# Generalized Sampling on Graphs With Subspace and Smoothness Priors

## TL;DR

This paper introduces a generalized sampling framework for graph signals that accommodates arbitrary signals and different sampling and reconstruction filters, utilizing priors like subspace and smoothness for improved recovery.

## Contribution

It extends classical sampling theory to graph signals with novel correction filter design and two priors, enabling flexible and effective sampling and reconstruction methods.

## Key findings

- Effective recovery with correction filters in graph domain
- Comparison shows improved performance over existing methods
- Numerical experiments validate the proposed framework

## 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.