  Many multi-dimensional signals appear in the real world, such as digital
images and data that has spatial and temporal dimensions. How to show the
spectrum of these multi-dimensional signals correctly is a key challenge in the
field of graph signal processing. This paper investigates the novel transform
for multi-dimensional signals defined on Cartesian product graph and studies
several related properties. Our work includes: (i) defining the
multi-dimensional graph fractional Fourier transform (MGFRFT) based on
Laplacian matrix and adjacency matrix; (ii) exploring the advantages of MGFRFT
in processing multi-dimensional signals in terms of spectrum and computational
time; (iii) applying the proposed transform to data compression to highlight
the utility and effectiveness of it.
