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About 423 papers for “compactly supported wavelets”

Compactly supported multi-wavelets

Wojciech Banaś · Opuscula Mathematica · 2012

In this paper we show some construction of compactly supported multi-wavelets in L 2 (R d ), d 2 which is based on the one-dimensional case, when d = 1. We also demonstrate that some methods, which are useful in the construction of wavelets with a compact support at d = 1, can be adapted to higher-dimensional cases if A M dd (Z) is an expansive matrix of a special form.

Cited by 2Open access

Compactly Supported (bi)orthogonal Wavelets

Generated By Interpolatory, Hui Ji, Zuowei Shen · 1997

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [D1] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [L] and [GM] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti-symmetric) wavelets generated by interpolatory refinable functions.

Cited by 0

Biorthogonal bases of compactly supported wavelets

Albert Cohen, Ingrid Daubechies, Jean-Christophe Feauveau · Communications on Pure and Applied Mathematics · 1992

Abstract Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.

Cited by 2,627

Asymptotic Regularity of Compactly Supported Wavelets

Hans Volkmer · SIAM Journal on Mathematical Analysis · 1995

We study the asymptotic regularity of orthonormal bases of compactly supported wavelets when their support width tends to infinity. We construct sequences of wavelets whose ratio of regularity to support width is greater than that in previously known examples.

Cited by 20

Orthonormal bases of compactly supported wavelets

Ingrid Daubechies · Communications on Pure and Applied Mathematics · 1988

Abstract 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.

Cited by 8,124

Nonlinear Bayesian estimation with compactly supported wavelets

Achim Hekler, Martin Kiefel, Uwe D. Hanebeck · KITopen · 2010

Bayesian estimation for nonlinear systems is still a challenging problem, as in general the type of the true probability density changes and the complexity increases over time. Hence, approximations of the occurring equations and/or of the underlying probability density functions are inevitable. In this paper, we propose an approximation of the conditional densities by wavelet expansions. This kind of representation allows a sparse set of characterizing coefficients, especially for smooth or piecewise smooth density functions. Besides its good approximation properties, fast algorithms operating on sparse vectors are applicable and thus, a good trade-off between approximation quality and run-time can be achieved. Moreover, due to its highly generic nature, it can be applied to a large class of nonlinear systems with a high modeling accuracy. In particular, the noise acting upon the system can be modeled by an arbitrary probability distribution and can influence the system in any way.

Cited by 5Open access

Construction of compactly supported biorthogonal wavelets II

Sherman D. Riemenschneider, Zuowei Shen · 1997

This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual refinable functions in L 2 (R s ). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given.

Cited by 0

Matrix parametrization of compactly supported orthonormal wavelets

M. Mansour · 2012

We derive a new set of necessary and sufficient conditions for the filter coefficients of the two-scale difference equation to yield an orthogonal wavelet of compact support. The conditions constitute a linear set of equations of an arbitrary decision vector of half the filter size. The vector of the filter coefficients is a differentiable function of the decision vector. The formulation enables the optimization of the filter design under any regular objective function. The proposed parametrization is used to design customized orthonormal wavelets and to reproduce the classical orthogonal wavelets as a solution of a nonlinear optimization problem.

Cited by 1

Tight frames of compactly supported affine wavelets

Wayne Lawton · Journal of Mathematical Physics · 1990

This paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by Daubechies [Commun. Pure Appl. Math. 41, 909 (1988)]. For each integer N≥1, a family of wavelet functions ψ having support [0,2N−1] is constructed such that {ψjk(x)=2j/2ψ(2jx−k) kj,k∈Z} is a tight frame of L2(R), i.e., for every f∈L2(R), f=c∑jk 〈ψjk‖f〉ψjk for some c>0. This family is parametrized by an algebraic subset VN of R4N. Furthermore, for N≥2, a proper algebraic subset WN of VN is specified such that all points in VN outside of WN yield orthonormal bases. The relationship between these tight frames and the theory of group representations and coherent states is discussed.

Cited by 138

Construction of Compactly Supported Biorthogonal Wavelets: I

Sherman D. Riemenschneider, Zuowei Shen · 1997

This paper presents a construction of compactly supported dual functions of a given box spline in L2(R s). In particular, a concrete method for the construction of compactly supported dual functions of bivariate box splines of increasing smoothness is provided.

Cited by 0

Compactly Supported Wavelets Derived From Legendre Polynomials: Spherical Harmonic Wavelets

Milde M. S. Lira, H. M. de Oliveira, Marcio Carvalho et al. · arXiv (Cornell University) · 2015

A new family of wavelets is introduced, which is associated with Legendre polynomials. These wavelets, termed spherical harmonic or Legendre wavelets, possess compact support. The method for the wavelet construction is derived from the association of ordinary second order differential equations with multiresolution filters. The low-pass filter associated with Legendre multiresolution analysis is a linear phase finite impulse response filter (FIR).

Cited by 0Open access

6. Orthonormal Bases of Compactly Supported Wavelets

Society for Industrial and Applied Mathematics eBooks · 1992

Except for the Haar basis, all the examples of orthonormal wavelet bases in the previous chapter consisted of infinitely supported functions, as a result of the orthogonalization trick (5.3.3). To construct orthonormal examples in which ψ is compactly supported, it pays to start from (or, equivalently, from the subband filtering scheme—see §5.6) rather than from ϕ or the . In §6.1 we show how to construct so that (5.1.20) is satisfied as well as (5.5.5) for some (a necessary condition to have some regularity for ψ). Not every such is associated to an orthonormal wavelet basis, however, an issue addressed in §§6.2 and 6.3. The main results of these two sections are summarized in Theorem 6.3.6, at the end of §6.3. Section 6.4 contains examples of compactly supported wavelets generating orthonormal bases. The orthonormal wavelet bases thus obtained cannot, in general, be written in a closed analytic form. Their graph can be computed with arbitrarily high precision, via an algorithm that I call the “cascade algorithm,” which is in fact a “refinement scheme” as used in computer aided design. All this is discussed in §6.5. A lot of this material goes back to Daubechies (1988b); for many of the results, better, simpler, or more general proofs have been found since, and I have given preference to these new ways of looking at things. These different approaches are borrowed mainly from Mallat (1989), Cohen (1990), Lawton (1990, 1991), Meyer (1990), and Cohen, Daubechies, and Feauveau (1992); for the link with refinement equations the references are Cavaretta, Dahmen, and Micchelli (1991) and Dyn and Levin (1990), as well as earlier papers by these authors (see §6.5).

Cited by 63