Two-Channel Filter Banks on Joint Time-Vertex Graphs with Oversampled Graph Laplacian Matrix

TL;DR

The paper introduces a joint time–vertex oversampled graph Laplacian to enable oversampled bipartite extensions that preserve all edges in the joint graph, facilitating redundant multiresolution representations. Building on this operator, it designs two-channel joint time–vertex oversampled filter banks with a K-coloring strategy and demonstrates perfect reconstruction in the oversampled setting. Through signal reconstruction, joint time–vertex signal denoising, and image/video denoising experiments, the approach consistently outperforms critically sampled and existing graph-filter-bank methods, highlighting gains in fidelity and robustness. The work also provides practical, efficient oversampling extensions in time and vertex domains via bipartite graph constructions and tensor-product joint extensions, with potential for broad applications in time-varying graph data.

Abstract

To address the limitations of conventional critically sampled graph filter banks in joint time-vertex signal processing, which require decomposing the joint graph into bipartite subgraphs and thus cannot fully exploit all temporal and spatial edges in a single-stage transform, we introduce the joint time-vertex oversampled graph Laplacian matrix. This operator enables the construction of bipartite extensions that preserve all edges of the original joint graph and supports redundant multiresolution representations. Based on this operator, we design two-channel joint time-vertex oversampled graph filter banks and develop efficient oversampling extensions using a $K$-coloring strategy. The proposed framework is applied to both graph signal and image/video denoising, modeling images as graph signals to leverage structural relationships. Extensive experiments demonstrate its effectiveness in decomposition, reconstruction, and denoising, achieving notable performance improvements over critically sampled and existing methods.

Figures (10)

• Figure 1: Two-channel filter bank based on critical graph sampling.
• Figure 2: Framework of joint time-vertex oversampled graph transform.
• Figure 3: Oversampled bipartite graph construction in the time and vertex domains. (a) The ring graph ($T=487$). (b) The Minnesota road network. The blue squares and red circles represent the low-pass and high-pass sets, respectively, and the light-blue edges indicate the newly added connections.
• Figure 4: Joint time-vertex signal $\mathbf{X}_1$ representation of epidemic spreading on the Minnesota Road Network. Panels (a) and (b) show the infection distributions of at the early and late stages, while (c) and (d) depict their corresponding GFT spectrum.
• Figure 5: The JFT of oversampled graph signals of infected populations under different epidemic realizations with varying models and contagion probabilities. (a) Low contagion with temporary immunity. (b) High contagion with temporary immunity. (c) Low contagion with permanent immunity. (d) High contagion with permanent immunity.

Theorems & Definitions (1)

• proof