Multi-dimensional Graph Fourier Transform
  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.
