Local-set-based Graph Signal Reconstruction

TL;DR

This paper introduces local-set-based iterative methods for reconstructing bandlimited graph signals from sampled data, leveraging frame theory and local sets to achieve faster convergence and robustness in various scenarios.

Contribution

It proposes novel local-set-based algorithms for graph signal reconstruction, with convergence proofs and improved speed over baseline methods.

Findings

Methods converge to the original signal under certain conditions.

Algorithms outperform baseline in convergence speed.

Effective in noisy and various sampling scenarios.

Abstract

Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios.