Convergence Of Cascade Algorithms Research Articles

Analysis and construction of a family of refinable functions based on generalized Bernstein polynomials

In this paper, we construct a new family of refinable functions from generalized Bernstein polynomials, which include pseudo-splines of Type II. A comprehensive analysis of the refinable functions is carried out. We then prove the convergence of cascade algorithms associated with the new masks and construct Riesz wavelets whose dilation and translation form a Riesz basis for $L_{2}(\mathbb{R})$ . Stability of the subdivision schemes, regularity and approximation orders are obtained. We also illustrate the symmetry of the corresponding refinable functions.

Compactly Supported Tight and Sibling Frames Based on Generalized Bernstein Polynomials

We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.

Generalized Pseudo-Butterworth Refinable Functions

In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L 2(ℝ) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.

Convergence and error estimate of cascade algorithms with infinitely supported masks in L p (ℝ s )

The cascade algorithm plays an important role in computer graphics and wavelet analysis. In this paper, we first investigate the convergence of cascade algorithms associated with a polynomially decaying mask and a general dilation matrix in Lp(ℝs) (1 ⩾ p ⩾ ∞) spaces, and then we give an error estimate of the cascade algorithms associated with truncated masks. It is proved that under some appropriate conditions if the cascade algorithm associated with a polynomially decaying mask converges in the Lp-norm, then the cascade algorithms associated with the truncated masks also converge in the Lp-norm. Moreover, the error between the two resulting limit functions is estimated in terms of the masks.

A CONSTRUCTION OF CONVERGENT CASCADE ALGORITHMS IN SOBOLEV SPACES

Cascade algorithms play an important role in wavelet analysis and computer graphics. The paper considers the convergence of cascade algorithms in Sobolev spaces. With the help of the factorization of matrix masks, we give a sufficient condition for the convergence. The condition is expressed in the time domain. More importantly, an algorithm for the construction of convergent cascade algorithms in Sobolev space starting from any matrix mask satisfying a mild condition is presented. Examples are given to illustrate our theorems.

Cascade algorithm and multiresolution analysis on the Heisenberg group

Abstract In this paper we investigate the relationship between the convergence of cascade algorithm and orthogonal (or biorthogonal) multiresolution analysis on the Heisenberg group. It is proved that the (strong) convergence of cascade algorithm together with the perfect reconstruction condition induces an orthogonal multiresolution analysis and vice versa. Similar results are also proved for biorthogonal multiresolution analysis.

Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function σ0, a cascade sequence (σn)∞n=1 is constructed by the iteration σn = Caσn-1,n = 1, 2, …, where Ca is defined by $$C_a g = \sum\limits_{\alpha \in \mathbb{Z}} {a(\alpha )g(2 \cdot - \alpha ), g \in L_p (\mathbb{R}).} $$ . In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e., ‖φn+1-φn‖Lp(ℝ)= O(ϱn) for some ϱ ∈ (0,1). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.

Multivariate Refinement Equations and Convergence of Cascade Algorithms in L p (0 < p < 1) Spaces

We consider the solutions of refinement equations written in the form $$ \begin{array}{*{20}c} {{\varphi {\left( x \right)} = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a{\left( \alpha \right)}\varphi {\left( {Mx - \alpha } \right)} + g{\left( x \right)}} },}} & {{x \in \mathbb{R}^{s} ,}} \\ \end{array} $$ where the vector of functions ϕ = (ϕ 1, ..., ϕ r ) T is unknown, g is a given vector of compactly supported functions on ℝ s , a is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite interval. The cascade algorithm with mask a, g, and dilation M generates a sequence ϕ n , n = 1, 2, ..., by the iterative process $$ \begin{array}{*{20}c} {{\varphi _{n} {\left( x \right)} = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a{\left( \alpha \right)}\varphi _{{n - 1}} {\left( {Mx - \alpha } \right)} + g{\left( x \right)}} },}} & {{x \in \mathbb{R}^{s} ,}} \\ \end{array} $$ from a starting vector of function ϕ 0. We characterize the L p -convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness property of the solutions of the refinement equations associated with the homogeneous refinement equation.

Convergence of Cascade Algorithms in Sobolev Spaces for Perturbed Refinement Masks

In this paper, the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks.

Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets

The cascade algorithm with mask a and dilation M generates a sequence \(\phi_n, n=1,2,\ldots, \) by the iterative process

Norms Concerning Subdivision Sequences and Their Applications in Wavelets

Let a:=( a(α)) α∈ Z s be a finitely supported sequence of r× r matrices and M be a dilation matrix. The subdivision sequence {( a n (α)) α∈ Z s : n∈ N } is defined by a 1= a and a n+1 (α)= ∑ β∈ Z s a n (β)a(α−Mβ), α∈ Z s , n∈ N. Let 1≤ p≤∞ and f=( f 1,…, f r ) T be a vector of compactly supported functions in L p ( R s ). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L p -norms of the combinations of the shifts of f with the subdivision sequence coefficients: ‖ ∑ α∈ Z s a n (α)f(x−α)‖ p . Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method.

Convergence of Cascade Algorithms in Sobolev Spaces Associated with Multivariate Refinement Equations

This paper is concerned with multivariate inhomogeneous refinement equations written in the form ϕ(x)=∑α∈Zsa(α)ϕ (Mx−α)+g(x),x∈Rs, where ϕ is the unknown function defined on the s-dimentional Euclidean space Rs,g is a given compactly supported function on Rs,a is a finitely supported sequence on Zs, and M is an s×s dilation matrix with m=|detM|. Let ϕ0 be an initial function in the Sobolev space Wk2(Rs). For n=1,2,…, define ϕn(x)=∑α∈Zsa(α)ϕn−1(Mx−α)+g(x),x∈Rs. In this paper, we give a characterization for the strong convergence in the Sobolev space Wk2(Rs)(k∈N) of the cascade sequence (ϕn)n∈N for the case in which M is isotropic.

Convergence of cascade algorithms introduced by I. Daubechies

Orthonormal wavelets are often constructed by an iteration procedure commonly referred to as thecascade algorithm. We obtain an asymptotic expansion and deduce necessary and sufficient conditions for convergence in Sobolev norms.