Multi-dimensional spectral graph wavelet transform

In this paper, we introduce the two-dimensional spectral graph wavelet transform (SGWT) of discrete functions defined on weighted Cartesian product graphs. The graphs we consider are undirected, non-planar, and can be cyclic with no multiple loops and no multiple edges. We build this transform with the help of spectral decomposition of N1N2 x N1N2 Laplacian matrix L of the Cartesian product graph G = G(1)rectangle G(2), where N1N2 is the number of vertices of the Cartesian product graph. We have established the inversion formula for SGWT for continuous scale parameters s(1) and s(2). By sampling the scale parameters at discrete values, we obtain discrete SGWT coefficients at different scales and vertices. We have obtained the frame bounds of the frame formed by the set of scaling and wavelet coefficients. Further, we have proved the localization result which holds for both normalized and non-normalized form of Laplacian. Towards the end, we have given the implementation details of SGWT through an example to show how to calculate the graph wavelet coefficients.