Joint filtering of graph and graph-signals

Joint filtering of signals indexed on a graph consists in filtering not only the signal, but also the graph by an appropriate downsampling. Existing methods for filtering and downsampling graph signals approximate graphs as sums of bipartite graphs or use nodal domains of the Laplacian. Here, a different method is introduced, and is based on the partitioning in meaningful subgraphs of the graph itself, e.g. network's communities; this partition may be interpreted as a coarsening of the graph and may also be tailored to be aware of the signal structure. A method is proposed to create filterbanks that compute, for graph signals, an approximation and several details using the partition to downsample the graph. This means that we jointly filter the graph and the graph signal; it leads to the design of a new subgraph-based filterbank for graph signals. This design is tested on simple examples for compression and denoising.