Graph signal processing and wavelet packets

Nowadays graphs became of significant importance given their use to describe complex system dynamics, with important applications to real world problems, e.g. graph representation of the brain, social networks, biological networks, spreading of a disease, etc.. What is missing in graph signal processing is a general definition of Spectral Graph Wavelet Packets Transform in the same fashion as for the classical framework where the equivalent of frequency is represented by the eigenvalues of the Laplacian matrix. Bremer and coauthors introduced diffusion wavelet packets transforms starting from diffusion wavelet definition, based on a diffusion operator T on a manifold or a graph. Cloninger et al., defined the natural graph wavelet packet dictionaries by introducing a set of novel multiscale basis transforms by considering the distance between graph Laplacian eigenvectors. In this paper we introduce a novel graph wavelet packets construction, to our knowledge different from the ones known in literature. Our work is inspired by the Spectral Graph Wavelet Transform (SGWT) defined by Hammond et al., and can be viewed as a generalization of their work. The result is a dictionary of frames particularly suitable for analyzing signals defined on graphs with a large number of nodes. We will give some concrete examples on how the wavelet packets can be used for compressing, denoising and reconstruction by considering a signal, given by the fRMI (functional magnetic resonance imaging) data, on the nodes of voxel-wise brain graph G with 900.760 nodes (representing the brain voxels).