Multi-dimensional graph fractional Fourier transform and its application to data compression

Many multi-dimensional signals appear in the real world, such as digital images and data that has spatial and temporal dimensions. Designing a transform method to process these multi-dimensional signals in graph fractional domain is a key challenge in graph signal processing. This paper investigates the novel transform for multi-dimensional graph signals defined on Cartesian product graph and studies several related properties. Our work includes: (i) proposing the two-dimensional graph fractional Fourier transforms using two basic graph signal processing methods i.e. based on Laplacian matrix and adjacency matrix; (ii) extending the two-dimensional transforms to multi-dimensional graph fractional Fourier transforms (MGFRFT). MGFRFT provides an additional fractional analysis tool for multi-dimensional graph signal processing; (iii) exploring the advantages of MGFRFT in terms of spectrum, directional characteristics and computational time; (iv) applying the proposed transform to data compression to highlight its utility and effectiveness.