Abstract
In this paper, we consider the dual wavelet frames in both continuum setting, i.e., on manifolds, and discrete setting, i.e., on graphs. Firstly, we give sufficient conditions for the existence of dual wavelet frames on manifolds by their corresponding masks. Then, we present the formula of the decomposition and reconstruction for the dual wavelet frame transforms on graphs. Finally, we give a numerical example to illustrate the validity of the dual wavelet frame transformation applied to the graph data.
Highlights
Interest in signal processing algorithms in various applications has increased in recent years
Leonardi and Van De Ville introduced Meyer-like wavelets and scaling kernels that result in tight spectral graph wavelet frames in [7, 8]
This paper is primarily concerned with the construction of the dual wavelet frames on manifolds and graphs using the machinery of extension principles
Summary
Interest in signal processing algorithms in various applications has increased in recent years. Examples of these signal processing algorithms include sensor networks, transportation networks, the Internet, and social networks In these applications, data are defined on topologically complicated domains, such as high-dimensional structures, irregularly sampled spaces, and manifolds. Readers can refer to [19,20,21,22] for more construction methods and properties regarding dual wavelet frameworks in L2(Rn) Motivated by these and other applications, in this paper, we introduce and study dual wavelet frames on manifolds. Let G fl {V, E, w} denote a undirected, connected, weighted graphs, where V fl {Vn ∈ M : n = 1, ⋅ ⋅ ⋅ , K} is a discretization of a given manifold M, E ⊂ V×V is an edge set, and w : E → R+ denotes a weight function.
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