Some New Inequalities for Wavelet Frames on Local Fields

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of science and engineering. Finding general and verifiable conditions which imply that the wavelet systems are wavelet frames is among the core problems in time-frequency analysis. In this article, we establish some new inequalities for wavelet frames on local fields of positive characteristic by means of the Fourier transform. As an application, an improved version of the Li-Jiang inequality for wavelet frames on local fields is obtained.

Multisensor image fusion algorithm using nonseparable wavelet frame transform

A muitisensor image fusion algorithm is described using 2-dimensional nonseparable wavelet frame （NWF） transform. The source muitisensor images are first decomposed by the NWF transform. Then, the NWF transform coefficients of the source images are combined into the composite NWF transform coefficients. Inverse NWF transform is performed on the composite NWF transform coefficients in order to obtain the intermediate fused image. Finally, intensity adjustment is applied to the intermediate fused image in order to maintain the dynamic intensity range. Experiment resuits using real data show that the proposed algorithm works well in muitisensor image fusion.

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.

On Characterization of Nonuniform Tight Wavelet Frames on Local Fields

In this article, we introduce a notion of nonuniform wavelet frames on local fields of positive characteristic. Furthermore, we gave a complete characterization of tight nonuniform wavelet frames on local fields of positive characteristic via Fourier transform. Our results also hold for the Cantor dyadic group and the Vilenkin groups as they are local fields of positive characteristic.

The Irregular Weighted Wavelet Frame Conjugate Gradient Algorithm

The dropping off of data during information transmission and the storage device’s damage etc.often leads the sampled data to be non-uniform.The paper, based on the stability theory of irregular wavelet frame and the irregular weighted wavelet frame operator,proposed an irregular weighted wavelet fame conjugate gradient iterative algorithm for the reconstruction of non-uniformly sampling signal. Compared the experiment results with the iterative algorithm of the Ref.[5],the new algorithm has remarkable advantages in approximation error,running time and so on.

ON NECESSARY AND SUFFICIENT CONDITIONS FOR DUALS OF WAVELET FRAMES

This paper is concerned with the characterization of the duals of wavelet frames of L(2)(R). The sufficient and necessary conditions for them are obtained.

Some Results on Wavelet Frame Packets

The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.

Pointwise Convergence and Uniform Convergence of Wavelet Frame Series

Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.

Characterizations of L_p(R) Using Tight Wavelet Frames

In this paper,a Littlewood-Paley function characterization of the spaces L p（R）,1〈p〈∞,is first established by means of the equivalent conditions of tight wavelet frames,wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles.With the above characterization,another characterization of L p（R）,1〈p〈∞,is also established in terms of the weighted l 2-norm of the wavelet frame coefficients,which can be a useful tool in harmonic analysis,approximation theory,and image processing.

A Class of Bidimensional FMRA Wavelet Frames

This paper addresses the construction of wavelet frame from a frame multiresolution analysis （FMRA） associated with a dilation matrix of determinant ±2. The dilation matrices of determinant ±2 can be classified as six classes according to integral similarity. In this paper, for four classes of them, the construction of wavelet frame from an FMRA is obtained, and, as examples, Shannon type wavelet frames are constructed, which have an independent value for their optimality in some sense.

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.

Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations,Ⅱ

The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.

PERTURBATION OF WAVELET AND GABOR FRAMES

In this work two aspects of theory of frames are presented: a side necessary condition on irregular wavelet frames is obtained, another perturbation of wavelet and Gabor frames is considered. Specifically, we present the results obtained on frame stability when one disturbs the mother of wavelet frame, or the parameter of dilatation, and in Gabor frames when the generating function or the parameter of translation are perturbed. In all cases we work without demanding compactness of the support, neither on the generating function, nor on its Fourier transform.

Conjugate Symmetric Complex Tight Wavelet Frames with Two Generators

Two algorithms for constructing a class of compactly supported conjugate symmetric complex tight wavelet framesψ={ψ1,ψ2}are derived.Firstly,a necessary and sufficient condition for constructing the conjugate symmetric complex tight wavelet frames is established.Secondly,based on a given conjugate symmetric low pass filter,a description of a family of complex wavelet frame solutions is provided when the low pass filter is of even length.When one wavelet is conjugate symmetric and the other is conjugate antisymmetric,the two wavelet filters can be obtained by matching the roots of associated polynomials.Finally,two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames which have some vanishing moments.

Irregular wavelet frames on L^2 (R^n)

In this paper, we present the conditions on dilation parameter {sj}j that ensure a discrete irregular wavelet system to be a frame on L2(Rn), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.

A class of irregular wavelet frames

THE inner product 【f, g】 of f and g in L2 （R） is defined by 【f, g】, and is the norm of f ; moreover, dx is the Fourier transform of f. ForΩ】O, we

On Parseval super-frame wavelets

Suppose that η1,...,η_n are measurable functions in L2（R）.We call the n-tuple（η1,...,ηn） a Parseval super frame wavelet of length n if {2^（k/2） η1（2~kt-l） ⊕···⊕2^（k/2） ηn（2kt-l）：k,l∈Z} is a Parseval frame for L2（R）⊕n.In high dimensional case,there exists a similar notion of Parseval super frame wavelet with some expansive dilation matrix.In this paper,we will study the Parseval super frame wavelets of length n,and will focus on the path-connectedness of the set of all s-elementary Parseval super frame wavelets in one-dimensional and high dimensional cases.We will prove the corresponding path-connectedness theorems.

Irregular wavelet frames

For the non-band-limited functionΨ, a sufficient condition is presented under which $\{ \sqrt {s_j } \psi (s_j \cdot - kb)\} $ is a frame for L2(R). The stability of these frames is studied. For the wavelets frequently used in signal processing, some concrete results are given.

Minimum-energy wavelet frame on the interval

The construction and properties of interval minimum-energy wavelet frame are systematically studied in this paper. They are as follows： 1） give the definition of interval minimum-energy wavelet frame; 2） give the necessary and sufficient conditions for the minimum-energy frames for L^2[0,1]; 3） present the construction algorithm for minimum-energy wavelet frame associated with refinable functions on the interval with any support y; 4） give the decomposition and reconstruction formulas of the minimum-energy frame on the interval [0,1],

A theory of constructing locally supported radial wavelet frame and its application

A new method for constructing locally supported radial wavelet frame or basis, which is different from the multiresolution analysis, is proposed. A continuously differentiable radial function with a local, support is chosen at first. Then a radial wavelet is obtained by the first and second derivatives of the radial function. If the radial function is both locally supported and infinitely differentiable, so is the radial wavelet. It is shown that the radial wavelet is a multidimensional dyadic one. I. Daubechies’ wavelet frame Theorem is extended from one dimension ton dimensions. It is proven that the family generated by dilations and translations from a single radial wavelet can constitute a frame inL 2(? n ). Consequently, it is concluded that the radial wavelet family generated by dilations and translations combined with their linear combination can constitute an orthonormal basis inL 2(? n ). Finally, an example of the radial wavelet, which is inseparable and with a local support and infinitely high regularity, is given based on the framework given here. As an application, a class of wavelet network for image denoising is designed, and an underrelaxation iterative fastlearning algorithm with varied learning rate is given as well.