Homogeneous approximation property for continuous wavelet transforms

The homogeneous approximation property (HAP) for frames is useful in practice and has been developed recently. In this paper, we study the HAP for the continuous wavelet transform. We show that every pair of admissible wavelets possesses the HAP in L² sense, while it is not true in general whenever pointwise convergence is considered. We give necessary and sufficient conditions for the pointwise HAP to hold, which depends on both wavelets and functions to be reconstructed.     

Homogeneous approximation property for wavelet frames

The homogeneous approximation property (HAP) of wavelet frames is useful in practice since it means that the number of building blocks involved in a reconstruction of f up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame generated with functions satisfying some moderate decay conditions possesses the HAP. Our result improves a recent work of Heil and Kutyniok’s. Moreover, for wavelet frames generated with separable time-scale parameters, i.e., wavelet frames of the form

On the Orthogonality of Frames and the Density and Connectivity of Wavelet Frames

We examine some recent results of Bownik on density and connectivity of the wavelet frames. We use orthogonality (strong disjointness) properties of frame and Bessel sequences, and also properties of Bessel multipliers (operators that map wavelet Bessel functions to wavelet Bessel functions). In addition we obtain an asymptotically tight approximation result for wavelet frames.     

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.     

Homogeneous Approximation Property for Multivariate Continuous Wavelet Transforms

The homogeneous approximation property (HAP) for the continuous wavelet transform is useful in practice because it means that the measure of the building area involved in a reconstruction of a function up to some error is essentially invariant under timescale shifts. For the univariate case, it was shown that the pointwise HAP holds if and only if the Fourier transforms of both wavelets and the function to be reconstructed are compactly supported on ??{0}. In this paper, we study the HAP for multivariate wavelet transforms. We show that similar results hold for this case. However, the above condition is only sufficient but not necessary if the dimension of the variable is greater than 1, which is different from the univariate case. We also get a convergence result on the inverse of wavelet transforms, which improves similar results by Daubechies and Holschneider and Tchamitchain.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

Approximation by radial bases and neural networks

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

A new approach to fuzzy wavelet system modeling

In this paper, we propose simple but effective two different fuzzy wavelet networks (FWNs) for system identification. The FWNs combine the traditional Takagi–Sugeno–Kang (TSK) fuzzy model and discrete wavelet transforms (DWT). The proposed FWNs consist of a set of if–then rules and, then parts are series expansion in terms of wavelets functions. In the first system, while the only one scale parameter is changing with it corresponding rule number, translation parameter sets are fixed in each rule. As for the second system, DWT is used completely by using wavelet frames. The performance of proposed fuzzy models is illustrated by examples and compared with previously published examples. Simulation results indicate the remarkable capabilities of the proposed methods. It is worth noting that the second FWN achieves high function approximation accuracy and fast convergence.     

小波框架的某些判定准则

本文首先给出了希尔伯特空间H上两个半小波框架序列成为小波框架的一个充分条件,该结论与框架的扰动理论有关.然后建立了通过膨胀与平移母函数生成L~2(R)上的小波框架的膨胀参数,平移参数以及母函数的一些充分条件.这些结果推广了小波框架理论中经典文献中的相关结论.我们还对贝塞尔序列进行了讨论,并得到了一些有趣的结论.     

Directional Haar wavelet frames on triangles

Traditional wavelets are not very effective in dealing with images that contain orientated discontinuities (edges). To achieve a more efficient representation one has to use basis elements with much higher directional sensitivity. In recent years several approaches like curvelets and shearlets have been studied providing essentially optimal approximation properties for images that are piecewise smooth and have discontinuities along C²-curves. While curvelets and shearlets have compact support in frequency domain, we construct directional wavelet frames generated by functions with compact support in time domain. Our Haar wavelet constructions can be seen as special composite dilation wavelets, being based on a generalized multiresolution analysis (MRA) associated with a dilation matrix and a finite collection of ‘shear’ matrices. The complete system of constructed wavelet functions forms a Parseval frame. Based on this MRA structure we provide an efficient filter bank algorithm. The freedom obtained by the redundancy of the applied Haar functions will be used for an efficient sparse representation of piecewise constant images as well as for image denoising.     

Approximations for Gabor and wavelet frames

Let be a frame vector under the action of a collection of unitary operators . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.

     

A new approach to fuzzy wavelet system modeling

In this paper, we propose simple but effective two different fuzzy wavelet networks (FWNs) for system identification. The FWNs combine the traditional Takagi–Sugeno–Kang (TSK) fuzzy model and discrete wavelet transforms (DWT). The proposed FWNs consist of a set of if–then rules and, then parts are series expansion in terms of wavelets functions. In the first system, while the only one scale parameter is changing with it corresponding rule number, translation parameter sets are fixed in each rule. As for the second system, DWT is used completely by using wavelet frames. The performance of proposed fuzzy models is illustrated by examples and compared with previously published examples. Simulation results indicate the remarkable capabilities of the proposed methods. It is worth noting that the second FWN achieves high function approximation accuracy and fast convergence.     

Wavelets on S3 and SO(3)—Their construction,relation to each other and Radon transform of wavelets on SO(3)

The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S³ and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L²‐functions on S³. Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S² for every fixed detection direction. Copyright © 2010 John Wiley & Sons, Ltd.     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.     

Periodic wavelet frames

In this paper, starting from any two functions satisfying some simple conditions, using a periodization method, we construct a dual pair of periodic wavelet frames and show their optimal bounds. The obtained periodic wavelet frames possess trigonometric polynomial expressions. Finally, we present two examples to explain our theory.     

Directional Frames for Image Recovery: Multi-scale Discrete Gabor Frames

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments.     

Adaptive wavelet methods for the stochastic Poisson equation

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ? ^d . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.     

多元小波变换及L^2（R^p）框架

提出了一般形式不可分离变量的多元小波变换,通常的小汉变换只是本的特例,进而相应地构造了多元小波框架以及多元函数小波框架展开式。     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.