Sparsity is good. We like sparsity. We can make signals more sparse by transforming them. This paper proposes a novel, two-parameter method for designing a stable wavelet basis. Our goal is to determine a basis that represents a given signal as sparsely as possible. We choose the Gini index as a measure of sparsity and sparsify a signal by iteratively lifting the wavelet basis and at each step ...

Existing work on the representation of operators in one-dimensional, compactly-supported, orthonormal wavelet bases is extended to two dimensions. The non-standard form of the representation of operators is given in separable two-dimensional, periodic, orthonormal wavelet bases. The matrix representation of the partial differential operators ∂x and ∂y are constructed and a closed form formula f...

Precomputed Radiance Transfer (PRT) aims at computing global illumination effects such as soft shadows and object interreflection in realtime. This is accomplished by precomputing the factors of the Rendering Equation and projecting them into a suitable basis which allows to combine them efficiently at runtime. Several basis function have been proposed for PRT. Recently, Haar wavelets have been...

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...

This paper describes a meshless method with waveletbased nodes for the two-dimensional time-dependent simulation of semiconductor devices. In this method the solution is approximated using global radial basis functions (RBF) and distributed waveletgenerated points. This allows the computation of problems with complex-shaped boundaries and forming fine and coarse points abundance in locations wh...

O(N) methods are based on the localization properties of the density matrix in real space, an effect refered to as nearsightedness. We show that, in addition to this real-space localization there is also a localization in Fourier space. Using a basis set with good localization properties in both real and Fourier space such as wavelets, one can exploit both localization properties to obtain a de...

In this paper we show that wavelets can be used as basis functions for Galerkin methods. For differential equations, finite element or finite difference methods lead to matrices that are already shown to be sparse, but they tend to be ill-conditioned, [4]. In this paper, we show that wavelet basis gives sparse matrix with low condition number for Galerkin methods. Numerical examples are present...