Results 1  10
of
56
Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
Abstract

Cited by 573 (8 self)
 Add to MetaCart
(Show Context)
ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
Abstract

Cited by 541 (16 self)
 Add to MetaCart
(Show Context)
. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Using the refinement equation for the construction of prewavelets II
, 1991
"... ..."
(Show Context)
On the construction of multivariate (pre)wavelets
, 1992
"... A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L2(IR d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermo ..."
Abstract

Cited by 108 (14 self)
 Add to MetaCart
A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L2(IR d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.
Stability and linear independence associated with wavelet decompositions
 Proc. Amer. Math. Soc
, 1993
"... Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask ..."
Abstract

Cited by 70 (17 self)
 Add to MetaCart
(Show Context)
Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
Abstract

Cited by 69 (32 self)
 Add to MetaCart
Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
Abstract

Cited by 45 (12 self)
 Add to MetaCart
(Show Context)
The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
WaveletGalerkinMethods: An Adapted Biorthogonal Wavelet Basis
, 1993
"... In this paper we construct a compactly supported biorthogonal wavelet basis adapted to some simple differential operators. Moreover, we estimate the condition numbers of the corresponding stiffness matrices. ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
In this paper we construct a compactly supported biorthogonal wavelet basis adapted to some simple differential operators. Moreover, we estimate the condition numbers of the corresponding stiffness matrices.
Compactly supported wavelet bases for Sobolev spaces
, 2003
"... In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and ˜φ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk: = 2j/2ψ(2j ·− ..."
Abstract

Cited by 22 (8 self)
 Add to MetaCart
In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and ˜φ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk: = 2j/2ψ(2j ·−k) (j,k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H m (R), then the derivatives 2j/2ψ (m) (2j ·−k) (j,k ∈ Z) also form a Riesz basis for L2(R). Consequently, {ψjk: j,k ∈ Z} is a stable wavelet basis for the Sobolev space H m (R). The pair of φ and ˜φ are not required to be biorthogonal or semiorthogonal. In particular, φ and ˜φ can be a pair of Bsplines. The added flexibility on φ and ˜φ allows us to construct wavelets with relatively small supports.