Reconstruction of graph signals: Percolation from a single seeding node

Abstract

Abstract-Schemes to reconstruct signals defined in the nodes of a graph are proposed. Our focus is on reconstructing bandlimited graph signals, which are signals that admit a sparse representation in a frequency domain related to the structure of the graph. The schemes, which are designed within the framework of linear shift-invariant graph filters, consider that the signal is injected at a single seeding node. After several sequential applications of the graph-shift operator -which computes linear combinations of the information available at neighboring nodes -the seeding signal percolates across the graph. We show that if the node is allowed to change the seeding signal with each application of the shift operator, the original bandlimited signal can be recovered.Conditions under which such a recovery is feasible are identified for two different reconstruction schemes. We illustrate both reconstruction schemes in synthetic graph signals and we analyze their performance in noisy real-world scenarios.

Graph Fourier transform based on directed Laplacian

“…Fortunately, recent work toward the development of important concepts and tools, extending classical signal processing theory, including sampling and interpolation on graphs [6]- [8], graph-based transforms [9]- [13], and graph filters [14], [15] have enriched the field of graph signal processing. These tools have been utilized in solving a variety of problems such as signal recovery on graphs [16]- [18], clustering and community detection [19], [20], graph signal denoising [21], and semisupervised classification on graphs [22].…”