Wavelet Subspaces Research Articles

A Quadrature Formula Based On Sampling In Meyer Wavelet Subspaces

In this paper we discuss a weighted trapezoidal rule based on sampling in Meyer wavelet subspaces. For a wide class of functions, we obtain convergence and error bounds. Some examples are given to construct sampling functions.

Application of Wavelet Shrinkage to Solving the Sideways Heat Equation

The sideways heat equation is considered in terms of the ill-posed operator equation Au = g, g ∈ R(A) ⊂ L2(ℝ), with a given noisy right hand side. For a reconstruction of the solution from indirect data the dual least square method generated by the family of Meyer wavelet subspaces is applied. An explicit relation between the truncation level of the wavelet expansion and the data error bound is found under which convergence results including error estimation are obtained. Next, a certain simple nonlinear modification of the method based on local refinements of the wavelet expansion of the noisy data is investigated. Moreover, it is shown that an accuracy of the solution reconstruction can be improved by adding some sufficiently large coefficients on the level higher than that indicated by the convergence theorem.

Wavelets generated by using discrete singular convolution kernels

This paper explores the connection between wavelet methods and an efficient computational algorithm - the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cases of the present DSC kernel-generated wavelets. A family of wavelet generators proposed in this paper are found to form an infinite-dimensional Lie group which has an invariant subgroup of translation and dilation. If DSC kernels form an orthogonal system, they are found to span a wavelet subspace in a multiresolution analysis.

Embedding anda prioriwavelet-adaptivity for Dirichlet problems

The accuracy of the domain embedding method from [A. Rieder, Model. Math. Anal. Numer. 32 (1998) 405-431] for the solution of Dirichlet problems suffers under a coarse boundary approximation. To overcome this drawback the method is furnished with an a priori (static) strategy for an adaptive approximation space refinement near the boundary. This is done by selecting suitable wavelet subspaces. Error estimates and numerical experiments validate the proposed adaptive scheme. In contrast to similar, but rather theoretical, concepts already described in the literature our approach combines a high generality with an easy-to-implement algorithm.

On the translation invariance of wavelet subspaces

An examination of the translation invariance of V0 under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.

Classification of Wavelet Bases by Translation Subgroups and Nonharmonic Wavelet Bases

The structure of the set S of shiftable points of wavelet subspaces is researched in this paper. We prove that S = ℝ or $$ S = \frac{1} {q}\mathbb{Z} $$ where q∈ℕ. The spectral and functional characterizations for the shiftability are given. Furthermore, the nonharmonic wavelet bases are discussed.

Sampling theorem for wavelet subspaces: error estimate and irregular sampling

The error estimate is useful in the application of the sampling theorem. For the classical Shannon sampling theorem, various errors are widely studied, but for the sampling theorem in general wavelet subspaces, only the aliasing error is studied. In this paper, we study three other errors: truncation error, amplitude error, and time-jitter error. With the same technique, a result on irregular sampling is improved.

Truncation Error on Wavelet Sampling Expansions

We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L2(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions.

Deconvolution Using Meyer Wavelets

In this paper, we have a procedure based on band limited orthogonal wavelets, the Meyer wavelets, to solve convolution equations of the first kind, which is usually an ill-posed problem. The problem will be converted into a well-posed problem in the scaling subspaces, provided that the kernel k ∈ L1(R )a ndˆ(ω) � . In the caseˆ(ω )h as a single zero, we search for a solution in the wavelet subspaces, which can be used to solve the problem numerically. Results related to the convergence rate and error bounds are obtained. However, the stability of the discrete system depends on the resolution level m.

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V0/L2 (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf e V0-Several examples are given.

Sampling theorem for multiwavelet subspaces

Sampling theorem on multiwavelet subspaces is established. Necessary and sufficient conditions are obtained. The result covers the Shannon’s sampling theorem and the early results on the sampling theorem for wavelet subspaces.

Irregular sampling theorems for wavelet subspaces

From the Paley-Wiener 1/4-theorem, the finite energy signal f(t) can be reconstructed from its irregularly sampled values f(k+/spl delta//sub /spl kappa//) if f(t) is band-limited and sup/sub /spl kappa//|/spl delta//sub /spl kappa//|<1/4. We consider the signals in wavelet subspaces and wish to recover the signals from its irregular samples by using scaling functions. Then the method of estimating the upper bound of sup/sub /spl kappa//|/spl delta//sub /spl kappa//| such that the irregularly sampled signals can be recovered is very important. Following the work done by Liu and Walter (see J. Fourier Anal. Appl., vol.2, no.2, p.181-9, 1995), we present an algorithm which can estimate a proper upper bound of sup/sub /spl kappa//|/spl delta//sub /spl kappa//|. Compared to Paley-Wiener 1/4-theorem, this theorem can relax the upper bound for sampling in some wavelet subspaces.

Three-band compactly supported orthogonal interpolatingscaling function

The authors construct a three-band compactly supported orthogonal scaling function with an integer-shifted sampling property. Walter&apos;s sampling theorem for wavelet subspaces corresponding to this scaling function has a compactly supported interplant. Therefore, the signals in multiresolution spaces can be reconstructed quickly and accurately without any truncation errors.

Suppression of speckle in synthetic aperture radar images using wavelet

A filter for suppressing speckle in synthetic aperture radar (SAR) images utilizing wavelet is proposed. The filter suppresses speckle by reducing the amplitude of the detail images in wavelet subspaces, while preserving edges by releasing the amplitude reduction around edges; information on edges, contained in the detail images, is utilized for edge detection. Simulations and application to SAR images have shown that the performance of the filter is satisfactory in both smoothing and edge preservation, and in generating visually-natural images as well.

Generalized sampling theorems in multiresolution subspaces

It is well known that under very mild conditions on the scaling function, multiresolution subspaces are reproducing kernel Hilbert spaces (RKHSs). This allows for the development of a sampling theory. In this paper, we extend the existing sampling theory for wavelet subspaces in several directions. We consider periodically nonuniform sampling, sampling of a function and its derivatives, oversampling, multiband sampling, and reconstruction from local averages. All these problems are treated in a unified way using the perfect reconstruction (PR) filter bank theory. We give conditions for stable reconstructions in each of these cases. Sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported, except in trivial cases. This restriction no longer applies for the generalizations we study here, due to the existence of FIR PR banks. In fact, with nonuniform sampling, oversampling, and reconstruction from local averages, we can guarantee compactly supported synthesizing functions. Moreover, local averaging schemes have additional nice properties (robustness to the input noise and compression capabilities). We also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis. After this, we extend the sampling theory to random processes. We require autocorrelation functions to belong to some subspace related to wavelet subspaces. It turns out that we cannot recover random processes themselves (unless they are bandlimited) but only their power spectral density functions. We consider both uniform and nonuniform sampling.

On gibb's phenomenon for sampling series in wavelet subspaces

Let {Vm be a multiresolution analysis of L2(R) such that a sampling function S for V0 exists. Then we show the sampling approximation of a function in H0,α1/2 onto Vm converges to it uniformly as m→∞. Also Gibbs phenomenon for this sampling expension is analyzed.

Irregular sampling for spline wavelet subspaces

Spline wavelets /spl psi//sub m/(t) are important in time-frequency localization due to (i) /spl psi//sub m/ can be arbitrarily close to the optimal case as m is sufficiently large, (ii) /spl psi//sub m/ has compact support and simple analytic expression, which lead to effective computation. Although the spline wavelet subspaces are so simple, Walter's well-known sampling theorem does not hold if the order of spline m is even. Moreover, when irregular sampling is considered in these spaces, it is hard to determine the sampling density, which is a serious problem in applications, in this correspondence, a general sampling theorem is obtained for m/spl ges/3 in the sense of iterative construction and the sampling density /spl delta//sub m/ is estimated.

On the convergence of wavelet-based iterative signal extrapolation algorithms

A generalized Papoulis-Gerchberg (PG) algorithm for signal extrapolation based on the wavelet representation has been recently proposed by Xia, Kuo and Zhang. In this research, we examine the convergence property and the convergence rate of several signal extrapolation algorithms in wavelet subspaces. We first show that the generalized PG algorithm converges to the minimum norm solution when the wavelet bases are semi-orthogonal (or known as the prewavelet). However, the generalized PG algorithm converges slowly in numerical implementation. To accelerate the convergence rate, we formulate the discrete signal extrapolation problem as a two-step process and apply the steepest descent and conjugate gradient methods for its solution. Numerical experiments are given to illustrate the performance of the proposed algorithms.

Irregular Sampling in Wavelet Subspaces

As a particular wavelet subspace, the Paley-Wiener space \(B_{\pi}\) has both regular and irregular sampling theorems. A regular sampling theorem in general wavelet subspaces has been established for several years. In this paper, we discuss the irregular sampling problem in wavelet subspaces.

Signal Extrapolation in Wavelet Subspaces

The Papoulis–Gerchberg (PG) algorithm is well known for band-limited signal extrapolation. The authors consider the generalization of the PG algorithm to signals in the wavelet subspaces in this research. The uniqueness of the extrapolation for continuous-time signals is examined, and sufficient conditions on signals and wavelet bases for the generalized PG (GPG) algorithm to converge are given. A discrete GPG algorithm is proposed for discrete-time signal extrapolation, and its convergence is investigated. Numerical examples are given to illustrate the performance of the discrete GPG algorithm.