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

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 1.5^N 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