Showing papers on "Biorthogonal system published in 2008"

Biorthogonal Systems in Banach Spaces

Abstract: The main theme of this book is the relation between the global structure of Banach spaces and the various types of generalized "coordinate systems" - or "bases" - they possess. This subject is not new and has been investigated since the inception of the study of Banach spaces. In this book, the authors systematically investigate the concepts of Markushevich bases, fundamental systems, total systems and their variants. The material naturally splits into the case of separable Banach spaces, as is treated in the first two chapters, and the nonseparable case, which is covered in the remainder of the book. This book contains new results, and a substantial portion of this material has never before appeared in book form. The book will be of interest to both researchers and graduate students. Topics covered in this book include: - Biorthogonal Systems in Separable Banach Spaces - Universality and Szlenk Index - Weak Topologies and Renormings - Biorthogonal Systems in Nonseparable Spaces - Transfinite Sequence Spaces - Applications Petr Hajek is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic. Vicente Montesinos is Professor of Mathematics at the Polytechnic University of Valencia, Spain. Jon Vanderwerff is Professor of Mathematics at La Sierra University, in Riverside, California. Vaclav Zizler is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic.

Biorthogonal multiple vector-valued multivariate wavelet packets associated with a dilation matrix☆

Abstract: In this paper, the notion of multiple vector-valued multiresolution analysis of space L 2 ( R s , C r × r ) is introduced A method for constructing biorthogonal multiple vector-valued wavelet packets in higher dimensions is presented and their properties is investigated by means of time–frequency analysis method, matrix theory and operator theory Three biorthogonality formulas concerning these wavelet packets are obtained Finally, new Riesz bases of space L2(Rs, Cr×r) is obtained by constructing a series of subspaces of biorthogonal multiple vector-valued wavelet packets

The Cauchy two-matrix model

Abstract: We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.

Theory of Dual-Tree Complex Wavelets

Abstract: We study analyticity of the complex wavelets in Kingsbury's dual-tree wavelet transform. A notion of scaling transformation function that defines the relationship between the primal and dual scaling functions is introduced and studied in detail. The analyticity property is examined and dealt with via the transformation function. We separate analyticity from other properties of the wavelet such as orthogonality or biorthogonality. This separation allows a unified treatment of analyticity for general setting of the wavelet system, which can be dyadic or M-band; orthogonal or biorthogonal; scalar or multiple; bases or frames. We show that analyticity of the complex wavelets can be characterized by scaling filter relationship and wavelet filter relationship via the scaling transformation function. For general orthonormal wavelets and dyadic biorthogonal scalar wavelets, the transformation function is shown to be paraunitary and has a linear phase delay of omega/2 in (0, 2pi).

Detection of QRS Complexes Based on Biorthogonal Spline Wavelet

Abstract: Biorthogonal spline wavelet is used to detect the QRS complex of ECG signal. Mallat algorithm is applied in the decomposition of ECG signal by using the equivalent filter of a biorthogonal spline wavelet. Lipschitz exponent is introduced to investigate the relationship between the signal singularity (R Peak) and the zero-crossing point of the modulus maximum pair of the signal's wavelet transform. Adaptive threshold, refractory period and expiating are applied to improve the anti-interference performance. Experimental results demonstrated that this method is robust against time varying characteristics of QRS complex and noise. A correct detection rate of 99.905% has been achieved when the MIT-BIH arrhythmia database is used to test the proposed QRS complex detection algorithm.

Image Denoising Using Matched Biorthogonal Wavelets

Abstract: Current denoising techniques use the classical ortho normal wavelets for decomposition of an image corrupted with additive white Gaussian noise, upon which various thresholding strategies are built. The use of available biorthogonal wavelets in image denoising is less common because of their poor performance. In this paper, we present a method to design image-matched biorthogonal wavelet bases and report on their potential for denoising. We have conducted experiments on various image datasets namely Natural, Satellite and Medical with the designed wavelets using two existing thresholding strategies. Test results from comparing the performance of matched and fixed biorthogonal wavelets show an average improvement of 35% in MSE for low SNR values (0 to 18 db) in every dataset. This improvement was also seen in the PSNR and visual comparisons. This points to the importance of matching when using wavelet-based denoising.

Minimal splines and wavelets

Abstract: This paper is dedicated to the memory of the prominent mathematician S.G. Mikhlin. Here, Mikhlin’s idea of approximation relations is used for construction of wavelet resolution in the case of spline spaces of zero height. These approximation relations allow one to establish the embedding of the spline spaces corresponding to nested grids. Systems of functionals which are biorthogonal to the basic splines are constructed using the relations; then the systems obtained are used for wavelet decomposition. It is established that, for a fixed pair of grids of which one is embedded into the other and for an arbitrary fixed (on the coarse grid) spline space, there exists a continuum of spline spaces (on the fine grid) which contain the aforementioned spline space on the coarse grid. The wavelet decomposition of such embedding is given and the corresponding formulas of decomposition and formulas of reconstruction are deduced. The space of ( Open image in new window , φ)-splines is introduced with three objects: the full chain of vectors, prescribed infinite grid on real axis and the preassigned vector-function φ with m + 1 components (m is called the order of the splines). Under certain assumptions, the splines belong to the class Cm − 1. The gauge relations between the basic splines on the coarse grid and the basic splines on the fine grid are deduced. A general method for construction of a biorthogonal system of functionals (to basic spline system) is suggested. In this way, a chain of nested spline spaces is obtained, and the wavelet decomposition of the chain is discussed. The spaces and chains of spaces are completely classified in the terms of manifolds.

A Lattice Structure of Biorthogonal Linear-Phase Filter Banks With Higher Order Feasible Building Blocks

Abstract: This paper proposes a lattice structure of biorthogonal linear-phase filter banks (BOLPFBs) using new building blocks which can obtain long filters with fewer number of building blocks than conventional ones. The structure is derived from a generalization of the building blocks of first-order LPFBs. Furthermore, the proposed building blocks are applicable for both even and odd number of channels. The resulting FBs have good performance in stopband attenuation and low implementation costs.

Efficient Design of High-Complexity Cosine Modulated Filter Banks Using $2 M$ th Band Conditions

Abstract: This paper presents several new properties of biorthogonal cosine modulated filter banks (CMFBs) and efficient algorithms for designing CMFBs with a very large number of subbands and very long filters. For a biorthogonal CMFB, we find the periodicity and symmetry of its overall transfer function and aliasing transfer functions which can be efficiently computed based on a decimated uniform discrete Fourier transform (DFT) analysis filter bank. By exploiting gradient information and 2M th band conditions, efficient algorithms are proposed for designing both orthogonal and biorthogonal CMFBs. In addition, an efficient matrix inversion algorithm with O(N 2 ) complexity is also presented. Several numerical examples and comparisons with many other existing methods are included to demonstrate the design performance and efficiency of the algorithms.

Coupled dynamic thermoviscoelasticity problem

Abstract: We construct a closed analytic solution of a coupled dynamic thermoviscoelasticity problem for bodies of canonical shape. The solution is represented in the form of spectral expansions in a biorthogonal eigenfunction system of the nonself-adjoint pencil of differential operators generated by the problem under study. The spectral expansions are obtained with the help of a special class of nonsymmetric integral transforms.

Biorthogonal method of moments of coupled‐cluster equations: Alternative derivation, further considerations, and application to a model magnetic system

Abstract: The energy expansion defining the biorthogonal method of moments of coupled-cluster equations (MMCC) [Piecuch and Wloch, J Chem Phys, 2005, 123, 224105 and Piecuch et al., Chem Phys Lett 2006, 418, 467], which leads to the size extensive completely renormalized (CR) coupled-cluster (CC) approach with singles, doubles, and noniterative triples employing the left eigenstates of the similarity-transformed Hamiltonian, termed CR-CC(2,3), is overviewed and rederived. The rederivation of the biorthogonal MMCC expansion presented in this work is based on a direct resummation and subsequent elimination of the many-body components of the exponential wave operator of CC theory that appear at individual moment contributions in the original MMCC energy expansion [Kowalski and Piecuch, J Chem Phys, 2000, 113, 18; Kowalski and Piecuch, J Chem Phys 2001, 115, 2966], enabling one to understand why the CR-CC(2,3) method using the biorthogonal MMCC theory is more accurate than the earlier CR-CCSD(T) approach. The superiority of the CR-CC(2,3) method over the CR-CCSD(T) and other previously developed single-reference CC methods with a noniterative treatment of triply excited clusters, including the widely used CCSD(T) approach and the triples corrections defining the CCSD(2) schemes, is illustrated by examining the singlet–triplet gap of the (HFH)− magnetic system in which two paramagnetic centers are linked via a polarizable diamagnetic bridge. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

FIR Filter Banks for Hexagonal Data Processing

Abstract: Images are conventionally sampled on a rectangular lattice. Thus, traditional image processing is carried out on the rectangular lattice. The hexagonal lattice was proposed more than four decades ago as an alternative method for sampling. Compared with the rectangular lattice, the hexagonal lattice has certain advantages which include that it needs less sampling points; it has better consistent connectivity and higher symmetry; the hexagonal structure is also pertinent to the vision process. In this paper, we investigate the construction of symmetric FIR hexagonal filter banks for multiresolution hexagonal image processing. We obtain block structures of FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. These block structures yield families of orthogonal and biorthogonal FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. In this paper, we also discuss the construction of orthogonal and biorthogonal FIR filter banks with scaling functions and wavelets having optimal smoothness. In addition, we present a few of such orthogonal and biorthogonal FIR filters banks.

L-CAMP: Extremely Local High-Performance Wavelet Representations in High Spatial Dimension

Abstract: A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new "local compression-alignment-modified- prediction (L-CAMP)" methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score 575 000 000 000.

Construction of biorthogonal two-direction refinable function and two-direction wavelet with dilation factor m

Abstract: An algorithm for constructing a pair of biorthogonal two-direction refinable function and the corresponding biorthogonal two-direction wavelet is presented. In particular, the algorithm can also be used to construct two-direction orthogonal refinable function. Finally, we give an example illustrating how to use our method to construct biorthogonal two-direction scaling function with dilation factor 3.

A combinatorial derivation with Schroder paths of a determinant representation of Laurent biorthogonal polynomials

Abstract: A combinatorial proof in terms of Schroder paths and other weighted plane paths is given for a determinant representation of Laurent biorthogonal polynomials (LBPs) and that of coefficients of their three-term recurrence equation. In this process, it is clarified that Toeplitz determinants of the moments of LBPs and their minors can be evaluated by enumerating certain kinds of configurations of Schroder paths in a plane.

On the computation of gramian matrices for refinable bases on the interval

Abstract: In this paper, we discuss a simple method for the numerical computation of Gramian matrices, that appears within the construction of multiresolution analysis on the interval. The presented approach covers all, the orthogonal, the semiorthogonal and the biorthogonal cases. We consider constructions, which are based on translates of a known pair of compactly supported functions ϕ, inside the interval, and supplemented by boundary functions. Using the given two-scale-coefficients of these boundary functions, we can reduce the problem to a linear system of equations. Furthermore, we discuss conditions providing that these systems are uniquely solvable. In particular, no integrals have to be computed numerically. Finally, using the Schoenberg spline basis on the interval as an example, we show how to apply the method to a well-known problem.

List Decoding of Biorthogonal Codes and the Hadamard Transform With Linear Complexity

Abstract: Let a biorthogonal Reed-Muller code RM (1,m) of length n = 2m be used on a memoryless channel with an input alphabet plusmn1 and a real-valued output R. Given any nonzero received vector y in the Euclidean space Rn and some parameter epsiisin(0,1), our goal is to perform list decoding of the code RM (1, m) and retrieve all codewords located within the angle arccos e from y. For an arbitrarily small epsi, we design an algorithm that outputs this list of codewords with the linear complexity order of n [ln2 isin] bit operations. Without loss of generality, let vector y be also scaled to the Euclidean length radic(n) of the transmitted vectors. Then an equivalent task is to retrieve all coefficients of the Hadamard transform of vector y whose absolute values exceed nisin. Thus, this decoding algorithm retrieves all ne-significant coefficients of the Hadamard transform with the linear complexity n [ln2 isin] instead of the complexity n In2n of the full Hadamard transform.

Greedy Type Bases in Banach Spaces

Abstract: Let (X, · ) be a (real) Banach space. We refer to [38] or [28] as some introduction to the general theory of Banach spaces. Note that, as usual in the case, all the results we discuss here remain valid for complex scalars with possibly different constants. Let I be a countable set with possibly some ordering we refer to whenever considering convergence with respect to elements of I (wich will be denoted by limi→∞). Definition 1 We say that countable system of vectors is biorthogonal if for i, j ∈ I we have

Normalized trigonometric splines of Lagrange type

Abstract: Bϕ-splines, which are a nonpolynomial generalization of the well-known B-splines, are investigated. Bϕ-splines arise from approximation relations regarded as a system of linear algebraical equations, from which both polynomial and nonpolynomial splines are derived. Third-order normalized trigonometric splines of Lagrange type (zero height) determined by the generating vector function ϕ(t) = (1, sin t, cos t, sin2 t)T are constructed. These splines are twice continuously differentiable and have minimal compact support. A system of functionals biorthogonal to Bϕ-splines is defined. The solution of the interpolation problem generated by the resulting biorthogonal system in the space of Bϕ-splines is found.

Existence and design of biorthogonal matrix-valued wavelets

Abstract: Biorthogonal matrix-valued wavelets have been employed to analyse matrix-valued signals based on matrix multiresolution analysis.The sufficient condition for existence of a biorthogonal matrix-valued scaling function has been established in terms of the corresponding two-scale matrix symbols.Two designs based on factorization of biorthogonal two-scale matrix symbols are presented.In particular,explicit constructing formulations for biorthogonal matrix-valued wavelets are given.With these formulations,highpass filters of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters.Examples of two-scale matrix filter banks are given.

Regularity of a vector potential problem and its spectral curve

Abstract: In this note we study a minimization problem for a vector of measures subject to a prescribed interaction matrix in the presence of external potentials. The conductors are allowed to have zero distance from each other but the external potentials satisfy a growth condition near the common points. We then specialize the setting to a specific problem on the real line which arises in the study of certain biorthogonal polynomials (studied elsewhere) and we prove that the equilibrium measures solve a pseudo-algebraic curve under the assumption that the potentials are real analytic. In particular the supports of the equilibrium measures are shown to consist of a finite union of compact intervals.

Some Formal Aspects of the Theory of Intermolecular Interactions and of the BSSE Problem

Abstract: Some results of mathematical character concerning the theory of intermolecular interactions and the BSSE problem are presented. It is shown that the concept of complete basis set may be introduced for intermolecular potential surfaces only by considering explicitly the limiting process in which the basis sets of both monomers approach completeness simultaneously. That does not lead to any overcompleteness problem if we do not postulate the existence of two complete basis sets from the outset. The intimate connection between the BSSE and the differences of some biorthogonal integrals and their "original" counterparts is also discussed. The operator of BSSE is given in terms of such differences. It is shown that in a special case, when only the overlap of the occupied orbitals is considered, the "bi-expectation" value of the energy coincides with the conventional expectation value for the single determinant wave function built up of the unperturbed orbitals of the individual monomers. It is discussed, by using a model of the biorthogonal perturbation theory, why the conceptually fully different a priori (CHA) and a posteriori (CP) schemes of BSSE correction usually give very close numerical results. (The necessary biorthogonal perturbation formalism is developed in the Appendices.) The results give justification for the additivity assumptions inherent in the CP method.