Elimination of Edge Effects Using Spline Wavelets Which Maintain a Uniform Two-Scale Relation

Use of the compactly supported B-spline wavelet of Chui and Wang is hindered by loss of accuracy on decomposition, through truncation of weight sequences which are countably infinite. Adaptations to finite intervals often encounter significant problems with error near boundaries, called edge effects. For multiresolution analysis on a finite interval which employ the piecewise linear B-wavelet the present research provides a frontal approach to decomposition which avoids truncation of weight sequences, experiences no error at boundaries, and which exhibits a factor of three increase in computational efficiency, over the usual approach characterized by truncation of infinite weight sequences. As a further modest contribution, a simple derivation of the piecewise linear B-spline wavelet for L\sb2(R) is given. The simple technique is then applied to the derivation of supplementary boundary wavelets, which are necessary in order to complete the piecewise linear B-wavelet basis on a finite interval. There is also presented a modification to the Chui and Quak piecewise-cubic spline multiresolution analysis for the finite interval. The modification is intended to simplify implementation. Boundary scaling functions with multiple nodes at interval endpoints are rejected, in favor of the classical B-spline scaling function restricted to the interval. This necessitates derivation of revised boundary wavelets. In addition, a direct method of decomposition results in significant bandwidth reduction on solving an associated linear systems. Image distortion is reduced by employing natural spline projection. Finally, a hybrid projection scheme is proposed, which particularly for large systems further lowers operation count. Numerical experiments which try the algorithm are performed: The problems of edge detection, data compression, and data smoothing by thresholding in the wavelet transform domain are examined. The cubic B-spline wavelet yields compression ratios as high as 40 to 1 in the numerical experiments

Wavelet representation of functions defined on tetrahedrical grids

In this paper, a method for representing scalar functions on volumes is presented. The method is based on wavelets and it can be used for representing volumetric data (geometric or scalar) defifined on non structured grids. The basic contribution is the extension of wavelets to represent scalar functions on volumetric domains of arbitrary topological type. This extension is made by constructing a wavelet basis defifined on any tetrahedrized volume. This basis construction is achieved using multiresolution analysis and the lifting schemeFacultad de Informátic

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Let A∈Rd×d, d≥1 be a dilation matrix with integer entries and |detA|=2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2d+d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d=2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.The first author was partially supported by MEC/MICINN Grant #MTM2011-27998 (Spain)