Showing papers on "Biorthogonal system published in 2012"
TL;DR: In this article, the authors studied the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets.
Abstract: We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schrodinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.
64 citations
TL;DR: In this article, a variationally consistent contact formulation is considered and an abstract framework for the a priori error analysis in the special case of frictionless contact and small deformations is provided.
Abstract: In this paper, a variationally consistent contact formulation is considered and we provide an abstract framework for the a priori error analysis in the special case of frictionless contact and small deformations. Special emphasis is put on quadratic mortar finite element methods. It is shown that under quite weak assumptions on the Lagrange multiplier space $${\mathcal{O} (h^{t-1}), 2 < t < \frac52}$$ , a priori results in the H 1-norm for the error in the displacement and in the H ?1/2-norm for the error in the surface traction can be established provided that the solution is regular enough. We discuss several choices of Lagrange multipliers ranging from the standard lowest order conforming finite elements to locally defined biorthogonal basis functions. The crucial property for the analysis is that the basis functions have a local positive mean value. Numerical results are exemplarily presented for one particular choice of biorthogonal (i.e. dual) basis functions and also comprise the case of finite deformation contact.
52 citations
TL;DR: The Meijer-G random field as mentioned in this paper is a two-level random point field with Laguerre weights, and it is shown that it can be expressed in terms of the Fredholm determinantal determinant.
Abstract: We apply the general theory of Cauchy biorthogonal polynomials developed previously by the authors, to the case associated with Laguerre measures. In particular, we obtain explicit formulae in terms of Meijer-G functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy two-matrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two--level random point field, the "Meijer-G random field". We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the Meijer-G random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble.
40 citations
TL;DR: A comparative study of different wavelet families for analysis of wrist motions from electromyography (EMG) signals finds that 'Biorthogonal' and 'Coiflets' wavelets families are more suitable for accurate classification of EMG signals of different wrist motions.
39 citations
TL;DR: A digital image watermarking algorithm based on Discrete Wavelet Transform - Discrete Cosine Transform -Singular Value Decomposition (DWT-DCT-SVD) with good performance with respect to robustness against various image processing operations is discussed.
Abstract: achieve the copyright protection in digital images, watermarking is identified as a major technology to protect digital images from illegal manipulation and geometric distortions. In this paper we discuss a digital image watermarking algorithm based on Discrete Wavelet Transform - Discrete Cosine Transform -Singular Value Decomposition (DWT-DCT-SVD). Here in this paper we examined and compared various wavelet families such as Haar, Daubechies, Biorthogonal and Coiflets for the watermarking algorithm. The difference in this method, from other traditional methods is that the watermark is embedded in high frequency band. Traditionally it is assumed that for having good robustness, a watermark should be embedded in low or mid frequency.DWT provides scalability, DCT provides compression and SVD offers minimum or no distortion. Choice of wavelets depends on the choice of wavelet function as DWT can be composed of any function that satisfies requirements of multiresolution analysis. In each of these wavelet families we analyzed effects of wavelets on image quality. The simulation results show good performance with respect to robustness against various image processing operations.
28 citations
TL;DR: In this article, a tensor product wavelet method is applied to solve various singularly perturbed boundary value problems, and the numerical results indicate robustness with respect to the singular perturbations.
Abstract: Locally supported biorthogonal wavelets are constructed on the unit interval with respect to which second-order constant coefficient differential operators are sparse. As a result, the representation of second-order differential operators on the hypercube with respect to the resulting tensor product wavelet coordinates is again sparse. The advantage of tensor product approximation is that it yields (nearly) dimension-independent rates. An adaptive tensor product wavelet method is applied to solve various singularly perturbed boundary value problems. The numerical results indicate robustness with respect to the singular perturbations. For a two-dimensional model problem this will be supported by theoretical results.
28 citations
TL;DR: Generalized block-lifting factorization of M-channel (M >; 2) biorthogonal filter banks (BOFBs) for lossy-to-lossless image coding is presented in this paper and achieves better result in both objective measure and perceptual visual quality for the images with a lot of high-frequency components.
Abstract: Generalized block-lifting factorization of M-channel (M >; 2) biorthogonal filter banks (BOFBs) for lossy-to-lossless image coding is presented in this paper. Since the proposed block-lifting structure is more general than the conventional lifting factorizations and does NOT require many restrictions such as paraunitary, number of channels, and McMillan degree in each building block unlike the conventional lifting factorizations, its coding gain is higher than that of the previous methods. Several proposed BOFBs are designed and applied to image coding. Comparing the results with conventional lossy-to-lossless image coding structures, including the 5/3- and 9/7-tap discrete wavelet transforms in JPEG 2000 and a 4 × 8 hierarchical lapped biorthogonal transform in JPEG XR, the proposed BOFBs achieve better result in both objective measure and perceptual visual quality for the images with a lot of high-frequency components.
26 citations
TL;DR: In this paper, the biorthogonal matrix extension problem with symmetry was investigated and a step-by-step algorithm for constructing the desired pair of extension matrices (P e, P ˜ e ) from the given pair of matrices was provided.
21 citations
TL;DR: A symmetric version of the nonsymmetric mixed finite element method for nearly incompressible elasticity that can statically condense out all auxiliary variables from the saddle point problem arriving at a symmetric and positive‐definite system based only on the displacement.
Abstract: We present a symmetric version of the nonsymmetric mixed finite element method presented in (Lamichhane, ANZIAM J 50 (2008), C324–C338) for nearly incompressible elasticity. The displacement–pressure formulation of linear elasticity is discretized using a Petrov–Galerkin discretization for the pressure equation in (Lamichhane, ANZIAM J 50 (2008), C324–C338) leading to a non-symmetric saddle point problem. A new three-field formulation is introduced to obtain a symmetric saddle point problem which allows us to use a biorthogonal system. Working with a biorthogonal system, we can statically condense out all auxiliary variables from the saddle point problem arriving at a symmetric and positive-definite system based only on the displacement. We also derive a residual based error estimator for the mixed formulation of the problem. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012
19 citations
TL;DR: In this article, Biane, Bougerol, and O'Connell give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite.
Abstract: Using the determinantal formula of Biane, Bougerol, and O’Connell, we give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well-ordered with a scale σ. We show that, at each time t > 0, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. Here, one-parameter extensions (θ-extensions) of the Stieltjes-Wigert polynomials, which are themselves q-extensions of the Hermite polynomials, play an essential role. The two parameters a and σ of the process combined with time t are mapped to the parameters q and θ of the biorthogonal polynomials. By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained. We study the determinantal structure of the matrix model and prove that, at each time t > 0, the present noncolliding Brownian motion with drift is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel. Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift.
15 citations
TL;DR: In this article, an extension of the classical Paley-Wiener space structure, based on bilinear expansions of integral kernels into biorthogonal sequences of functions, is studied.
TL;DR: In this paper, the construction of biorthogonal p-wavelet packets on R+ related to the Walsh polynomials and their properties are investigated by means of Walsh-Fourier transform.
Abstract: This paper deals with the construction of biorthogonal p -wavelet packets on R+ related to the Walsh polynomials and their properties are investigated by means of Walsh-Fourier transform. Three biorthogonal formulas regarding these p -wavelet packets are derived. Moreover, it is shown how to obtain several new Riesz bases of the space L2(R+) by constructing a series of subspaces of these p -wavelet packets. Mathematics subject classification (2010): 42C40, 42C15, 42C10.
TL;DR: In this article, multiscale representations of discrete manifold-valued data are discussed and a stability result is obtained for definable multi-scale decompositions of manifold analogs of biorthogonal wavelets.
Abstract: We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold analogs of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result.
TL;DR: In this article, the Lagrange multipliers are used to restore field continuity across nonconforming surfaces in 3D problems, making it possible to implement the relative motion of stator and rotor without remeshing in the 3D Finite Element (FE) modeling of electrical machines.
Abstract: This paper discusses the application of Lagrange multipliers to restore field continuity across nonconforming surfaces in 3-D problems. The method makes it in particular possible to implement the relative motion of stator and rotor without remeshing in the 3-D Finite Element (FE) modeling of electrical machines. The choice of a special set of biorthogonal shape functions for the Lagrange multiplier makes it possible to preserve the positive definiteness of the FE system. It is shown that such a biorthogonal basis cannot be constructed canonically for a 3-D magnetic vector potential formulation. For a 3-D magnetic scalar potential formulation, however, the situation is different and a biorthogonal basis can be found.
TL;DR: This paper proposes GTD (generalized triangular decomposition) filter banks as a subband coder for optimizing the theoretical coding gain and shows that in both theory and numerical simulations, the optimal GTD subband coders have superior performance than optimal traditional sub band coders.
Abstract: Filter bank optimization for specific input statistics has been of great interest in both theory and practice in many signal processing applications. In this paper we propose GTD (generalized triangular decomposition) filter banks as a subband coder for optimizing the theoretical coding gain. We focus on perfect reconstruction orthonormal GTD filter banks and biorthogonal GTD filter banks. We show that in both cases there are two fundamental properties in the optimal solutions, namely, total decorrelation and spectrum equalization. The optimal solutions can be obtained by performing the frequency dependent GTD on the Cholesky factor of the input power spectrum density matrices. We also show that in both theory and numerical simulations, the optimal GTD subband coders have superior performance than optimal traditional subband coders. In addition, the uniform bit loading scheme can, with no loss of optimality, be used in the optimal biorthogonal GTD coders, which solves the granularity problem in the conventional optimum bit loading formula. We then extend the use of GTD filter banks to wireless communication systems, where linear precoding and zero-forcing decision feedback equalization is used in frequency selective channels. We consider the quality of service (QoS) problem of minimizing the transmitted power subject to the bit error rate and total bit rate constraints. Optimal systems with orthonormal precoder and unconstrained precoder are both derived and shown to be related to the frequency dependent GTD of the channel frequency response.
TL;DR: In this article, the inverse problem which arises in the study of the integrable PDE proposed by V Novikov is solved for a class of discrete densities using Cauchy biorthogonal polynomials.
Abstract: The inverse problem which arises in the study of the integrable PDE proposed by V Novikov is solved for a class of discrete densities. The method of solution relies on the use of Cauchy biorthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformations to the ‘string’-type boundary value problems known from prior works.
TL;DR: This paper investigates the use of biorthogonal interpolating wavelets as a basis for this projection of the large-eddy simulation (LES) equations, placing special emphasis on the wavelet-based differential operators that define this mapping.
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TL;DR: The outstanding quantitative analysis of the specified wavelets is done to investigate the signal quality, mean square error, entropy and peak-to-peak SNR at multiscale stage-4 for both 1D voice signal and 2D image.
Abstract: Mutiscale analysis represents multiresolution scrutiny of a signal to improve its signal quality. Multiresolution analysis of 1D voice signal and 2D image is conducted using DCT, FFT and different wavelets such as Haar, Deubachies, Morlet, Cauchy, Shannon, Biorthogonal, Symmlet and Coiflet deploying the cascaded filter banks based decomposition and reconstruction. The outstanding quantitative analysis of the specified wavelets is done to investigate the signal quality, mean square error, entropy and peak-to-peak SNR at multiscale stage-4 for both 1D voice signal and 2D image. In addition, the 2D image compression performance is significantly found 93.00% in DB-4, 93.68% in bior-4.4, 93.18% in Sym-4 and 92.20% in Coif-2 during the multiscale analysis.
06 Nov 2012
TL;DR: This project presents a general Matlab library for rational function systems and their applications, which contains Blaschke functions, MT systems and biorthogonal systems and built in methods for finding the poles automatically.
Abstract: There is a wide range of applications of rational function systems. Including in system, control theories and signal processing. A special class of rational functions, the so-called Blaschke functions and the orthonormal Malmquist--Takenaka (MT) systems are effectively used for representing signals especially electrocardiograms. We present our project on a general Matlab library for rational function systems and their applications. It contains Blaschke functions, MT systems and biorthogonal systems. We implemented not only the continuous but the discrete versions as well, since in applications the latter one is needed. The complex and real interpretations are both available. We also built in methods for finding the poles automatically. Also, some interactive GUIs were implemented for visual demonstration that help the users in understanding the roles of certain parameters such as poles, multiplicity etc.
TL;DR: With the aid of fixed-point theorem and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a linear Fredholm-Volterra integro-differential equation is turned into a numerical algorithm, so that it can be solved numerically.
Abstract: With the aid of fixed-point theorem (an equivalent version for the linear case) and biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a linear Fredholm-Volterra integro-differential equation is turned into a numerical algorithm, so that it can be solved numerically.
TL;DR: Using the determinantal formula of Biane, Bougerol, and O'Connell, this article gave multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite.
Abstract: Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multitime joint probability densities to the noncolliding Brownian motion with drift, where the number of particles is finite We study a special case such that the initial positions of particles are equidistant with a period $a$ and the values of drift coefficients are well-ordered with a scale $\sigma$ We show that, at each time $t >0$, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz Here one-parameter extensions ($\theta$-extensions) of the Stieltjes-Wigert polynomials, which are themselves $q$-extensions of the Hermite polynomials, play an essential role The two parameters $a$ and $\sigma$ of the process combined with time $t$ are mapped to the parameters $q$ and $\theta$ of the biorthogonal polynomials By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained We study the determinantal structure of the matrix model and prove that, at each time $t >0$, the present noncolliding Brownian motion with drift is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift
TL;DR: In this paper, an image analysis method was developed with higher detection accuracy for rice fissures compared with using the classical Canny and Sobel methods, which is obtained using a common scanning machine with resolution of 600 dpi.
Abstract: An image analysis method was developed with higher detection accuracy for rice fissures compared with using the classical Canny and Sobel methods. The rice images are obtained using a common scanning machine with resolution of 600 dpi. The scanning images are enhanced by the gamma correction and smoothed using the anisotropic nonlinear diffusion PDEs. The diffusion process is stopped when the peak signal to noise ratio is lower than 30 dB or changes slowly. After that the wavelet coefficients of the smoothed images are calculated using continuous wavelet transform with the biorthogonal B-spline wavelets bior1.5 in multi-resolution spaces. The wavelet coefficients in y forward direction are used as the magnitudes. Finally, the magnitudes are standardized and used for the judgment of the fissures as the local maxima. Two different kinds of rice kernels are used for the test of the effectiveness of the proposed algorithm, including 30 long- and 20 medium-cracked grains with 1, 2, 3, or 4 fissures. The results demonstrate a satisfying performance of the fissure detecting systems, and even the faint lines of the fissures can also be detected.
03 Jul 2012
TL;DR: This project presents a general MATLAB library for rational function systems and their applications, which contains Blaschke functions, MT systems and biorthogonal systems and built in methods for finding the poles automatically.
Abstract: There is a wide range of applications of rational function systems. Including in system, control theories and signal processing. A special class of rational functions, the so-called Blaschke functions and the orthonormal Malmquist-Takenaka (MT) systems are effectively used for representing signals especially electrocardiograms. We present our project on a general MATLAB library for rational function systems and their applications. It contains Blaschke functions, MT systems and biorthogonal systems. We implemented not only the continuous but the discrete versions as well, since in applications the latter one is needed. The complex and real interpretations are both available. We also built in methods for finding the poles automatically. Also, some interactive GUIs were implemented for visual demonstration that help the users in understanding the roles of certain parameters such as poles, multiplicity etc.
TL;DR: It is shown that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions.
Abstract: In this paper, we present an explicit method to construct directly in the x-domain compactly supported scaling functions corresponding to the wavelets adapted to a sum of differential operators with constant coefficients. Here the adaptation to an operator is taken to mean that the wavelets give a diagonal form of the operator matrix. We show that the biorthogonal compactly supported wavelets adapted to a sum of differential operators with constant coefficients are closely connected with the representation of the null-space of the adjoint operator by the corresponding scaling functions. We consider the necessary and sufficient conditions (actually the Strang–Fix conditions) on integer shifts of a compactly supported function (distribution) f ∈ S'(ℝ) to represent exactly any function from the null-space of a sum of differential operators with constant coefficients.
TL;DR: In this paper, the authors studied the convergence of a vanishing viscosity approximation of the one-dimensional linear wave equation to a control of the conservative limit equation with respect to moment problems and biorthogonal sequences.
Abstract: The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? Our viscous term contains the fractional power of the Dirichlet Laplace operator and it is multiplied by a small parameter devoted to tend to zero. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to the vanishing parameter.
Journal Article•
TL;DR: In this paper, the authors propose a buildling scheme of fast parametrized biorthogonal transforms taking advantage of data-flow graphs for calculation of fast orthonormal transforms and two-point biorithogonal butterfly operators.
Abstract: In this paper the authors propose the buildling scheme of fast parametrized biorthogonal transforms taking advantage of data-flow graphs for calculation of fast orthonormal transforms and two-point biorthogonal butterfly operators. Streszczenie. W pracy zaproponowano schemat budowy szybkich parametryzowanych przeksztalce ´ n biortogonalnych oparty o diagramy przeply- wowe dla szybkich przeksztalce ´ n ortonormalnych i dwupunktowe biortogonalne operatory motylkowe. ( Szybkie parametryzowane transformacje bi-ortogonalne)
Journal Article•
TL;DR: In this article, the separation property for Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces is established. But the separation properties are restricted to spaces with an M-basis.
TL;DR: For the minimal splines of arbitrary order on a non-uniform grid, a system of linear functionals biorthogonal to the system of coordinate splines is constructed in this article.
Abstract: For the minimal splines of arbitrary order on a nonuniform grid, a system of linear functionals biorthogonal to the system of coordinate splines is constructed. The matrices of refining and sparsing decompositions are obtained for the spaces of splines of arbitrary order associated with infinite and finite nonuniform grids on an interval and on a segment, respectively.
TL;DR: In this article, a compactly supported totally interpolating biorthogonal stable multi-wavelet system is studied and the necessary and sufficient conditions for such systems to have given approximation orders are stated in simple equations.
Abstract: In this article, compactly supported totally interpolating biorthogonal multiwavelet systems are studied. Necessary and sufficient conditions for such systems to have given approximation orders are stated in simple equations. It is shown that the shorter nontrivial filter component that has the minimum possible length for a given approximation order is uniquely determined up to a discrete parameter. Among systems with such property, we provide all totally interpolating biorthogonal stable multiwavelet systems of approximation orders 2 and 3 with minimal total length whose scaling vectors have minimal lengths as well.