Multiwavelet Frames Research Articles

Approximation by frame-like multiwavelets

Approximation properties of multivariate quasi-projection operators [Formula: see text] generated by vectors of compactly supported functions [Formula: see text], [Formula: see text] are studied. Error estimates in [Formula: see text]-norm are obtained for a wide class of such operators. For refinable function vectors [Formula: see text], [Formula: see text] quasi-projection operators are related to dual multiwavelet systems. Although the general scheme for the construction of dual multiwavelet frames in the multivariate case is known, its realization in practice is a difficult task because of the necessity of providing some additional properties. The notion of frame-like multiwavelets is a relaxed version of multiwavelet frames. Frame-like multiwavelets retain frame-type decompositions waiving the usual frame condition. This simplifies the problem of frame-like multiwavelets construction. Approximation properties of frame-like multiwavelets are established. Algorithms for constructing frame-like multiwavelets with the desired approximation order are suggested.

Multivariate Symmetric Interpolating Dual Multiwavelet Frames

The construction of symmetric multiwavelets in the multivariate case with useful in applications properties is a challenging task, mainly due to the complexity of the matrix extension problem. Nevertheless, for the interpolating case, a general technique can be developed. For an appropriate pair of symmetry group H and matrix dilation M and for a given H-symmetric interpolating refinable matrix mask, a method for the construction of H-symmetric dual refinable matrix masks with a preassigned order of sum rule is suggested. Wavelet matrix masks are constructed using a certain explicit matrix extension algorithm, and their symmetry properties are studied via its polyphase components. The resulting multiwavelet systems form dual multiwavelet frames, where wavelet functions have symmetry properties, preassigned order of vanishing moments and preassigned order of the balancing property. Several examples are presented.

A Characterization of Multi-Wavelet Dual Frames in Sobolev Spaces

For the past few years, wavelet and multi-wavelet frames have attracted interest from researchers. In this paper, we address some of these problems in the setting of the Sobolev space, and characterize of multi-wavelet dual frames in these spaces by using a pair of equations.

A尺度二元最小能量多小波框架的构造

a尺度二元最小能量多小波框架的构造

Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor

Characterization of matrix Fourier multiwavelet frames multipliers with integer dilation factor

Performance Comparison of Multiwavelet and Multicontourlet Frame Based Features for Improving Classification Accuracy in Remote Sensing Images

Performance Comparison of Multiwavelet and Multicontourlet Frame Based Features for Improving Classification Accuracy in Remote Sensing Images

Characterizations of dual multiwavelet frames of periodic functions

We prove characterizations of dual multiwavelet frames for [Formula: see text] constructed from a pair of [Formula: see text]-periodic multirefinable generators. Based on the notion of the Mixed Fundamental sequence we derive Mixed Extension Principles dictating the construction of dual wavelet masks with respect to given refinement masks and we show that these Principles characterize dual framelets.

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

In this work we prove that any pair of homogeneous dual multiwavelet frames of $$L_2(\mathbb {R}^s)$$ constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].

Characterization of matrix Fourier multipliers for A-dilation Parseval multi-wavelet frames

Let [Formula: see text] be a [Formula: see text] expansive integral matrix with [Formula: see text]. This paper investigates matrix Fourier multipliers for [Formula: see text]-dilation Parseval multi-wavelet frames, which are [Formula: see text] matrices with [Formula: see text] function entries, map [Formula: see text]-dilation Parseval multi-wavelet frames of length [Formula: see text] to [Formula: see text]-dilation Parseval multi-wavelet frames of length [Formula: see text], where [Formula: see text]. We completely characterize all matrix Fourier multipliers for [Formula: see text]-dilation Parseval multi-wavelet frames and construct several numerical examples. As Fourier wavelet frame multiplier, matrix Fourier multipliers can be used to derive new [Formula: see text]-dilation Parseval multi-wavelet frames and can help us better understand the basic properties of frame theory.

Minimum-Energy Multiwavelet Frame on the Interval[0,1]

Drawing inspiration from the idea of combining multiwavelets on the interval with frame theory organically, we study minimum-energy multiwavelet frame on the interval[0,1](MEMWFI). Firstly, left boundary multiscaling functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward, and the equivalent characterizations of MEMWFI are given. Then, two algorithms of constructing MEMWFI are proposed. Finally, the decomposition formula, reconstruction formulas, and numerical examples are given.

Construction of two-direction tight wavelet frames

We investigate the construction of two-direction tight wavelet frames. First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.

Semi-orthogonal multiwavelet frames in <italic>L</italic><sup>2</sup>(R<sup><italic>n</italic></sup>)

We investigate semi-orthogonal multiwavelet frames with matrix dilation that |det A |>1 in L 2 (R n ) and obtain some characterizations of strictly semi-orthogonal multiwavelet frames. We nd the equivalence of semi-orthogonal Parseval multiwavelet frames and Generalized Multiresolution Analysis (GMRA) multiwavelet Parseval frames. In particular, we discuss the semi-orthogonality of minimally support frequency (MSF) multi- wavelet frames, and construct some examples of semi-orthogonal multiwavelet frames by wavelet sets.

Symmetric Multiwavelet Frames with General Lattice

The multiwavelet has more desired properties than any single wavelet, such as short support, symmetry, and smoothness. In this paper, symmetric multiwavelet frames with general lattice and with any symmetric points are constructed from multiwavelet frames given. The existing result is generalized to the case of general lattice.

Symmetric Multiwavelet Frames with General Composite Dilation in Higher Dimensions

Frames play an important role in signal processing, image processing, data compression and sampling theory. In this paper, symmetric multiwavelet frames with general composite dilation and with any symmetric points are constructed from composite dilation multiwavelet frames given. At last, an example is provided to prove the theory.

Matrix Fourier multipliers for Parseval multi-wavelet frames

A Fourier multiplier for orthonormal wavelets is an L∞-function that sends every orthonormal wavelet to an orthonormal wavelet. This type of multipliers plays an important role in the study of basic properties of wavelets including some geometrical and topological properties of the wavelet theory. Matrix Fourier multipliers are matrices with L∞-function entries that map Parseval multi-wavelet frames to Parseval multi-wavelet frames. Like Fourier wavelet multiplier, matrix Fourier multipliers can be used to derive new multi-wavelet frames and can help us better understand the basic theory of multi-wavelet frame theory. In this paper we characterize all the matrix Fourier multipliers for Parseval multi-wavelet frames.

Minimum‐Energy Multiwavelet Frames with Arbitrary Integer Dilation Factor

In order to organically combine the minimum‐energy frame with the significant properties of multiwavelets, minimum‐energy multiwavelet frames with arbitrary integer dilation factor are studied. Firstly, we define the concept of minimum‐energy multiwavelet frame with arbitrary dilation factor and present its equivalent characterizations. Secondly, some necessary conditions and sufficient conditions for minimum‐energy multiwavelet frame are given. Thirdly, the decomposition and reconstruction formulas of minimum‐energy multiwavelet frame with arbitrary integer dilation factor are deduced. Finally, we give several numerical examples based on B‐spline functions.

Orthogonal Multiwavelet Frames in L2(Rd)

We characterize the orthogonal frames and orthogonal multiwavelet frames in L2(Rd) with matrix dilations of the form , where A is an arbitrary expanding d × d matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames. Finally, we give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.

Multiwavelet sampling theorem in Sobolev spaces

This paper is to establish the multiwavelet sampling theorem in Sobolev spaces. Sampling theorem plays a very important role in digital signal communication. The most classical sampling theorem is Shannon sampling theorem, which works for bandlimited signals. Recently, sampling theorems in wavelets or multiwavelets subspaces are extensively studied in the literature. In this paper, we firstly propose the concept of dual multiwavelet frames in dual Sobolev spaces (Hs(ℝ), H−s(ℝ)). Then we construct a special class of dual multiwavelet frames, from which the multiwavelet sampling theorem in Sobolev spaces is obtained. That is, for any f ∈ Hs(ℝ) with s > 1/2, it can be exactly recovered by its samples. Especially, the sampling theorem works for continuous signals in L2(ℝ), whose Sobolev exponents are greater than 1/2.

Tight Multiwavelet Frames with Different Matrix Dilations and Matrix Translations

The characterization of tight multiwavelet frames with different matrix dilations and matrix translations for L 2(R d ) is established. The result contains and further extends the generalizations that have appeared in the literature. Two sufficient conditions for affine frames are also presented.

NEW LOOK AT THE CONSTRUCTIONS OF MULTIWAVELET FRAMES

Using the fiberization technique of a shift-invariant space and the matrix characterization of the decomposition of a shift-invariant space of finite length into an orthogonal sum of singly generated shift-invariant spaces, we show that the main result in [13] can be interpreted as a statement about the length of a shift-invariant space, and give a more natural construction of multiwavelet frames from a frame multiresolution analysis of <TEX>$L^2(\mathbb{R}^d)$</TEX>.