Generalized Sampling on Graphs With Subspace and Smoothness Priors

Abstract

We consider 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 can be different. We present design methods of the correction filter that compensates for these differences and can be obtained in closed form in the graph frequency domain. This paper considers 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 recovery methods both for the case when the recovery filter can be optimized and in the 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" in SI subspaces. We compare our approach with existing sampling methods in graph signal processing. The effectiveness of the proposed generalized sampling is validated numerically through several experiments.

Graph Signal Compression via Task-Based Quantization

“…In this section, we design a joint graph signal sampling and quantization scheme based on the identified relationship between such setups and task-based quantization in (P2). Directly solving (P2) is difficult due to the coupling between its optimization variables, combined with the non-linear relationship between these parameters and the statistical model of the quantization error, observed in (6). Therefore, in the following, we first design the sampling matrix based on (P2) for a fixed bit allocation, i.e., when {Mi} is given.…”

“…The basic graph sampling theory focuses on bandlimited graph signals and relates the spectral support to the number of samples required for representing the signal in a manner which allows its complete reconstruction [5]. Recently, sampling theorems for non-bandlimited graph signals, exploiting sparsity in domains other than the spectral domain, were proposed in [6], [7].…”

Design of Graph Signal Sampling Matrices for Arbitrary Signal Subspaces

“…In this case, A = UVB. Periodic graph spectrum (PGS) model [20] assumes the periodicity of the graph spectrum as follows:…”

“…Other sampling matrices have also been proposed like aggregation sampling [5] and graph frequency domain sampling [4]. Regardless of the choice of S T (and A), the best possible recovery is always given by [1,17,20] x…”

“…The sampling-then-reconstruction framework for generalized sampling of graph signals is the same as that in Hilbert spaces [20,21]. For generalized sampling, correction operation is performed between sampling and reconstruction to compensate the JH, KY, and YT are supported in part by JST PRESTO under grant JP-MJPR1935, JSPS KAKENHI under grants 19K22864 and 20H02145.…”