Abstract
We propose a framework for generalized sampling of graph signals that parallels sampling in shift-invariant (SI) subspaces. This framework allows for arbitrary input signals, which are not constrained to be bandlimited. Furthermore, the sampling and reconstruction filters may be different. We present design methods of the correction filter that compensate for these differences and lead to closed form expressions in the graph frequency domain. In this study, we consider two priors on graph signals: The first is a subspace prior, where the signal is assumed to lie in a periodic graph spectrum (PGS) subspace. The PGS subspace is proposed as a counterpart of the SI subspace used in standard sampling theory. The second is a smoothness prior that imposes a smoothness requirement on the graph signal. We suggest the use of recovery techniques for when the recovery filter can be optimized and under a setting in which a predefined filter must be used. Sampling is performed in the graph frequency domain, which is a counterpart of "sampling by modulation" used in SI subspaces. We compare our approach with existing sampling techniques on graph signal processing. The effectiveness of the proposed generalized sampling approach is validated numerically through several experiments.
Highlights
Sampling theory for graph signals has been recently studied with the goal of building parallels of sampling results in standard signal processing [1]–[10]
In the context of subspace sampling with a periodic graph spectrum (PGS) prior, our results allow for a perfect recovery of graph signals beyond those that are bandlimited for almost all signal and sampling spaces
A PGS subspace of a given graph G is a space of graph signals that can be expressed as a graph Fourier transform (GFT) spectrum filtered by a given generator: XPGS =
Summary
Sampling theory for graph signals has been recently studied with the goal of building parallels of sampling results in standard signal processing [1]–[10]. We expand on [40] by building a generalized graph sampling framework that allows for (perfect) recovery of graph signals beyond bandlimited signals, and parallels SI sampling for time domain signals. Following the work on general Hilbert space sampling, we show how to design graph filters that allow to best approximate the input signal under several different criteria [19], [46]–[48]. Under all recovery criteria considered, we show that reconstruction is given by the spectral graph filters, the response of which has a closed form solution that depends on the generator function, smoothness, and sampling/reconstruction filters. In the context of subspace sampling with a PGS prior, our results allow for a perfect recovery of graph signals beyond those that are bandlimited for almost all signal and sampling spaces.
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