On wavelets interpolated from a pair of wavelet sets

We show that any wavelet, with the support of its Fourier transform small enough, can be interpolated from a pair of wavelet sets. In particular, the support of the Fourier transform of such wavelets must contain a wavelet set, answering a special case of an open problem of Larson. The interpolation procedure, which was introduced by X. Dai and D. Larson, allows us also to prove the extension property.

     

On interpolatory divergence-free wavelets

We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

     

一种利用Gabor小波特征的目标跟踪方法

小波特征跟踪是一种新的目标跟踪方法。这种方法利用Gabor小波提取目标的边界特征来组成特征模板,并由小波滤波器系数组成特征向量。在跟踪过程中不断优化特征模板的旋转、尺度及平移等几何变形参数,使得小波特征向量值在最小平方和的意义上与初始值相匹配,从而实现目标旋转、尺度及平移等几何变形时的自动跟踪。实验表明,该方法能够有效地跟踪飞机、车辆等目标,并且能够抵抗全局亮度的变化和局部遮挡的干扰。     

Sparse-grid finite-volume multigrid for 3D-problems

We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset of all possible meshes in the hierarchy, a sparse grid finite-volume discretisation can be constructed.The multigrid algorithm consists of a simple damped point-Jacobi relaxation as the smoothing procedure, while the coarse grid correction is made by interpolation from several coarser grid levels.The combination of sparse grids and multigrid with semi-coarsening leads to a relatively small number of degrees of freedom,N, to obtain an accurate approximation, together with anO(N) method for the solution. The algorithm is symmetric with respect to the three coordinate directions and it is fit for combination with adaptive techniques.To analyse the convergence of the multigrid algorithm we develop the necessary Fourier analysis tools. All techniques, designed for 3D-problems, can also be applied for the 2D case, and — for simplicity — we apply the tools to study the convergence behaviour for the anisotropic Poisson equation for this 2D case.     

Dual frames in connected with generalized sampling in shift-invariant spaces

The aim of this article is to derive stable generalized sampling in a shift-invariant space by using some special dual frames in L²(0,1). These sampling formulas involve samples of filtered versions of the functions in the shift-invariant space. The involved samples are expressed as the frame coefficients of an appropriate function in L²(0,1) with respect to some particular frame in L²(0,1). Since any shift-invariant space with stable generator is the image of L²(0,1) by means of a bounded invertible operator, our generalized sampling is derived from some dual frame expansions in L²(0,1).     

Multiscale kernels

This paper reconstructs multivariate functions from scattered data by a new multiscale technique. The reconstruction uses standard methods of interpolation by positive definite reproducing kernels in Hilbert spaces. But it adopts techniques from wavelet theory and shift-invariant spaces to construct a new class of kernels as multiscale superpositions of shifts and scales of a single compactly supported function φ. This means that the advantages of scaled regular grids are used to construct the kernels, while the advantages of unrestricted scattered data interpolation are maintained after the kernels are constructed. Using such a multiscale kernel, the reconstruction method interpolates at given scattered data. No manipulations of the data (e.g., thinning or separation into subsets of certain scales) are needed. Then, the multiscale structure of the kernel allows to represent the interpolant on regular grids on all scales involved, with cheap evaluation due to the compact support of the function φ, and with a recursive evaluation technique if φ is chosen to be refinable. There also is a wavelet-like data reduction effect, if a suitable thresholding strategy is applied to the coefficients of the interpolant when represented over a scaled grid. Various numerical examples are presented, illustrating the multiresolution and data compression effects.     

Fully adaptive multiresolution finite volume schemes for conservation laws

The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity.

     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Vieta–Fibonacci wavelets: Application in solving fractional pantograph equations

In this paper, the Vieta–Fibonacci wavelets as a new family of orthonormal wavelets are generated. An operational matrix concerning fractional integration of these wavelets is extracted. A numerical scheme is established based on these wavelets and their fractional integral matrix together with the collocation technique to solve fractional pantograph equations. The presented method reduces solving the problem under study into solving a system of algebraic equations. Several examples are provided to show the accuracy of the method.     

Numerical stability of biorthogonal wavelet transforms

For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including efficient numerical estimates if the numberL of decomposition levels is small, as well as growth estimates forL . These estimates allow easy determination of numerical stability directly from the wavelet coefficients. Examples show that many biorthogonal wavelets are in fact numerically well behaved.