Multi-dimensional Spectral Graph Wavelet Transform

In this paper, we introduce the two-dimensional spectral graph wavelet transform (SGWT) of discrete functions defined on Cartesian product graphs. We build this transform with the help of spectral decomposition of N1N2×N1N2 Laplacian matrix L of the Cartesian product graph G = G1□G2, where N1N2 is the number of vertices of the Cartesian product graph. We have established the inversion formula for SGWT for continuous scale parameter s. By sampling the scale parameter s at discrete values, we obtain discrete SGWT coeffients at different scales and vertices.We have found the frame bounds of the frame formed by the set of scaling and wavelet coeffients. 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. Mathematics Subject Classification: 42C40; 42C15; 44A15; 44A35.