 2-D univariate versions.