Narrow Results

Let $p\in (\frac{n}{n+1},1]$ and $f\in h^p({ℝ}^n)$ be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of $h^p({ℝ}^n)$ and their dual spaces. More precisely, the authors prove that $h^1({ℝ}^n)\times \mathrm{\text{bmo}}({ℝ}^n)=L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and, for any $p\in (\frac{n}{n+1},1)$, $h^p({ℝ}^n)\times {\mathrm{\Lambda }}_{\alpha }({ℝ}^n)=L^1({ℝ}^n)+h^p({ℝ}^n)$, where $\mathrm{\text{bmo}}({ℝ}^n)$ denotes the local BMO space, ${\mathrm{\Lambda }}_{\alpha }({ℝ}^n)$, for any $p\in (\frac{n}{n+1},1)$ and $\alpha :=n(\frac{1}{p}-1)$, the inhomogeneous Lipschitz space and $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ a variant of the local Orlicz–Hardy space related to the Orlicz function $\mathrm{\Phi }(t):=\frac{t}{\mathrm{\text{log}}(e+t)}$ for any $t\in [0,\infty )$ which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃²(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

This paper considers a class of robust estimation problems for varying coefficient dynamic models via wavelet techniques, which can adapt to local features of the underlying functions and has less restriction to the smoothness of the functions. The convergence rates and asymptotic distributions of the robust wavelet-based estimator are established when the design variables are stationary short-range dependent (SRD) and the errors are long-range dependent (LRD). Particularly, a rate of convergence ${(nlogn)}^{-1/3}$ in terms of estimation consistency can be achievable when the true components satisfy certain smoothness for a LRD process. Furthermore, an asymptotic property of the proposed estimator is given to indicate the confidence level of our proposed method for varying coefficient models with LRD.

This paper presents a fusion scheme for wavelets and fractal image compression based on the self-similarity of the space-frequency plane of sub-bands after wavelet transformation of images. Various kinds of wavelet transform are examined for the characteristics of their self-similarity and evaluated for the adoption of fractal encoder. The aim of this paper is to reduce the information of the two sets of blocks involved in the fractal image compression by using the self-similarity of images. And also, the new video encoder using the fusion method of wavelets and fractal adopts the similar manner as the motion compensation technique of MPEG encoder. Experimental results show almost the same PSNR and bits rate as conventional fractal image encoder by depending on the sampled images through computer simulations.

In this paper, we propose how to embed the watermarking to an image under the wavelet transform precisely utilizing an appropriate filter for specifying the embedded position and clustering the variance of multi-resolution representation of an image. This system has a feature of the robustness against alteration attacks of malicious users utilizing the precise detection of embedded watermarks.

A Generalized Multiresolution Analysis (GMRA) associated with a wavelet is a sequence of nested subspaces of the function space ℒ²(ℝ), with specific properties, and arranged in such a way that each of the subspaces corresponds to a scale 2^m over all time-shifts n. These subspaces can be expressed in terms of a generating-wandering subspace — of the dyadic-scaling operator — spanned by orthonormal wavelet-functions — generated from the wavelet. In this paper we show that a GMRA can also be expressed in terms of subspaces for each time-shift n over all scales 2^m. This is achieved by means of "elementary" reducing subspaces of the dyadic-scaling operator. Consequently, Time-Shifts GMRA associated with wavelets, as well as "sub-GMRA" associated with "sub-wavelets" will then be introduced.

We present a preliminary investigation of compression of segmented 3D seismic volumes for the rendering purposes. Promising results are obtained on the base of 3D discrete cosine transforms followed by the SPIHT coding scheme. An accelerated version of the algorithm combines 1D discrete cosine transform in vertical direction with the 2D wavelet transform of horizontal slices. In this case the SPIHT scheme is used for coding the mixed sets of cosine-wavelet coefficients.

We study the statistical performance of multiresolution-based estimation procedures for the scaling exponents of multifractal processes. These estimators rely on the computation of multiresolution quantities such as wavelet, increment or aggregation coefficients. Estimates are obtained by linear fits performed in log of structure functions of order q versus log of scale plots. Using various and recent types of multiplicative cascades and a large variety of multifractal processes, we study and benchmark, by means of numerical simulations, the statistical performance of these estimation procedures. We show that they all undergo a systematic linearisation effect: for a range of orders q, the estimates account correctly for the scaling exponents; outside that range, the estimates significantly depart from the correct values and systematically behave as linear functions of q. The definition and characterisation of this effect are thoroughly studied. In contradiction with interpretations proposed in the literature, we provide numerical evidence leading to the conclusion that this linearisation effect is neither a finite size effect nor an infiniteness of moments effect, but that its origin should be related to the deep nature of the process itself. We comment on its importance and consequences for the practical analysis of the multifractal properties of empirical data.

In Global Positioning System (GPS), receivers use FFT-based convolvers to acquire the signals. This paper shows a robust substitute algorithm for calculating the convolution that is less sensitive to additive noise.

An operator splitting type preconditioner is presented for fast solution of linear systems obtained by Galerkin discretization of the Burton and Miller formulation for the Helmholtz equation. Our approach differs from usual boundary element treatments of the three-dimensional scattering problem because we use a basis of biorthogonal wavelets. Such wavelets result in a sparse linear system and that facilitates preconditioning and makes matrix vector products cheap to form. In this Part I of our work, we implement a biorthogonal wavelet transform on a closed surface in three dimensions. Numerical results demonstrate the gains in efficiency that are already achievable with this convenient but non-optimal implementation.

Natural boundary element approach is a promising method to solve boundary value problems of partial differential equations. This paper addresses the Neumann exterior problem of Stokes equations using the wavelet natural boundary element method. The Stokes exterior problem is reduced into an equivalent Hadamard-singular Natural Integral Equation (NIE). By virtue of the wavelet-Galerkin algorithm, the simple and accurate computational formulae of stiffness matrix are obtained. The 2^J+3 × 2^J+3 stiffness matrix is sparse and determined only by its 2^J + 3J + 1 entries. It greatly decreases the computational complexity. Also, the condition number of stiffness matrix is , where N is the discrete node number. This indicates that the proposed algorithm is more stable than that of classical finite element method. The error estimates are established for the wavelet-Galerkin approximate solution. Several numerical examples are given to evaluate the performance of our method with encouraging results.

Structure distortion evaluation allows us to directly measure the similarity between signature patterns without classification using feature vectors, which usually suffers from limited training samples. In this paper, we incorporate the merits of both global and local alignment algorithms to define structure distortion using signature skeletons identified by a robust wavelet thinning technique. A weak affine model is employed to globally register two signature skeletons and structure distortion between two signature patterns, which are determined by applying an elastic local alignment algorithm. Similarity measurement is evaluated in the form of Euclidean distance of all found corresponding feature points. Experimental results showed that the proposed similarity measurement was able to provide sufficient discriminatory information in terms of equal error rate being 18.6% with four training samples.

A financial time series analysis method based on the theory of wavelets is proposed. It is based on the transformation of data of the series in the corresponding wavelet coefficients and in the analysis of the latter, which represent the local characteristics of the series better. In particular, an algorithm for short term previsions is defined.

In this paper, we design a new family of biorthogonal wavelet transforms that are based on polynomial and discrete splines. The wavelet transforms are constructed via lifting steps, where the prediction and update filters are derived from various types of interpolatory and quasi-interpolatory splines. The transforms use finite and infinite impulse response (IIR) filters and are implemented in a fast lifting mode. We analyze properties of the generated scaling functions and wavelets. In the case when the prediction filter is derived from a polynomial interpolatory spline of even order, the synthesis scaling function and wavelet are splines of the same order. We formulate conditions for the IIR filter to generate an exponentially decaying scaling function.

We construct a familiy of band-limited wavelets with Fourier transform discontinuous at the origin.

Cascade algorithms play an important role in wavelet analysis and computer graphics. The paper considers the convergence of cascade algorithms in Sobolev spaces. With the help of the factorization of matrix masks, we give a sufficient condition for the convergence. The condition is expressed in the time domain. More importantly, an algorithm for the construction of convergent cascade algorithms in Sobolev space starting from any matrix mask satisfying a mild condition is presented. Examples are given to illustrate our theorems.

Let the operators D and T be the dilation-by-2 and translation-by-1 on , which are both bilateral shifts of infinite multiplicity. If ψ(·) in is a wavelet, then {D^mTⁿψ(·)}_{(m,n)∈ℤ²} is an orthonormal basis for the Hilbert space but the reversed set {TⁿD^mψ(·)}_{(n,m)∈ℤ²} is not. In this paper we investigate the role of the reversed functions TⁿD^mψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.

Based on the wavelet transform theory and its well emerging properties of universal approximation and multiresolution analysis, the new notion of the wavelet network is proposed as an alternative to feed forward neural networks and neuro-fuzzy for approximating arbitrary nonlinear functions. Earlier, two types of neuron models, namely, Wavelet Synapse (WS) neuron and Wavelet Activation (WA) functions neuron have been introduced. Derived from these two neuron models with different non-orthogonal wavelet functions, neural network and neuro-fuzzy systems are presented. Comparative study of wavelets with NN and NF are also presented in this paper.

Wavelet transform has emerged as a powerful tool for time-frequency analysis and signal coding favored for the interrogation of complex non-stationary signals such as the ECG signal. Measurement of timing intervals of ECG signal by automated system is highly superior to its subjective analysis. The timing interval is found from the onset and offset of the wave components of the ECG signal. Since the Daubechies wavelet is similar to the shape of the ECG signal, better detection is achieved. Discrete Wavelet Transform is easier to implement, provides multiresolution and also reduces the computational time, and thus, is used. In the pre-processing step, the base line wandering is removed from the ECG signal. Then the R peak and the QRS complexes are detected. Twenty five records from the MIT-BIH arrhythmia database are used to evaluate the proposed method. Sensitivity and positive prediction are used as performance measures. This method is very simple and detects all the R peaks (sensitivity = 100% and positive prediction = 99.86%). That is, false positive detection is very negligible and false negative detection is zero. The performance of the proposed method is better than other methods that exist in the literature.

Theoretical modeling of dynamic processes in chemical engineering often implies the numeric solution of one or more partial differential equations. The complexity of such problems is increased when the solutions exhibit sharp moving fronts. An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. This method is based on multiresolution analysis and interpolating wavelets, that dynamically adapts the collocation grid so that higher resolution is automatically attributed to domain regions where sharp features are present. Space derivatives were computed in an irregular grid by cubic splines method. The effectiveness of the method is demonstrated with some relevant examples in a chemical engineering context.