The windowed two-dimensional graph fractional Fourier transform

In the vibrant landscape of image and signal processing, the research on multi-dimensional graph signals has been making remarkable strides. However, despite the progress, the in-depth exploration and analysis of graph signals defined on two-dimensional (2-D) Cartesian product graphs still present certain gaps and challenges. This paper presents a comprehensive investigation into the generalization of the windowed graph fractional Fourier transform (WGFRFT) in the context of 2-D Cartesian product graphs. Firstly, the 2-D WGFRFT and its inverse transform are meticulously derived and defined, accompanied by the proposal of a fast algorithm. Subsequently, through an experiment, the advantages of the 2-D WGFRFT over the WGFRFT in processing two-dimensional graph signals are verified. Moreover, the effectiveness of the fast algorithm is rigorously validated through vertex-frequency analysis. And lastly, based on the 2-D WGFRFT, a novel filter learning method is put forward, and its potential in image classification is demonstrated.