Piecewise-Constant Representation and Sampling of Bandlimited Signals on Graphs

Signal representations on graphs are at the heart of most graph signal processing techniques, allowing for targeted signal models for tasks such as denoising, compression, sampling, reconstruction and detection. This paper studies the piecewise-constant representation of bandlimited graph signals, thereby establishing the relationship between the bandlimited graph signal and the piecewise-constant graph signal. For this purpose, we first introduce the concept of
$\epsilon$
-level piecewise-constant representation for a general signal space. Then, using a distance matrix, a single-layer piecewise-constant representation algorithm is proposed to find an
$\epsilon$
-level piecewise-constant representation for bandlimited graph signals. On this basis, we further propose a multi-layer piecewise-constant representation algorithm, which can find a node partition with as few pieces as possible to represent bandlimited graph signals piecewise within a preset error bound. Finally, as an application, we apply the node partition obtained by the multi-layer algorithm to establish a sampling theory for bandlimited signals, which does not need to compute the eigendecomposition of a variation operator in both sampling and signal reconstruction. Numerical experiments show that the proposed algorithms have good performance.