Pair Of Wavelets Research Articles

Daubechies Wavelets as Approximate Hilbert-Pairs?

A Hilbert-pair is a pair of wavelets that are Hilbert transforms of each other. A perfect reconstruction multirate filter bank defines a wavelet if the infinite product formula converges. If one chooses two filter banks arbitrarily, in general, the Hilbert transform relationship is not well approximated. This letter reports on an interesting discovery about the celebrated family of orthonormal Daubechies filters with maximum vanishing moments. It is found that if two filters whose lengths differ by four are chosen from this family, a Hilbert-pair of reasonable approximation quality is obtained. An explanation for this discovery is provided, and lessons that can be learned are discussed.

Computation of differential operators in aggregated wavelet frame coordinates

Adaptive wavelet algorithms for solving operator equations have been shown to converge with the best possible rates in linear complexity. For the latter statement, all costs are taken into account, i.e. also the cost of approximating entries from the infinite stiffness matrix with respect to the wavelet basis using suitable quadrature. A difficulty is the construction of a suitable wavelet basis on the generally non-trivially shaped domain on which the equation is posed. In view of this, recently corresponding algorithms have been proposed that require only a wavelet frame instead of a basis. By employing an overlapping decomposition of the domain, where each subdomain is the smooth parametric image of the unit cube, and by lifting a wavelet basis on this cube to each of the subdomains, the union of these collections defines such a frame. A potential bottleneck within this approach is the efficient approximation of entries corresponding to pairs of wavelets from different collections. Indeed, such wavelets are piecewise smooth with respect to mutually non-nested partitions. In this paper, considering partial differential operators and spline wavelets on the subdomains, we propose an easy implementable quadrature scheme to approximate the required entries, which allows the fully discrete adaptive frame algorithm to converge with the optimal rate in linear complexity.

Construction of trivariate biorthogonal compactly supported wavelets

Abstract In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonseparable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scaling functions.

Orthonormal Hilbert-Pair of Wavelets With (Almost) Maximum Vanishing Moments

An orthonormal Hilbert-pair consists of a pair of conjugate-quadrature-filter (CQF) banks such that the equivalent wavelet function of both banks are approximate Hilbert transforms of each other. We found that the celebrated orthonormal wavelets of Daubechies, which have maximum vanishing-moment (VM), cannot be used to construct good Hilbert-pairs. In this letter, we reduce the number of VM by one and construct a Hilbert-pair with almost maximum VM. Each pair of wavelets are time-reverse versions of each other, and the individual wavelets are of the least asymmetric type (i.e., approximate linear phase CQF)

Image analysis using a dual-tree M-band wavelet transform

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary pre-processing stage in the M-band case. While several reconstructions are possible because of the redundancy of the representation, we propose a new optimal signal reconstruction technique, which minimizes potential estimation errors. The effectiveness of the proposed M-band decomposition is demonstrated via denoising comparisons on several image types (natural, texture, seismics), with various M-band wavelets and thresholding strategies. Significant improvements in terms of both overall noise reduction and direction preservation are observed.

Numerical solution of the integral equation of the second kind by using wavelet bases of Hermite cubic splines

In this paper, We use the wavelet bases of Hermite cubic splines to solve the second kind integral equations x ( t ) - ∫ 0 1 K ( t , s ) x ( s ) d s = y ( t ) , t ∈ [ 0 , 1 ] . A pair of wavelets are constructed on the basis of Hermite cubic splines. This wavelets are in C 1 and supported on [0, 2]. Moreover, one wavelet is symmetric, and the other is anti-symmetric. This spline wavelets are then adapted to the interval [0, 1]. The computational results demonstrate the advantage of the wavelet basis.

Biorthogonal two-direction refinable function and two-direction wavelet

The concept of two-direction refinable function and two-direction wavelet is generalized to the biorthogonal setting. The definition of biorthogonal two-direction refinable function is introduced, and the biorthogonality criteria for two-direction refinable function is established. We present an algorithm for constructing a pair of biorthogonal two-direction refinable function and the corresponding biorthogonal two-direction wavelet. Finally, we give two examples illustrating how to use our method to construct a pair of biorthogonal refinable function and its corresponding biorthogonal two-direction wavelet pair.

Wavelet bases of Hermite cubic splines on the interval

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on [−1,1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0,1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. The wavelet basis is used to solve the Sturm–Liouville equation with the Dirichlet boundary condition. Numerical examples are provided. The computational results demonstrate the advantage of the wavelet basis.

The Mexican hat wavelet family: application to point-source detection in cosmic microwave background maps

We propose a new detection technique in the plane based on an isotropic wavelet family. This family is naturally constructed as an extension of the Gaussian–Mexican hat wavelet pair and for that reason we call it the Mexican hat wavelet family (MHWF). We show the performance of these wavelets when dealing with the detection of extragalactic point sources in cosmic microwave background (CMB) maps: a very important issue within the most general problem of the component separation of the microwave sky. Specifically, flat two-dimensional simulations of the microwave sky comprising all astrophysical components plus instrumental noise have been analysed for the channels at 30, 44 and 70 GHz of the forthcoming ESA Planck mission Low Frequency Instrument (LFI). We adopt up-to-date cosmological evolution models of extragalactic sources able to fit well the new data on high-frequency radio surveys and we discuss our current results on point-source detection by comparing them with those obtained using the Mexican hat wavelet (MHW) technique, which has been already proven a suitable tool for detecting point sources. By assuming a 5 per cent reliability level, the first new members of the MHWF, at their ‘optimal scale’, provide three point-source catalogues on half of the sky (at Galactic latitude |b| > 30°) at 30, 44 and 70 GHz of 639, 387 and 340 extragalactic sources, respectively. The corresponding flux detection limits are 0.38, 0.45 and 0.47 Jy. By using the same simulated sky patches and at the same frequencies as before, the MHW at its optimal scale provides 543, 322 and 311 sources with flux detection limits of 0.44, 0.51 and 0.50 Jy, respectively (5 per cent reliability level). These results show a clear improvement when we use the new members of the MHWF and, in particular, the MHW2 with respect to the MHW.

Time-varying spectral analysis in neurophysiological time series using Hilbert wavelet pairs

An analytic wavelet transform, based on Hilbert wavelet pairs, is applied to bivariate time-varying spectral estimation for neurophysiological time series. Under the assumption of an underlying block stationary process, both single-trial and ensemble studies are amenable to this method. A bootstrap procedure, which samples with replacement blocks centered around the events of interest, is proposed to identify time points for which the event-averaged magnitude squared coherence is non-zero. Clinical data sets are used to compare the wavelet-based technique with the classical Fourier-based spectral measures and highlight its ability to detect time-varying coherence and phase properties.

MULTIVARIATE SPECTRAL ANALYSIS USING HILBERT WAVELET PAIRS

We investigate the use of Hilbert wavelet pairs (HWPs) in the non-decimated discrete wavelet transform for the time-varying spectral analysis of multivariate time series. HWPs consist of two high-pass and two low-pass compactly supported filters, such that one high-pass filter is the Hilbert transform (approximately) of the other. Thus, common quantities in the spectral analysis of time series (e.g., power spectrum, coherence, phase) may be estimated in both time and frequency. Compact support of the wavelet filters ensures that the frequency axis will be partitioned dyadically as with the usual discrete wavelet transform. The proposed methodology is used to analyze a bivariate time series of zonal (u) and meridional (v) winds over Truk Island.

The Discrete Wavelet Transform for a Symmetric-Antisymmetric Multiwavelet Family on the Interval

The Chui-Lian multiwavelet family of approximation order three is extended to serve as a basis for an orthogonal discrete multiwavelet transform on the interval. This requires, in addition to the original symmetric and antisymmetric pairs of scaling functions and wavelets, four new functions of each type at each edge of the interval, i.e., one more than required just to preserve the approximation order. The criteria are given that such edge functions must satisfy in order to allow multiresolution analysis of data sequences without special constraints on endpoint behavior or use of periodic boundary conditions, and a specific choice is examined in applications to both noiseless and noisy data. These are accomplished through the development of interval-basis extensions for both an earlier multiwavelet projection-based prefilter/postfilter combination and a multiwavelet vector thresholding algorithm.

The Double-Density Dual-Tree DWT

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.

The phaselet transform-an integral redundancy nearly shift-invariant wavelet transform

This paper introduces an approximately shift invariant redundant dyadic wavelet transform - the phaselet transform - that includes the popular dual-tree complex wavelet transform of Kingsbury (see Phil. R. Soc. London A, Sept. 1999) as a special case. The main idea is to use a finite set of wavelets that are related to each other in a special way - and hence called phaselets - to achieve approximate shift-redundancy; the bigger the set, the better the approximation. A sufficient condition on the associated scaling filters to achieve this is that they are fractional shifts of each other. Algorithms for the design of phaselets with a fixed number vanishing moments is presented - building on the work of Selesnick (see IEEE Trans. Signal Processing) for the design of wavelet pairs for Kingsbury's dual-tree complex wavelet transform. Construction of two-dimensional (2-D) directional bases from tensor products of one-dimensional (1-D) phaselets is also described. Phaselets as a new approach to redundant wavelet transforms and their construction are both novel and should be interesting to the reader, independent of the approximate shift invariance property that this paper argues they possess.

Quasi-Biorthogonal Frame Multiresolution Analyses and Wavelets

We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses.

The design of approximate Hilbert transform pairs of wavelet bases

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

Characterizations of Biorthogonal Wavelets Which Are Associated with Biorthogonal Multiresolution Analyses

We first characterize the Riesz wavelets which are associated with multiresolution analyses (MRAs) and the Riesz wavelets whose duals are also Riesz wavelets. The characterizations show that if a Riesz wavelet is associated with an MRA, then it has a dual Riesz wavelet. We then improve Wang's characterization for a pair of biorthogonal wavelets to be associated with biorthogonal MRAs by showing that one of the two conditions in his characterization is redundant.

Biorthogonal Wavelet Space: Parametrization and Factorization

In this paper we study the algebraic and geometric structure of the space of compactly supported biorthogonal wavelets. We prove that any biorthogonal wavelet matrix pair (which consists of the scaling filters and wavelet filters) can be factored as the product of primitive para-unitary matrices, a pseudo identity matrix pair, an invertible matrix, and the canonical Haar matrix. Compared with the factorization results of orthogonal wavelets, it now becomes apparent that the difference between orthogonal and biorthogonal wavelets lies in the pseudo identity matrix pair and the invertible matrix, which in the orthogonal setting will be the identity matrix and a unitary matrix. Thus by setting the pseudo identity matrix pair to be the identity matrix and using the Schmidt orthogonalization method on the invertible matrix, it is very straightforward to convert a biorthogonal wavelet pair into an orthogonal wavelet.

On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. A dyadic orthonormal (or orthogonal) wavelet is a function 4 E L2 (R) (Lebesgue measure), with the property that the set {20'(2't 1): n, l E 2} is an orthonormal basis for L2 (R). For certain measurable sets E C R the normalized characteristic function 72==7rXE is the Fourier transform of such a wavelet. There are several characterizations of such sets. In [DL] they are called wavelet sets, and the corresponding wavelet is called an s-elementary wavelet. In [FW],[HWW1],[HWW2] they are the support sets of MFS (minimal frequency supported) wavelets. In [DL], a method of operator-theoretic interpolation between certain special pairs of wavelets, and between single wavelets and special families of them is developed. The interpolation pairs of wavelet sets are extensively discussed and the method yields a new construction of a class of wavelets due to Meyer. An interpolation pair is the simplest case of an interpolation family (see definitions below). Affiliated to an interpolation family is a group (under composition) of measure preserving transformations of R called the interpolation maps. Problem E in [DL] asked for a characterization of all groups which are isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. They showed, by example, that the cyclic groups 22 and 23 of order 2 and 3 could be realized in this way, but they proved no cases of other groups. The purpose of this paper is to prove that every finite group can be realized in this way. Before introducing the characterization of wavelet sets, we will need some more terminology. In [DL], two measurable sets E, F are called translation congruent modulo 2wr if there is a measurable bijection 0: E -> F such that s(s) s is an integral multiple of 2wr for each s E E. Analogously, two measurable sets G, H are called dilation congruent modulo 2 if there is a measurable bijection T: G -> H such that for each s E G there is an integer n depending on s, such that T (s) = 2s. We note in passing that if Ei and Fi are measurable subsets of R such that Ei and Received by the editors July 7, 1998 and, in revised form, November 24, 1998. 2000 Mathematics Subject Classification. Primary 42C40.

A new class of biorthogonal wavelet systems for image transform coding

We construct general biorthogonal Coifman wavelet systems, a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair. A time-domain design method is employed and closed-form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived. The resulting filter coefficients are all dyadic fractions, which is an attractive feature in the realization of multiplication-free discrete wavelet transform. Even-ordered systems in this family are symmetric, which correspond to linear-phase dual filters. In particular, three filterbanks (FBs) in this family are systematically verified to have competitive compression potential to the 9-7 tap biorthogonal wavelet FB by Cohen et al., which is currently the most widely used one in the field of wavelet transform coding. In addition, the proposed FB's have much smaller computational complexity in terms of floating-point operations required in transformation, and therefore indicate a better tradeoff between compression performance and computational complexity.