Cascade Algorithms in Wavelet Analysis † Rong -
نویسنده
چکیده
In this paper we survey some recent results on cascade algorithms. Let a be a finitely supported sequence on ZZ. The cascade operator Qa is the the linear operator on Lp(IR) (1 ≤ p ≤ ∞) given by Qaf := ∑ j∈Z a(j)f(2 · − j), f ∈ Lp(IR). The iteration scheme Qaf (n = 1, 2, . . .) is called the cascade algorithm associated with a. The Lp convergence of a cascade algorithm is characterized in terms of the p-norm joint spectral radius of two matrices associated with the corresponding mask. For the special case p = 2, convergence of a cascade algorithm is characterized in terms of the spectrum of the transition matrix associated with the mask. Then the basic theory on cascade algorithms is employed to give a unified treatment of orthogonal wavelets, biorthogonal wavelets, and fundamental refinable functions. Furthermore, we give a comprehensive review of biorthogonal wavelet bases. Our methods can be used to deal with more complicated problems such as biorthogonal wavelet bases on bounded domains. Finally, we extend our study of cascade algorithms to high dimensional spaces. † Research was supported in part by NSERC Canada under Grants # OGP 121336. §
منابع مشابه
Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets
The cascade algorithm with mask a and dilation M generates a sequence φ n , n = 1, 2,. .. , by the iterative process φ n (x) = α∈Z s a(α)φ n−1 (Mx − α) x ∈ R s , from a starting function φ 0 , where M is a dilation matrix. A complete characterization is given for the strong convergence of cascade algorithms in Sobolev spaces for the case in which M is isotropic. The results on the convergence o...
متن کاملPointwise Convergence of Bounded Cascade Sequences
The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function φ0, a cascade sequence (φn)n=0 is constructed by the iteration φn = Caφn−1, n = 1, 2, . . . , where Ca is defined by Cag = ∑ α∈Z a(α)g(2 · −α), g ∈ Lp(R). In this paper, under a condition that the sequence (φn)n=0 is bounded in L∞(R), we prove that the following three statements are ...
متن کاملRefinable Functions and Cascade Algorithms in Weighted Spaces with Hölder Continuous Masks
Refinable functions and cascade algorithms play a fundamental role in wavelet analysis, which is useful in many applications. In this paper we shall study several properties of refinable functions, cascade algorithms and wavelets, associated with Hölder continuous masks, in the weighted subspaces L2,p,γ(R) of L2(R), where 1 6 p 6 ∞, γ > 0 and f ∈ L2,p,γ(R) means ‖f‖L2,p,γ(R) := ∥∥∥∥∥ ∑ k∈Z ∣∣êγ...
متن کاملWavelets and Their Associated Operators
This article is devoted to the study of wavelets based on the theory of shift-invariant spaces. It consists of two, essentially disjoint, parts. In the rst part, the berization of the analysis operator of a shift-invariant system is discussed. That berization applies to wavelet systems via the notion of quasi-wavelet systems, and leads to the theory of wavelet frames. Highlights in this theory ...
متن کاملComparative Analysis of Image Denoising Methods Based on Wavelet Transform and Threshold Functions
There are many unavoidable noise interferences in image acquisition and transmission. To make it better for subsequent processing, the noise in the image should be removed in advance. There are many kinds of image noises, mainly including salt and pepper noise and Gaussian noise. This paper focuses on the research of the Gaussian noise removal. It introduces many wavelet threshold denoising alg...
متن کامل