Construction of higher multiplicity wavelets adapted to a particular task is discussed. The proposed method uses parameterization of wavelet matrices. Large classes of wavelet matrices are searched to minimize a given cost function on training data. The method is applied to a problem of detecting and possibly removing a disturbance of a certain kind from a signal. The importance of choosing the...

The theory of [RS2] is applied to yield compactly supported tight affine frames (wavelets) in L2(IR ) from box splines. The wavelets obtained are smooth piecewisepolynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness. Two bivariate constr...

We construct orthonormal wavelet bases of L2(IR) with compact support for dilation matrices of determinant 2. The key idea is to describe the set H2 of all two-dimensional (2D) scaling coefficients satisfying the orthogonality condition as an implicit function. This set includes the scaling coefficients for induced 1D wavelets. We compute the tangent space of H2 at HN , the scaling coefficients...

Tomographic reconstruction from PET data is an ill-posed problem that requires regularization. Recently, Daubechies et al. [1] proposed an l (1) regularization of the wavelet coefficients that can be optimized using iterative thresholding schemes. In this paper, we extend this approach for the reconstruction of dynamic (spatio-temporal) PET data. Instead of using classical wavelets in the tempo...

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C and supported on [−1, 1]. Moreover, one wavelet is symmetric, and the other is anti-symmetric. These spline wavelets are then adapted to the interval [0, 1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. ...

We construct classes of nonstationary wavelets generated by what we call spherical basis functions (SBFs), which comprise a subclass of Schoen-berg's positive deenite functions on the m-sphere. The wavelets are intrinsically deened on the m-sphere, and are independent of the choice of coordinate system. In addition, they may be orthogonalized easily, if desired. We will discuss decomposition, r...

The numerical solutions to the nonlinear integral equations of Hammerstein-type y(t) =f(t) + 11 k(t,s)g(s,y(s))ds, t E [0,1] with using Daubechies wavelets are investigated. A general kernel scheme basing on Daubechies wavelets combined with a collocation method is presented. The approach of creating Daubechies interval wavelets and their main properties are briefly mentioned. Also we present a...

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on [−1, 1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0, 1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. ...

The theory of RS2] is applied to yield compactly supported tight aane frames (wavelets) in L 2 (IR d) from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of \mother wavelets", however, increases with the increase of the required smoothness. Two bivariate const...

This paper is on the angle-frequency localization of periodic scaling functions and wavelets. It is shown that the uncertainty products of uniformly local, uniformly regular and uniformly stable scaling functions and wavelets are uniformly bounded from above by a constant. Results for the construction of such scaling functions and wavelets are also obtained. As an illustration, scaling function...