  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.
