Graph Linear Canonical Transform: Definition, Vertex-Frequency Analysis and Filter Design

TL;DR

This paper introduces the graph linear canonical transform (GLCT), a novel method for graph signal processing that combines fractional Fourier, scale, and chirp transforms to improve local information capture and filter design.

Contribution

The paper defines the GLCT, explores vertex-frequency analysis in this domain, and develops filter design methods, including learning-based approaches, for enhanced graph signal processing.

Findings

GLCT enables adjustable smoothing modes for graph signals.

Vertex-frequency analysis in GLCT captures local information effectively.

Filter design methods improve image classification performance.

Abstract

This paper proposes a graph linear canonical transform (GLCT) by decomposing the linear canonical parameter matrix into fractional Fourier transform, scale transform, and chirp modulation for graph signal processing. The GLCT enables adjustable smoothing modes, enhancing alignment with graph signals. Leveraging traditional fractional domain time-frequency analysis, we investigate vertex-frequency analysis in the graph linear canonical domain, aiming to overcome limitations in capturing local information. Filter design methods, including optimal design and learning with stochastic gradient descent, are analyzed and applied to image classification tasks. The proposed GLCT and vertex-frequency analysis present innovative approaches to signal processing challenges, with potential applications in various fields.