A Brief Survey of the Graph Wavelet Frame

Abstract

In recent years, the research of wavelet frames on the graph has become a hot topic in harmonic analysis. In this paper, we mainly introduce the relevant knowledge of the wavelet frames on the graph, including relevant concepts, construction methods, and related theory. Meanwhile, because the construction of graph tight framelets is closely related to the classical wavelet framelets on $ℝ$ , we give a new construction of tight frames on $ℝ$ . Based on the pseudosplines of type II, we derive an MRA tight wavelet frame with three generators ${\psi }_1,{\psi }_2$ , and ${\psi }_3$ using the oblique extension principle (OEP), which generate a tight wavelet frame in $L_2\left(ℝ\right)$ . We analyze that three wavelet functions have the highest possible order of vanishing moments, which matches the order of the approximation order of the framelet system provided by the refinable function. Moreover, we introduce the construction of the Haar basis for a chain and analyze the global orthogonal bases on a graph $\mathcal{G}$ . Based on the sequence of framelet generators in $L_2\left(ℝ\right)$ and the Haar basis for a coarse-grained chain, the decimated tight framelets on graphs can be constructed. Finally, we analyze the detailed construction process of the wavelet frame on a graph.