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. Our previous Generalized
Haar-Walsh Transform (GHWT) is a multiscale transform for signals on graphs,
which is a generalization of the classical Haar and Walsh-Hadamard Transforms.
This article proposes the extended Generalized Haar-Walsh Transform (eGHWT).
The eGHWT and its associated best-basis selection algorithm for graph signals
will 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 previous GHWT/best-basis algorithm seeks the
most suitable orthonormal basis for a given task among more than 1.5N possible bases, the eGHWT/best-basis algorithm can find
a better one by
searching through more than 0.618 ⋅ (1.84)N possible bases. This article
describes the details of the eGHWT/basis-basis algorithm and demonstrates its
superiority using several examples including genuine graph signals as well as
conventional digital images viewed as graph signals.
Keywords:
Multiscale basis dictionaries, wavelets on graphs,
graph signal processing, adapted time-frequency analysis,
the best-basis algorithm