In this paper, a nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter is presented, which refines 2D point-normal pairs. The construction of the scheme is built upon the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter, by replacing the weighted binary arithmetic average in the linear scheme with the circle average. For such a nonlinear scheme, we investigate its smoothness and get that it can reach $C^{1}$ C 1 with suitable choices of the tension parameter when degree $m\geq 3$ m ≥ 3 . Besides, the nonlinear scheme can reconstruct the circle and the selection of parameters and initial normal vectors can effectively control the shape of the limit curves.
In this paper, a family of non-stationary combined ternary 5-point subdivision schemes with multiple variable parameters is proposed. The construction of the scheme is based on the generalized ternary subdivision scheme of order 4, which is built upon refinement of a family of generalized B-splines, using the variable displacements. For such a non-stationary scheme, we study its smoothness and get that it can generate C 2 interpolating limit curves and C 4 approximating limit curves. Besides, we investigate the exponential polynomial generation/reproduction property and approximation order. It can generate/reproduce certain exponential polynomials with suitable choices of the variable parameters, and reach approximation order 5.
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 ψ 1 , ψ 2 , and ψ 3 using the oblique extension principle (OEP), which generate a tight wavelet frame in L 2 ℝ . 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 G . Based on the sequence of framelet generators in L 2 ℝ 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.
In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.
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