eGHWT: The Extended Generalized Haar-Walsh Transform

TL;DR

The paper introduces the extended Generalized Haar-Walsh Transform (eGHWT), a multiscale graph signal transform that improves basis selection efficiency and performance over previous methods, with applications to images and matrix data.

Contribution

The eGHWT extends the GHWT by incorporating both graph-domain and sequency-domain partitions, enhancing basis selection and signal approximation capabilities.

Findings

eGHWT outperforms GHWT in basis selection and signal approximation.

The algorithm maintains $O(N \,\log N)$ computational complexity.

Demonstrated effectiveness on graph signals, images, and matrix data.

Abstract

Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important and much research has been done recently. The Generalized Haar-Walsh Transform (GHWT) developed by Irion and Saito (2014) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh-Hadamard Transforms. We propose the extended Generalized Haar-Walsh Transform (eGHWT), which is a generalization of the adapted time-frequency tilings of Thiele and Villemoes (1996). The eGHWT examines not only the efficiency of graph-domain partitions but also that of "sequency-domain" partitions simultaneously. Consequently, the eGHWT and its associated best-basis selection algorithm for graph signals significantly improve the performance of the previous GHWT with the similar computational cost, $O(N \log N)$, where $N$ is the number of nodes of an input graph. While the GHWT best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than $(1.5)^N$ possible orthonormal bases in $\mathbb{R}^N$, the eGHWT best-basis algorithm can find a better one by searching through more than $0.618\cdot(1.84)^N$ possible orthonormal bases in $\mathbb{R}^N$. This article describes the details of the eGHWT best-basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Furthermore, we also show how the eGHWT can be extended to 2D signals and matrix-form data by viewing them as a tensor product of graphs generated from their columns and rows and demonstrate its effectiveness on applications such as image approximation.