Construction Of Biorthogonal Wavelets Research Articles

Divergence-free Wavelets for Coherent Vortex Extraction in 3D homogeneous isotropic turbulence

In this paper we investigate the use of divergence-free wavelet bases for the Coherent Vortex Extraction (CVE) of turbulent flows. We begin with a short presentation of the construction of 3D divergence-free biorthogonal wavelets. We apply the CVE decomposition to a homogeneous isotropic turbulent flow, computed by a Direct Numerical Simulation (DNS) at resolution 2403 and upsampled to N=2563. First, the CVE is applied to the vorticity field. Using the divergence-free wavelets for the vorticity field makes sense since the vorticity also verifies an incompressibility condition when the velocity does. The coherent part of the vorticity field is reconstructed from the largest wavelet coefficients, corresponding to $ 3\% N$, while the complement constitutes the incoherent part. We show that the coherent part corresponds to the vortex tubes of the flow and retains most of the energy and enstrophy. These results are then compared to those obtained using non-divergent free wavelets, both orthogonal and biorthogonal ones. Then we also apply the CVE method, using divergence-free wavelets, to decompose the velocity field and subsequently compute the corresponding vorticity fields. The results show that the decomposition of velocity exhibit large smooth vortex structures in contrast to what is obtained with the decomposition of the vorticity.

General Solutions of Matrix Extension Associated with Multiple Channel Biorthogonal Wavelet Filters

This paper is concerned with seeking the general solutions of matrix equation \(\tilde M\left( \xi \right)M^ * \left( \xi \right) = I_s\)for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.

Efficient wavelet construction with Catmull–Clark subdivision

This paper presents an efficient biorthogonal wavelet construction with the generalized Catmull–Clark subdivision based on the lifting scheme. The subdivision wavelet construction scheme is applicable to all variants of Catmull–Clark subdivision, so it is more universal than the previous wavelet construction for the generalized bicubic B-spline subdivision. Because the analysis and synthesis algorithms of the wavelets are composed of a series of local and in-place lifting operations, they can be performed in linear time. The experiments have demonstrated the stability of the proposed wavelet analysis based on the ordinary Catmull–Clark subdivision. Moreover, the resulting Catmull–Clark subdivision wavelets have better fitting quality than the generalized bicubic B-spline subdivision wavelets at a similar computation cost.

Wavelets on manifolds: An optimized construction

A key ingredient of the construction of biorthogonal wavelet bases for Sobolev spaces on manifolds, which is based on topological isomorphisms is the Hestenes extension operator. Here we firstly investigate whether this particular extension operator can be replaced by another extension operator. Our main theoretical result states that an important class of extension operators based on interpolating boundary values cannot be used in the construction setting required by Dahmen and Schneider. In the second part of this paper, we investigate and optimize the Hestenes extension operator. The results of the optimization process allow us to implement the construction of biorthogonal wavelets from Dahmen and Schneider. As an example, we illustrate a wavelet basis on the 2-sphere.

The lifting scheme based on the second generation wavelets

The lifting scheme is a custom-design construction of Biorthogonal wavelets, a fast and efficient method to realize wavelet transform, which provides a wider range of application and efficiently reduces the computing time with its particular frame. This paper aims at introducing the second generation wavelets, begins with traditional Mallat algorithms, illustrates the lifting scheme and brings out the detail steps in the construction of Biorthogonal wavelets. Because of isolating the degrees of freedom remaining the biorthogonality relations, we can fully control over the lifting operators to design the wavelet for a particular application, such as increasing the number of the vanishing moments.

Construction of biorthogonal wavelets from pseudo-splines

Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1–46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163–181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.

Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation

In this paper, two new wavelet bases are developed for discretising the angular term of the first-order Boltzmann transport equation. The wavelets proposed are based on Sweldens second generation wavelets [Sweldens, W., 1993. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. 1, 54], which are constructed through the lifting procedure [Sweldens, W., 1995. The lifting scheme: a new philosophy in biorthogonal wavelet construction. Wavelet Applications in Signal and Image Processing III]. In this paper, the wavelets are built on an octahedral domain, Fig. 2, and the angular flux approximation takes the form of finite element linear and quadratic representations. Full details of the meshing over the octahedron and derivation of the wavelet functions are given. The wavelets discussed are similar to the wavelets developed in Buchan [Buchan, A., 2003 c. Angular discretisation of the first order Boltzmann transport equation. Part 2: linear spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.] and [Buchan, A., 2003b. Angular discretisation of the first order Boltzmann transport equation. Part 3: quadratic spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.], in this paper the bases use a new fundamental amendment for mitigating the inaccuracies observed with the earlier bases. The performance of the new angular discretisation techniques are demonstrated using 2 one-dimensional and 4 two-dimensional test problems. These problems demonstrate the accuracy and susceptibility to ray effects of the proposed methods. Comparisons of all calculations are made with the conventional S N and P N approximations. Benchmark solutions are provided by the established code EVENT.

Generalized B-spline subdivision-surface wavelets for geometry compression

We present a new construction of lifted biorthogonal wavelets on surfaces of arbitrary two-manifold topology for compression and multiresolution representation. Our method combines three approaches: subdivision surfaces of arbitrary topology, B-spline wavelets, and the lifting scheme for biorthogonal wavelet construction. The simple building blocks of our wavelet transform are local lifting operations performed on polygonal meshes with subdivision hierarchy. Starting with a coarse, irregular polyhedral base mesh, our transform creates a subdivision hierarchy of meshes converging to a smooth limit surface. At every subdivision level, geometric detail can be expanded from wavelet coefficients and added to the surface. We present wavelet constructions for bilinear, bicubic, and biquintic B-Spline subdivision. While the bilinear and bicubic constructions perform well in numerical experiments, the biquintic construction turns out to be unstable. For lossless compression, our transform can be computed in integer arithmetic, mapping integer coordinates of control points to integer wavelet coefficients. Our approach provides a highly efficient and progressive representation for complex geometries of arbitrary topology.

Biorthogonal Loop-Subdivision Wavelets

We present a biorthogonal wavelet construction for Loop subdivision, based on the lifting scheme. Our wavelet transform uses scaling functions that are recursively defined by Loop subdivision for arbitrary manifold triangle meshes. We orthogonalize our wavelets with respect to local scaling functions. This way, the wavelet analysis computes locally a least squares fit when reducing the resolution and converting geometric detail into sparse wavelet coefficients. The contribution of our approach is local computation of both, wavelet analysis and synthesis in linear time. We provide numerical examples supporting the stability of our scheme.

Biorthogonal Wavelets with Certain Regularities

In this paper, we will discuss the construction of biorthogonal wavelets that possess the largest possible regularities and required vanishing moments. For the sake of applications, we also give a general Daubechies' iteration method of constructing biorthogonal wavelets by using biorthogonal splines.

Parallel Computation of Wavelet Transforms Using the Lifting Scheme

The lifting scheme l14r is a method for construction of biorthogonal wavelets and fast computation of the corresponding wavelet transforms. This paper describes a message–passing parallel implementation in which high efficiency is achieved by a modified data–swapping approach allowing communications to overlap computations. The method illustrated by application to Haar and Daubechies (D4) wavelets. Timing and speed–up results for the Cray T3E and the Fujitsu AP3000 are presented.

Method for Construction of Biorthogonal Wavelets Based on the Theory of Poles

A method for construction of the biorthogonal wavelets based on interpolator, which are obtained with application of the theory of poles, is suggested. Results of the method implementation are presented as tables and graphs.

The Wavelet Element Method Part II. Realization and Additional Features in 2D and 3D

The Wavelet Element Method (WEM) provides a construction of multiresolution systems and biorthogonal wavelets on fairly general domains. These are split into subdomains that are mapped to a single reference hypercube. Tensor products of scaling functions and wavelets defined on the unit interval are used on the reference domain. By introducing appropriate matching conditions across the interelement boundaries, a globally continuous biorthogonal wavelet basis on the general domain is obtained. This construction does not uniquely define the basis functions but rather leaves some freedom for fulfilling additional features. In this paper we detail the general construction principle of the WEM to the 1D, 2D, and 3D cases. We address additional features such as symmetry, vanishing moments, and minimal support of the wavelet functions in each particular dimension. The construction is illustrated by using biorthogonal spline wavelets on the interval.

Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms.

Analysis and Construction of Optimal Multivariate Biorthogonal Wavelets with Compact Support

In applications, it is well known that high smoothness, small support, and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of $L_p\, (1\le p \le \infty)$ smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1-2r,2r-1]s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br. Here the sequence br on ${\Bbb Z}$ is the unique univariate interpolatory refinement mask which is supported on [1-2r,2r-1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal scaling function, whose mask is supported on [0,2r-1]s for some positive integer r and satisfies the sum rules of optimal order r, has Lp smoothness not exceeding that of the univariate Daubechies orthogonal scaling function whose mask is supported on [0,2r-1]. We also demonstrate that a similar result holds true for biorthogonal wavelets. Examples are provided to illustrate the general theory. Finally, a general CBC (cosets by cosets) algorithm is presented to construct all the dual refinement masks of any given interpolatory refinement mask with the dual masks satisfying arbitrary order of sum rules. Thus, for any scaling function which is fundamental, this algorithm can be employed to generate a dual scaling function with arbitrary approximation order. This CBC algorithm can be easily implemented. As a particular application of the general CBC algorithm, a TCBC (triangle cosets by cosets) algorithm is proposed. For any positive integer k and any interpolatory refinement mask a such that a is symmetric about all the coordinate axes, such a TCBC algorithm provides us with a dual mask of a such that the dual mask satisfies the sum rules of order 2k and is also symmetric about all the coordinate axes. As an application of this TCBC algorithm, a family of optimal bivariate biorthogonal wavelets is presented with the scaling function being a spline function.

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈Zs∖{0}].Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.

Construction of biorthogonal wavelets starting from any two multiresolutions

Starting from any two given multiresolution analyses of L/sub 2/, {V/sub j//sup 1/}/sub j/spl isin/Z/ and {V/sub j//sup 2/}/sub j/spl isin/Z/, we construct biorthogonal wavelet bases that are associated with this chosen pair of multiresolutions. Thus, our construction method takes a point of view opposite to the one of Cohen-Daubechies-Feauveau (1992), which starts from a well-choosen pair of biorthogonal discrete filters. In our construction, the necessary and sufficient condition is the nonperpendicularity of the multiresolutions.

Reconstruction and Decomposition Algorithms for Biorthogonal Multiwavelets

We construct biorthogonal multiwavelets (abbreviated to wavelets) in a weighted Hilbert space L^2(E,\rho) where E is a compact subset in \bR^d. A recursive formula for biorthogonal wavelet construction is presented. The construction of the initial wavelets is reformulated as the solution of a certain matrix completion problem. A general solution of the matrix completion problem is identified and expressed conveniently in terms of any given particular solution. Several particular solutions are proposed. Reconstruction and decomposition algorithms are developed for the biorthogonal wavelets. Special results for the univariate case E=[0,1] are obtained.

The Lifting Scheme: A Custom-Design Construction of Biorthogonal Wavelets

We present the lifting scheme, a new idea for constructing compactly supported wavelets with compactly supported duals. The lifting scheme uses a simple relationship between all multiresolution analyses with the same scaling function. It isolates the degrees of freedom remaining after fixing the biorthogonality relations. Then one has full control over these degrees of freedom to custom design the wavelet for a particular application. The lifting scheme can also speed up the fast wavelet transform. We illustrate the use of the lifting scheme in the construction of wavelets with interpolating scaling functions.

Overview of wavelets and applications

Construction of orthogonal and biorthogonal wavelets as well as redundant wavelet frames will be reviewed. Fast computational wavelet‐based algorithms will be outlines and their performance will be compared with Fourier‐based algorithms. Various applications of wavelet analysis in speech compression and speech recognition, as well as broader applications in acoustic signal processing including denoising and transient analysis, will be discussed.