  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.
