CONVERGENCE OF CASCADE ALGORITHMS AND SMOOTHNESS OF REFINABLE DISTRIBUTIONS

In this paper, the author at first develops a method to study convergence of the cascadealgorithm in a Banach space without stable assumption on the initial (see Theorem 2.1), andthen applies the previous result on the convergence to characterizing compactly supportedrefinable distributions in fractional Sobolev spaces and Holder continuous spaces (see Theorems3.1, 3.3, and 3.4). Finally the author applies the above characterization to choosing appropriateinitial to guarantee the convergence of the cascade algorithm (see Theorem 4.2).

Multivariate Refinement Equations and Convergence of Cascade Algorithms in L_(p)(0

We consider the solutions of refinement equations written in the form$$\varphi \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi \left( {Mx - \alpha } \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$where the vector of functions } = (}1, ..., }r)T is unknown, g is a given vector of compactly supported functions on A^s, a is a finitely supported sequence of r 2 r matrices called the refinement mask, and M is an s 2 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 }n, n = 1, 2, ..., by the iterative process$$\varphi _n \left( x \right) = \sum\limits_{\alpha \in \Zopf^s} {a\left( \alpha \right)\varphi _{n - 1} \left(Mx - \alpha \right) + g\left( x \right),\,\,\,x \in \Ropf^s} $$from a starting vector of function }0. We characterize the Lp-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 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 L p (R s) (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 L p-norm,then the cascade algorithms associated with the truncated masks also converge in the L p-norm.Moreover,the error between the two resulting limit functions is estimated in terms of the masks.

Vector Cascade Algorithms with Infinitely Supported Masks in Weighted L_2 -Spaces

In this paper, we shall study the solutions of functional equations of the formΦ=∑α∈Zsa（α）Φ（M.-α）where is an r × 1 column vector of functions on the s-dimensional Euclidean space,a ：= （a（a））α∈Zs is an exponentially decaying sequence of r × r complex matrices called refinement mask and M is an s × s integer matrix such that limn∞ M-n =0. We axe interested in the question, for a mask a with exponential decay, if there exists a solution ~ to the functional equation with each function φj, j = 1,... ,r, belonging to L2（Rs） and having exponential decay in some sense？ Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by

A NOTE ON VECTOR CASCADE ALGORITHM

The focus of this paper is on the relationship between accuracy of multivariate refinable vector and vector cascade algorithm. We show that, if the vector cascade algorithm (1.5) with isotropic dilation converges to a vector-valued function with regularity, then the initial function must satisfy the Strang-Fix conditions.

Multiscale edge detection of noisy images using wavelets

The classical edge detectors work fine with the high quality pictures, but often are not good enough for noisy images because they cannot distinguish edges of different significance. The paper presented a novel approach to multiscale edge detection for noisy images using wavelet transforms based on Lipschitz regularity coefficients and a cascade algorithm. The relationship between wavelet transform and Lipschitz regularity was established. The proposed wavelet based edge detection algorithm combined the coefficients of wavelet transforms along with a cascade algorithm which significantly improves the result. The comparison between the proposed method and the classical edge detectors was carried out. The algorithm was applied to various images and its performance was discussed. The results of edge detection of contaminated images using the proposed algorithm show that it works better than the classical edge detectors.

Some Sufficient Conditions for Convergent Multivariate Subdivision Schemes with Nonnegative Finite Masks

It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes.