This paper presents a construction of compactly supported biorthogonal spline wavelets in L2(IR ). In particular, a concrete method for the construction of bivariate compactly supported biorthogonal wavelets from box splines of increasing smoothness is provided. Several examples are given to illustrate the method. Key-Words:multivariate biorthogonal wavelets, multivariate wavelets, box splines,...

The use of kernels is a key factor in the success of many classification algorithms by allowing nonlinear decision surfaces. Radial basis function (RBF) kernels are commonly used but often associated with dense Gram matrices. We consider a mathematical operator to sparsify any RBF kernel systematically, yielding a kernel with a compact support and sparse Gram matrix. Having many zero elements i...

Two items will be presented in the talk. The rst one is modiication of classic Daubechies compactly supported wavelets preserving localization in time with the growth of smoothness. The second one is nonstation-ary wavelets with the use of scaling lters that vary from one scale to the next ner one. Wavelets are a tool for decomposing functions in various applications. It is closely connected wi...

We prove that a compactly supported spline function of degree k satisses the scaling equation (x) = P N n=0 c(n)(mx?n) for some integer m 2, if and only if (x) = P n p(n)B k (x?n) where p(n) are the coeecients of a polynomial P(z) such that the roots of P(z)(z ? 1) k+1 are mapped into themselves by the mapping z ! z m , and B k is the uniform B-spline of degree k. Furthermore, the shifts of for...

We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.

We present a new local Craya–Herring decomposition of three-dimensional vector fields using compactly supported biorthogonal wavelets. Therewith vector-valued function spaces are split into two orthogonal components, i.e., curl-free and divergencefree spaces. The latter is further decomposed into toroidal and poloidal parts to decorrelate horizontal from vertical contributions which are of part...

In this paper we study scaling functions of a given regularity for arbitrary dilation factor q. We classify symmetric scaling functions and study the smoothness of some of them. We also introduce a new class of continuous symmetric scaling functions, the \Batman" functions, that have very small support. Their smoothness is established.

We analyze the detection and classification of singularities of functions f = χB , where B ⊂ R and d = 2, 3. It will be shown how the set ∂B can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ∂S is a d−1 dimensional piecewise smooth manifold with d = 2 or 3, we will classify smooth and non-smooth components of ∂S. This improves pre...