Wavelet Bi-frames On Manifolds

In this paper,we introduce wavelet bi-frame systems on manifolds with multiresolution analysis(MRA)structure and related filter banks.In particular,we are interested in properties of wavelet bi-frames on manifolds in terms of their corresponding bi-framelet filter banks.Firstly,we not only provide complete characterizations of a sequence of nonhomogeneous wavelet bi-frames for L2(M)in terms of its associated filter bank in both the finite impulse response(FIR)and infinite impulse response(?R or bandlimited)settings,but also demonstrate that using infinite impulse response filter banks enables the discretization of the continuous framelets via polynomial-exact quadrature rules,and a multi-level decomposition and reconstruction algorithm of wavelet bi-frames on manifolds via sampling operators is proposed.Secondly,the sufficient conditions for the existence of homogeneous wavelet bi-frames on manifolds are given in the FIR setting.According to the consistency of the Laplace-Beltrami operator on manifolds to the graph Laplacian,implement the transition from continuous wavelets on manifolds to discrete wavelets on graph,the formulas of the decomposition and reconstruction for the wavelet bi-frame transforms on graphs are presented.Especially,to efficiently process and analyze graph data,we show a fast algorithm for calculating wavelet bi-frame transforms on graph by low-degree Chebyshev polynomials.Finally,numerical simulations are given to demonstrate the feasibility of the proposed wavelet bi-frame transforms in graph data.