Discretizing continuous wavelet transforms using integrated wavelets

We present integrated wavelets as a method for discretizing the continuous wavelet transform. Using the language of group theory, the results are presented for wavelet transforms over semidirect product groups. We obtain tight wavelet frames for these wavelet transforms. Further integrated wavelets yield tight families of convolution operators independent of the choice of discretization of scale and orientation parameters. Thus these families can be adapted to specific problems. The method is more flexible than the well-known dyadic wavelet transform. We state an exact algorithm for implementing this transform. As an application the enhancement of digital mammograms is presented.     

Bayesian Inference with Wavelets: Density Estimation

Abstract

We propose a prior probability model in the wavelet coefficient space. The proposed model implements wavelet coefficient thresholding by full posterior inference in a coherent probability model. We introduce a prior probability model with mixture priors for the wavelet coefficients. The prior includes a positive prior probability mass at zero which leads to a posteriori thresholding and generally to a posteriori shrinkage on the coefficients. We discuss an efficient posterior simulation scheme to implement inference in the proposed model. The discussion is focused on the density estimation problem. However, the introduced prior probability model on the wavelet coefficient space and the Markov chain Monte Carlo scheme are general.     

Numerical stability of biorthogonal wavelet transforms

For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including efficient numerical estimates if the numberL of decomposition levels is small, as well as growth estimates forL . These estimates allow easy determination of numerical stability directly from the wavelet coefficients. Examples show that many biorthogonal wavelets are in fact numerically well behaved.     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

A two-dimensional wavelet-packet transform for matrix compression of integral equations with highly oscillatory kernel

We examine the use of wavelet packets for the fast solution of integral equations with a highly oscillatory kernel. The redundancy of the wavelet packet transform allows the selection of a basis tailored to the problem at hand. It is shown that a well chosen wavelet packet basis is better suited to compress the discretized system than wavelets. The complexity of the matrix–vector product in an iterative solution method is then substantially reduced. A two-dimensional wavelet packet transform is derived and compared with a number of one-dimensional transforms that were presented earlier in literature. By means of some numerical experiments we illustrate the improved efficiency of the two-dimensional approach.     

Nonlinear approximation schemes associated with nonseparable wavelet bi-frames

In the present paper, we study nonlinear approximation properties of multivariate wavelet bi-frames. For a certain range of parameters, the approximation classes associated with best N-term approximation are determined to be Besov spaces and thresholding the wavelet bi-frame expansion realizes the approximation rate. Our findings extend results about dyadic wavelets to more general scalings. Finally, we verify that the required linear independence assumption is satisfied for particular families of nondyadic wavelet bi-frames in arbitrary dimensions.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation

Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example. Received June 25, 1998 / Revised version received June 5, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

On the estimation of wavelet coefficients

In wavelet representations, the magnitude of the wavelet coefficients depends on both the smoothness of the represented function f and on the wavelet. We investigate the extreme values of wavelet coefficients for the standard function spaces A_k=f| ∥f^k)∥₂ ≤ 1}, k∈N. In particular, we compare two important families of wavelets in this respect, the orthonormal Daubechies wavelets and the semiorthogonal spline wavelets. Deriving the precise asymptotic values in both cases, we show that the spline constants are considerably smaller. This revised version was published online in June 2006 with corrections to the Cover Date.     

Wavelet Approximation of Periodic Functions

We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and L_p norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough.     

On wavelets interpolated from a pair of wavelet sets

We show that any wavelet, with the support of its Fourier transform small enough, can be interpolated from a pair of wavelet sets. In particular, the support of the Fourier transform of such wavelets must contain a wavelet set, answering a special case of an open problem of Larson. The interpolation procedure, which was introduced by X. Dai and D. Larson, allows us also to prove the extension property.

     

The high frequency behaviour of continuous wavelet transforms

The high frequency behaviour of continuous wavelet transforms is characterized by the number of vanishing moments of the corresponding basic wavelets. As a consequence we give satisfying answers to the following questions of theoretical as well as practical interest:

What are the differences or similarities between transforms to different wavelets?

Why do wavelet transforms react so sensitively to abrupt signal changes ?     

Testing heteroscedasticity by wavelets in a nonparametric regression model

In the nonparametric regression models, a homoscedastic structure is usually assumed. However, the homoscedasticity cannot be guaranteed a priori. Hence, testing the heteroscedasticity is needed. In this paper we propose a consistent nonparametric test for heteroscedasticity, based on wavelets. The empirical wavelet coefficients of the conditional variance in a regression model are defined first. Then they are shown to be asymptotically normal, based on which a test statistic for the heteroscedasticity is constructed by using Fan's wavelet thresholding idea. Simulations show that our test is superior to the traditional nonparametric test.     

Biorthogonal bases of compactly supported wavelets

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.     

Numerical stability of biorthogonal wavelet transforms

Biorthogonal wavelets are essential tools for numerous practical applications. It is very important that wavelet transforms work numerically stable in floating point arithmetic. This paper presents new results on the worst-case analysis of roundoff errors occurring in floating point computation of periodic biorthogonal wavelet transforms, i.e. multilevel wavelet decompositions and reconstructions. Both of these wavelet algorithms can be realized by matrix–vector products with sparse structured matrices. It is shown that under certain conditions the wavelet algorithms can be remarkably stable. Numerous tests demonstrate the performance of the results.      

Construction of Biorthogonal Discrete Wavelet Transforms Using Interpolatory Splines

We present a new family of biorthogonal wavelet and wavelet packet transforms for discrete periodic signals and a related library of biorthogonal periodic symmetric waveforms. The construction is based on the superconvergence property of the interpolatory polynomial splines of even degrees. The construction of the transforms is performed in a “lifting” manner that allows more efficient implementation and provides tools for custom design of the filters and wavelets. As is common in lifting schemes, the computations can be carried out “in place” and the inverse transform is performed in a reverse order. The difference with the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform. Our algorithm allows a stable construction of filters with many vanishing moments. The computational complexity of the algorithm is comparable with the complexity of the standard wavelet transform. Our scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas. In addition, these filters yield perfect frequency resolution.     

Dimension Functions of Rationally Dilated GMRAs and Wavelets

In this paper we study properties of generalized multiresolution analyses (GMRAs) and wavelets associated with rational dilations. We characterize the class of GMRAs associated with rationally dilated wavelets extending the result of Baggett, Medina, and Merrill. As a consequence, we introduce and derive the properties of the dimension function of rationally dilated wavelets. In particular, we show that any mildly regular wavelet must necessarily come from an MRA (possibly of higher multiplicity) extending Auscher’s result from the setting of integer dilations to that of rational dilations. We also characterize all 3 interval wavelet sets for all positive dilation factors. Finally, we give an example of a rationally dilated wavelet dimension function for which the conventional algorithm for constructing integer dilated wavelet sets fails.     

Using wavelet methods to solve noisy Abel-type equations with discontinuous inputs

One way of estimating a function from indirect, noisy measurements is to regularise an inverse of its Fourier transformation, using properties of the adjoint of the transform that degraded the function in the first place. It is known that when the function is smooth, this approach can perform well and produce estimators that have optimal convergence rates. When the function is unsmooth, in particular when it suffers jump discontinuities, an analogue of this approach is to invert the wavelet transform and use thresholding to decide whether wavelet terms should be included or excluded in the final approximation. We evaluate the performance of this approach by applying it to a large class of Abel-type transforms, and show that the smoothness of the target function and the smoothness of the transform interact in a particularly subtle way to determine the overall convergence rate. The most serious difficulties arise when the target function has a jump discontinuity at the origin; this has a considerably greater, and deleterious, impact on performance than a discontinuity elsewhere. In the absence of a discontinuity at the origin, the rate of convergence is determined principally by an inequality between the smoothness of the function and the smoothness of the transform.     

Matrix Thresholding for Multiwavelet Image Denoising

Vector thresholding is a recently proposed technique for the denoising of one-dimensional signals by means of multiwavelet shrinkage. It is more suited both to dealing with the multiwavelet vector coefficients and to taking into account the correlations which can be introduced among the starting vector coefficients by the use of a suitable prefilter. Motivated by the successful results of the multiwavelet transform when used in image processing, the aim of this paper is to extend vector thresholding to the two-dimensional case by introducing the notion of matrix thresholding. This new method allows us to easily exploit the matrix nature of the two-dimensional multiwavelet transform, and represents the natural extension of vector thresholding to the 2-D case. Afterwards, as the choice of the threshold level is very important in the practical application of thresholding methods, we propose a first attempt to extend the recently introduced method of H-curve to a multiple wavelet setting. The results of extensive numerical simulations confirm the effectiveness of our proposals and encourage us to keep going in this direction with further studies.     

Gibbs Phenomenon in Wavelet Analysis

We prove Dirichlet-type pointwise convergence theorems for the wavelet transforms and series of discontinuous functions and we examine the Gibbs ripples close to the jump location. Examples are given of wavelets without ripples, and an example (the Mexican hat) shows that the Gibbs ripple in continuous wavelet analysis can be 3.54% instead of 8.9% of the Fourier case. For the discrete case we show that there exist two Meyer type wavelets the first one has maximum ripple 3.58% and the second 9.8%. Moreover we describe several examples and methods for estimating Gibbs ripples both in continuous and discrete cases. Finally we discuss how a wavelet transform generates a summability method for the Fourier case.