Sampling and reconstruction of sparse signals on circulant graphs – an introduction to graph-FRI

Abstract

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any Ksparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, with subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.

Analysis vs Synthesis - An Investigation of (Co)sparse Signal Models on Graphs

“…Since this requires the extension of classical signal processing theory to the graph domain, the development of a rigorous theoretical foundation is paramount. Circulant graphs have been noted for providing a link between the classical (Euclidean) and graph domain of signal processing, which previously motivated the development of sparse graph wavelet analysis and sampling theory [5], [8]. As we will discover in Sect.…”

“…Contributions: The proposed theoretical analysis is primarily conducted in an effort to capture signal sparsity in the light of the connectivity of graphs and contribute to a more rigorous foundation of GSP, while simultaneously developing crucial insight for generalized signal models beyond. Its study was initiated in a previous body of work which developed a framework for sparse graph wavelet analysis and sampling on circulant graphs and beyond [2], [5], which, however, only offered partial characterization of the underlying signal model when the graph at hand is circulant. This work extends and upgrades the study of sparse representations on graphs, by providing an intuitive and complete characterization in the context of GSP as well as discovering more wide-ranging implications for Union of Subspaces (UoS) signal models.…”

Average sampling and average splines on combinatorial graphs

“…During the last decade signal processing on graphs was developed in a number of papers, for example, in [3], [6], [12]- [20]. Many of the papers on this list considered what can be called as a "point-wise sampling".…”

Detecting Localized Categorical Attributes on Graphs

“…To simultaneously bound both type-1 and type-2 errors, the right hand side of (17) should be larger than the right hand side of (16). Thus, we obtain We can reformulate the above formula and obtain Theorem 2.…”

“…It extends classical signal processing concepts such as signals, filters, Fourier transform, frequency response, low-and highpass filtering, from signals residing on regular lattices to data residing on general graphs; for example, a graph signal models the data value assigned to each node in a graph. Recent work involves sampling for graph signals [9], [10], [11], [12], recovery for graph signals [13], [14], [15], [16], representations for graph signals [17], [18] principles on graphs [19], [20], stationary graph signal processing [21], [22], graph dictionary construction [23], graph-based filter banks [24], [25], [26], [27], denoising on graphs [24], [28], community detection and clustering on graphs [29], [30], [31], distributed computing [32], [33] and graph-based transforms [34], [35], [36]. We here consider detecting localized categorical attributes on graphs.…”