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1.
MRA wavelets have been widely studied in recent years due to their applications in signal processing. In order to understand the properties of the various MRA wavelets, it makes sense to study the topological structure of the set of all MRA wavelets. In fact, it has been shown that the set of all MRA wavelets (in any given dimension with a fixed expansive dilation matrix) is path-connected. The current paper concerns a class of functions more general than the MRA wavelets, namely normalized tight frame wavelets with a frame MRA structure. More specifically, it focuses on the parallel question on the topology of the set of all such functions (in the given dimension with a fixed dilation matrix): is this set path-connected? While we are unable to settle this general path-connectivity problem for the set of all frame MRA normalized tight frame wavelets, we show that this holds for a subset of it. An s-elementary frame MRA normalized tight frame wavelets (associated with a given expansive matrix A as its dilation matrix) is a normalized tight frame wavelet whose Fourier transform is of the form $\frac{1}{\sqrt{2\pi}}\chi_{E}$ for some measurable set E?? d . In this paper, we show that for any given d×d expansive matrix A, the set of all (A-dilation) s-elementary normalized tight frame wavelets with a frame MRA structure is also path-connected. 相似文献
2.
It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly
supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction
of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C
1 orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how
to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric
wavelets for all the examples given in this paper.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
Sets K in d-dimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic
function 1
K
is a single dyadic orthonormal wavelet. The construction is characterized by its generality in the procedure, by its computational
implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1K
1, ..., 1K
L
are a family of orthonormal wavelets is treated in [27]. 相似文献
4.
Biswaranjan Behera 《Proceedings Mathematical Sciences》2001,111(4):439-463
The orthonormal basis generated by a wavelet ofL
2(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets.
We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL
2(ℝ
d
) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further,
we show how to construct various orthonormal bases ofL
2(ℝ
d
) from the multiwavelet packets. 相似文献
5.
Abderrazek Karoui 《Central European Journal of Mathematics》2008,6(4):504-525
The construction of nonseparable and compactly supported orthonormal wavelet bases of L
2(R
n
); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction
of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method
can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating
multidimensional multiwavelet bases.
相似文献
6.
Maciej Paluszyński Hrvoje Šikić Guido Weiss Shaoliang Xiao 《Journal of Geometric Analysis》2001,11(2):311-342
A tight frame wavelet ψ is an L
2(ℝ) function such that {ψ jk(x)} = {2j/2
ψ(2
j
x −k), j, k ∈ ℤ},is a tight frame for L
2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight
frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained
from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions
and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained
by other authors. 相似文献
7.
Under very minimal regularity assumptions, it can be shown that 2n−1 functions are needed to generate an orthonormal wavelet basis for L2(ℝn). In a recent paper by Dai et al. it is shown, by abstract means, that there exist subsets K of ℝn such that the single function ψ, defined by
, is an orthonormal wavelet for L2(ℝn). Here we provide methods for construucting explicit examples of these sets. Moreover, we demonstrate that these wavelets
do not behave like their one-dimensional couterparts. 相似文献
8.
This paper addresses periodic wavelet bi-frames associated with general expansive matrices. Periodization is an important method to obtain periodic wavelets from wavelets on Rd. MEP and MOEP provide us with criteria for the construction of wavelet bi-frames on Rd. Based on periodization techniques, MEP and MOEP, periodic wavelet bi-frames associated with the dyadic matrix have been constructed. However, the problem of constructing periodic wavelet bi-frames associated with general expansive matrices is still open. The geometry of a general expansive matrix is much more complicated than the dyadic matrix. In this paper, with the help of quasi-norms, MEP and MOEP we construct periodic wavelet bi-frames associated with general expansive matrices. 相似文献
9.
Frame Wavelets with Compact Supports for L^2(R^n) 总被引:1,自引:0,他引:1
De Yun YANG King Wei ZHOU Zhu Zhi YUAN 《数学学报(英文版)》2007,23(2):349-356
The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry. 相似文献
10.
Marcin Bownik 《Acta Appl Math》2009,107(1-3):195-201
We study properties of the closure of the set of tight frame wavelets. We give a necessary condition and a sufficient condition for a function to be in this closure. In particular, we show that the collection of tight frame wavelets is not dense in L 2(? n ), which answers a question posed by D. Han and D. Larson (Preprint, 2008). 相似文献
11.
A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor
frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional
Gabor frame multipliers. We prove that a L∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular
and
is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which
there is a function ∈ L∞(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame. 相似文献
12.
Sandra Saliani 《Constructive Approximation》2011,33(1):15-39
Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal
bases of L
2(ℝ), each one corresponding to a different splitting of L
2(ℝ) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and
Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet
packet system can be an orthonormal basis for L
2(ℝ) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This answer to this
question affects the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative
by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant
system machinery developed by Ron and Shen. 相似文献
13.
Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions 总被引:7,自引:0,他引:7
This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions.
It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the
trivial case; and also shown in [10,18] 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.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
14.
Demetrio Labate 《Journal of Geometric Analysis》2002,12(3):469-491
This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame
generators for shift-invariant subspaces of L2(ℝn). A number of applications of this general result are then obtained, among which are the characterization of tight frames
and dual frames for Gabor and wavelet systems. 相似文献
15.
The objective of this paper is to establish a complete characterization of tight frames, and particularly of orthonormal wavelets, for an arbitrary dilation factor a>1, that are generated by a family of finitely many functions in L2:=L2(
). This is a generalization of the fundamental work of G. Weiss and his colleagues who considered only integer dilations. As an application, we give an example of tight frames generated by one single L2 function for an arbitrary dilation a>1 that possess “good” time-frequency localization. As another application, we also show that there does not exist an orthonormal wavelet with good time-frequency localization when the dilation factor a>1 is irrational such that aj remains irrational for any positive integer j. This answers a question in Daubechies' Ten Lectures book for almost all irrational dilation factors. Other applications include a generalization of the notion of s-elementary wavelets of Dai and Larson to s-elementary wavelet families with arbitrary dilation factors a>1. Generalization to dual frames is also discussed in this paper. 相似文献
16.
We construct directional wavelet systems that will enable building efficient signal representation schemes with good direction selectivity. In particular, we focus on wavelet bases with dyadic quincunx subsampling. In our previous work (Yin, in: Proceedings of the 2015 international conference on sampling theory and applications (SampTA), 2015), we show that the supports of orthonormal wavelets in our framework are discontinuous in the frequency domain, yet this irregularity constraint can be avoided in frames, even with redundancy factor <2. In this paper, we focus on the extension of orthonormal wavelets to biorthogonal wavelets and show that the same obstruction of regularity as in orthonormal schemes exists in biorthogonal schemes. In addition, we provide a numerical algorithm for biorthogonal wavelets construction where the dual wavelets can be optimized, though at the cost of deteriorating the primal wavelets due to the intrinsic irregularity of biorthogonal schemes. 相似文献
17.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
18.
We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H
2
(ℝ)and symmetric MSF wavelets of L
2
(ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets
of L
2
(ℝ)and H
2
(ℝ).We also enumerate some of the symmetric wavelet sets of L
2
(ℝ)and all wavelet sets of H
2
(ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L
2
(ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin. 相似文献
19.
Ljiljana Arambašic Damir Bakic Rajna Rajic 《Journal of Fourier Analysis and Applications》2007,13(3):331-356
The article is devoted to dimension functions of orthonormal wavelets on the real line with dyadic dilations. We describe
properties of dimension functions and prove several characterization theorems. In addition, we provide a method of construction
of dimension functions. Various new examples of dimension functions and orthonormal wavelets are included. 相似文献
20.
Alfredo L. Gonz��lez Mar��a del Carmen Moure 《Journal of Fourier Analysis and Applications》2011,17(6):1119-1137
(Γ,a)-crystallographic multiwavelets are a finite set of functions Y = { yi}i=1L\Psi= \{ \psi ^{i}\}_{i=1}^{L}, which generate an orthonormal basis, a Riesz basis or a Parseval frame for L
2(ℝ
d
), under the action of a crystallographic group Γ, and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution
analysis of multiplicity n ((Γ,a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type (Γ,a)-MRA’s in terms of (Γ,a)-multireptiles is given. Examples of (Γ,a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided. 相似文献