Signal-adapted decomposition of graph signals

Analysis of signals defined on complex topologies modeled by graphs is a topic of increasing interest. Signal decomposition plays a crucial role in the representation and processing of such information, in particular, to process graph signals based on notions of scale (e.g., coarse to fine). The graph spectrum is more irregular than for conventional domains; i.e., it is influenced by graph topology, and, therefore, assumptions about spectral representations of graph signals are not easy to make. Here, we propose a tight frame design that is adapted to the graph Laplacian spectral content of given classes of graph signals. The design is based on using the ensemble energy spectral density, a notion of spectral content of given signal sets that we determine either directly using the graph Fourier transform or indirectly through a polynomial-based approximation scheme. The approximation scheme has the benefit that (i) it does not require eigendecomposition of the Laplacian matrix making the method feasible for large graphs, and (ii) it leads to a smooth estimate of the spectral content. A prototype system of spectral kernels each capturing an equal amount of energy is initially defined and subsequently warped using the signal set's ensemble energy spectral density such that the resulting subbands each capture an equal amount of ensemble energy. This approach accounts at the same time for graph topology and signal features, and it provides a meaningful interpretation of subbands in terms of coarse-to-fine representations. We also show how more simplified designs of signal-adapted decomposition of graph signals can be adopted based on ensemble energy estimates. We show the application of proposed methods on the Minnesota road graph and three different designs of brain graphs derived from neuroimaging data.