eGHWT: The Extended Generalized Haar–Walsh Transform

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, $$O(N \log N)$$ 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 $$(1.5)^N$$ ( 1.5 ) N possible orthonormal bases in $$\mathbb {R}^N$$ R N , the eGHWT best-basis algorithm can find a better one by searching through more than $$0.618\cdot (1.84)^N$$ 0.618 · ( 1.84 ) N possible orthonormal bases in $$\mathbb {R}^N$$ 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.

Multiscale Transforms for Signals on Simplicial Complexes

“…An alternative path to the creation of wavelet-like dictionaries and transforms is to first develop a hierarchical block decomposition of the domain and then use this to develop multiscale transforms [18,17,40]. These techniques rely on recursively computing bipartitions of the domain and then generating localized bases on the subsets of the domain.…”

“…For more details see on hierarchical partitioning, (specifically for the κ = 0 case), see [22,Chap. 3] and [40]. Figure 4 demonstrates such a hierarchical bipartition tree for a simple 2-complex consisting of triangles.…”

“…Unfortunately, these definitions do not generalize to non-homogeneous domains due to the lack of appropriate translation operators and dilation operators [44]. Instead, several methods have been proposed to generate similar bases, and overcomplete dictionaries to apply more abstract domains such as graphs and discretized manifolds [17,45,40]. Here, we describe a method to compute similar, piecewise-constant locally supported bases for κ-simplex valued functional spaces, which we call the (orthonormal) κ-Haar bases.…”

“…Each level of the dictionary contains an ONB whose vectors have the support of roughly half the size of the previous level. There are roughly (1.5) n possible ONBs formed by selecting different covering sets of regions from the hierarchical bipartition tree; see, e.g., [49,40] for more about the number of possible ONBs in such a hierarchical bipartition tree. Finally, we note that the computational cost of generating the entire dictionary is O (n 3 ) and that any valid hierarchical bipartition tree can be used to create a similar dictionary.…”

“…Our group have developed the graph versions of the block/local cosine and wavelet packet dictionaries for analysis of graph signals sampled at nodes so far. These include the Generalized Haar-Walsh Transform (GHWT) [17], the Hierarchical Graph Laplacian Eigen Transform (HGLET) [18], the Natural Graph Wavelet Packets (NGWPs) [7], and their relatives [20,45,40]; see also [19,21]. Some of these will be briefly reviewed in the later sections.…”