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Showing papers on "Biorthogonal system published in 2011"


Journal ArticleDOI
TL;DR: In this paper, the controllability of a system of n one-dimensional parabolic equations with m control is proved. But the controLL of the system is restricted to a part of the boundary by means of m controls.

85 citations


Journal ArticleDOI
TL;DR: With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.

47 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a general approach for nonlinear biorthogonal decomposition of random fields based on a fully symmetric operator framework that unifies different types of expansions and allows for a simple formulation of necessary and sufficient conditions for their completeness.

39 citations


Journal ArticleDOI
TL;DR: An algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+.
Abstract: In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared.

37 citations


Journal ArticleDOI
TL;DR: This study evaluated and compared three different wavelet families i.e. Daubechies, Coiflets, Biorthogonal and discussed important features of wavelet transform in compression of images.
Abstract: The objective of this paper is to evaluate a set of wavelets for image compression. Image compression using wavelet transforms results in an improved compression ratio. Wavelet transformation is the technique that provides both spatial and frequency domain information. These properties of wavelet transform greatly help in identification and selection of significant and non-significant coefficients amongst the wavelet coefficients. DWT (Discrete Wavelet Transform) represents image as a sum of wavelet function (wavelets) on different resolution levels. So, the basis of wavelet transform can be composed of function that satisfies requirements of multiresolution analysis. The choice of wavelet function for image compression depends on the image application and the content of image. A review of the fundamentals of image compression based on wavelet is given here. This study also discussed important features of wavelet transform in compression of images. In this study we have evaluated and compared three different wavelet families i.e. Daubechies, Coiflets, Biorthogonal. Image quality is measured, objectively using peak signal-to-noise ratio, Compression Ratio and subjectively using visual image quality.

34 citations


Journal ArticleDOI
TL;DR: In this paper, a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles was proved for disordered bosons.
Abstract: We prove a large deviations principle for the empirical measures of a class of biorthogonal and multiple orthogonal polynomial ensembles that includes biorthogonal Laguerre, Jacobi, and Hermite ensembles, the matrix model of Lueck, Sommers, and Zirnbauer for disordered bosons, the Stieltjes-Wigert matrix model of Chern-Simons theory, and Angelesco ensembles.

31 citations


Journal ArticleDOI
TL;DR: In this article, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested and several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.
Abstract: In this paper, some algorithms for constructing orthogonal and biorthogonal compactly supported wavelets on Vilenkin groups are suggested. As application, several examples of p-adic wavelets, which correspond to the refinable functions presented recently by the first author, are given.

30 citations


Journal ArticleDOI
TL;DR: In this article, a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues) is constructed, admitting a large family of monodromy-preserving deformations.
Abstract: We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painleve equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.

30 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2 and obtained the limiting distribution of the zeros of the polynomials p_(n,n) as n → ∞.
Abstract: We consider the two sequences of biorthogonal polynomials (p_(k,n))^∞_(k=0) and (q_(k,n)) ^∞_(k=0) related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_(n,n) as n → ∞. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.

29 citations


Journal ArticleDOI
TL;DR: In this article, the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators, is studied.
Abstract: We consider the problem on the convergence of biorthogonal expansions in a system of eigenfunctions and associated functions for a wide class of operators, whose special cases include nonself-adjoint differential operators. We introduce the notion of almost basis property of systems of root functions of a linear operator. We demonstrate the necessity to use a new method, earlier introduced by the authors, for defining associated functions.

27 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_\Theta$ of the Hardy space.
Abstract: Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems of reproducing kernels in a reproducing kernel Hilbert space $\mathcal{H}$ of analytic functions. We study the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_\Theta$ of the Hardy space. We find a class of spaces where systems biorthogonal to complete systems of reproducing kernels are always complete, and show that in general this is not true. In particular we answer the question posed by N.K. Nikolski and construct a model subspace with a non-complete biorthogonal system.

Journal ArticleDOI
TL;DR: Using a superconvergence property of a gradient recovery operator, it is proved an optimal a priori estimate for the finite element discretization for a class of meshes.

Journal ArticleDOI
TL;DR: In this paper, a new family of global A -biorthogonal methods by using short two-term recurrences and formal orthogonal polynomials, which contain the global bi-conjugate residual (Gl-BCR) algorithm and its improved version, was presented.

Journal ArticleDOI
TL;DR: This work proves optimal a priori estimates for both stream function and vorticity, and presents numerical results to demonstrate the efficiency of the approach.
Abstract: We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.

Journal ArticleDOI
TL;DR: The six-fold axial symmetric filter banks constructed in this paper result in algorithm templates with desirable symmetry for triangle surface processing, which represent multiresolution analysis and synthesis algorithms as templates.
Abstract: This paper discusses the construction of highly symmetric compactly supported wavelets for hexagonal data/image and triangle surface multiresolution processing. Recently, hexagonal image processing has attracted attention. Compared with the conventional square lattice, the hexagonal lattice has several advantages, including that it has higher symmetry. It is desirable that the filter banks for hexagonal data also have high symmetry which is pertinent to the symmetric structure of the hexagonal lattice. The high symmetry of filter banks and wavelets not only leads to simpler algorithms and efficient computations, it also has the potential application for the texture segmentation of hexagonal data. While in the field of computer-aided geometric design (CAGD), when the filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms for regular vertices have high symmetry, which make it possible to design the corresponding multiresolution algorithms for extraordinary vertices. In this paper we study the construction of six-fold axial symmetric biorthogonal filter banks and the associated wavelets, with both the dyadic and -refinements. The constructed filter banks have the desirable symmetry for hexagonal data processing. By associating the outputs (after one-level multiresolution decomposition) appropriately with the nodes of the regular triangular mesh with which the input data is associated (sampled), we represent multiresolution analysis and synthesis algorithms as templates. The six-fold axial symmetric filter banks constructed in this paper result in algorithm templates with desirable symmetry for triangle surface processing.

Journal ArticleDOI
TL;DR: This work builds a new biorthogonal pair of multi-resolution analyses on the interval, by integration or differentiation, and uses both pairs to construct isotropic or anisotropic divergence-free wavelet bases on the hypercube.

01 Jan 2011
TL;DR: In this article, the authors start from a given rational function system and take the linear space spanned by it and construct a rational function systems that is biorthogonal to the original one.
Abstract: In this paper we start from a given rational function system and take the linear space spanned by it. Then in this linear space we construct a rational function system that is biorthogonal to the original one. By means of biorthogonality expansions in terms of the original rational functions can be easily given. For the discrete version we need to choose the points of discretization and the weight function in the discrete scalar product in a proper way. Then we obtain that the biorthogonality relation holds true for the discretized systems as well.

Proceedings ArticleDOI
27 Jun 2011
TL;DR: The concept of a biorthogonal eigenfunction system (BES) of linear stability equations has been utilized for receptivity problems in boundary layers, wall jets, pipe ow, and detonations.
Abstract: The concept of a biorthogonal eigenfunction system (BES) of linear stability equations has been utilized for receptivity problems in boundary layers, wall jets, pipe ow, and detonations. Other applications of the BSE include the analysis of experimental and computational studies of perturbations. In the present survey, applications of the BSE to receptivity problems and to the analysis of experimental and computational data illustrate the main features of the method.

Journal ArticleDOI
TL;DR: It is found that Biorthogonal wavelets outperform the orthogonal ones in both the criteria and objectively peak signal to noise ratio and subjectively visual quality of image.
Abstract: e present work we analyze the performance of orthogonal and Biorthogonal wavelet filters for image compression on variety of test images. The test images are of different size and resolution. The compression performance is measured, objectively peak signal to noise ratio and subjectively visual quality of image and it is found that Biorthogonal wavelets outperform the orthogonal ones in both the criteria. General Terms Image compression algorithms

Journal ArticleDOI
TL;DR: It is demonstrated that 4-fold symmetric filter banks result in multiresolution algorithms with the required symmetry for quad surface processing, and some filter banks constructed in this paper result in very simpleMultiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1–2):29–39) and Wang et al. (Vis Comput 22(9–11):874–884, 2006).
Abstract: Surface multiresolution processing is an important subject in CAGD. It also poses many challenging problems including the design of multiresolution algorithms. Unlike images which are in general sampled on a regular square or hexagonal lattice, the meshes in surfaces processing could have an arbitrary topology, namely, they consist of not only regular vertices but also extraordinary vertices, which requires the multiresolution algorithms have high symmetry. With the idea of lifting scheme, Bertram (Computing 72(1---2):29---39, 2004) introduces a novel triangle surface multiresolution algorithm which works for both regular and extraordinary vertices. This method is also successfully used to develop multiresolution algorithms for quad surface and $\sqrt 3$ triangle surface processing in Wang et al. (Vis Comput 22(9---11):874---884, 2006; IEEE Trans Vis Comput Graph 13(5):914---925, 2007) respectively. When considering the biorthogonality, these papers do not use the conventional $L^2({{\rm I}\kern-.2em{\rm R}}^2)$ inner product, and they do not consider the corresponding lowpass filter, highpass filters, scaling function and wavelets. Hence, some basic properties such as smoothness and approximation power of the scaling functions and wavelets for regular vertices are unclear. On the other hand, the symmetry of subdivision masks (namely, the lowpass filters of filter banks) for surface subdivision is well studied, while the symmetry of the highpass filters for surface processing is rarely considered in the literature. In this paper we introduce the notion of 4-fold symmetry for biorthogonal filter banks. We demonstrate that 4-fold symmetric filter banks result in multiresolution algorithms with the required symmetry for quad surface processing. In addition, we provide 4-fold symmetric biorthogonal FIR filter banks and construct the associated wavelets, with both the dyadic and $\sqrt 2$ refinements. Furthermore, we show that some filter banks constructed in this paper result in very simple multiresolution decomposition and reconstruction algorithms as those in Bertram (Computing 72(1---2):29---39, 2004) and Wang et al. (Vis Comput 22(9---11):874---884, 2006; IEEE Trans Vis Comput Graph 13(5):914---925, 2007). Our method can provide the filter banks corresponding to the multiresolution algorithms in Wang et al. (Vis Comput 22(9---11):874---884, 2006) for dyadic multiresolution quad surface processing. Therefore, the properties of the scaling functions and wavelets corresponding to those algorithms can be obtained by analyzing the corresponding filter banks.

Journal ArticleDOI
TL;DR: 6-fold symmetric bi-frames yield frame decomposition and reconstruction algorithms (for regular vertices) with high symmetry, which is required for the design of the corresponding frame multiresolution algorithms for extraordinary vertices on the triangular mesh.

Proceedings ArticleDOI
19 Dec 2011
TL;DR: In this article, texture-based features from gray level cooccurrence matrix (GLCM) and various wavelet packet energies were used to classify retinal vasculature for biometric identification.
Abstract: Noting the advantages of texture-based features over the structural descriptors of vascular trees, we investigated texture-based features from gray level cooccurrence matrix (GLCM) and various wavelet packet energies to classify retinal vasculature for biometric identification. Wavelet packet energy features were generated by Daubechies, Coiflets and Reverse Biorthogonal wavelets. Two different entropy methods, Shannon and logarithm of energy, were used to prune wavelet packet decomposition trees. Next, wrapper methods were used for classification-guided feature selection. Features were ranked based on area under the receiver operating curves, Bhattacharya, and t-test metrics. Using the ranked lists, wrapper methods were used in conjunction with Naive Bayesian, k-nearest neighbor (k-NN), and Support Vector Machine (SVM) classifiers. Best results were achieved by using features from Reverse Biorthogonal 2.4 wavelet packet decomposition in conjunction with a nearest neighbor classifier, yielding a 3-fold cross validation accuracy of 99.42% with a sensitivity and specificity of 98.33% and 99.47% respectively.

Journal ArticleDOI
TL;DR: In this article, the Vilenkin-Chrestenson transforms are used to construct new orthogonal wavelet bases defined by finite collections of parameters in the spaces of complex periodic sequences.
Abstract: In the spaces of complex periodic sequences, we use the Vilenkin-Chrestenson transforms to construct new orthogonal wavelet bases defined by finite collections of parameters. Earlier similar bases were defined for the Cantor and Vilenkin groups by means of generalized Walsh functions. It is noted that similar constructions can be realized for biorthogonal wavelets as well as for the space l2(ℤ+).

Journal ArticleDOI
TL;DR: In this paper, the four-element linear functional equation is studied in the class of functions analytic beyond an isosceles trapezoid and vanishing at infinity, where a system of entire functions is constructed of completely regular growth biorthogonal with a piecewise exponential weight to the system of powers on the coordinate axis.
Abstract: The four-element linear functional equation is studied in the class of functions analytic beyond an isosceles trapezoid and vanishing at infinity. Some system of entire functions is constructed of completely regular growth biorthogonal with a piecewise exponential weight to the system of powers on the coordinate axis.

Journal ArticleDOI
TL;DR: In this paper, a parametric generalization of the classical Poincare-Perron theorem on stabilizing recurrence relations is proposed, where the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values.
Abstract: We prove a parametric generalization of the classical Poincare-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.

Journal ArticleDOI
TL;DR: A new frequency warping biorthogonal frame operator for non-smooth warping maps is introduced in this work, based on a mathematical model which has been previously introduced for computational purposes.
Abstract: Frequency warping is theoretically designed to be a unitary operator of infinite input and output dimensions, thus performing the resolution of identity. In real implementations finite dimensions have to be considered, then perfect reconstruction cannot be fulfilled. The accuracy of reconstruction is particularly compromised in case of non-smooth warping maps, which are more useful for practical applications. In order to overcome this limitation, a new frequency warping biorthogonal frame operator for non-smooth warping maps is introduced in this work. The proposed transformation is based on a mathematical model which has been previously introduced for computational purposes. By adding some redundancy with respect to the truncation of the infinite dimensions operator, the effect of an infinite output dimension can be taken into account in a compressed way, based on an analytical factorization. In the reconstruction process, the additional redundant samples are expanded, thus guaranteeing near perfect reconstruction.

Journal ArticleDOI
TL;DR: The approach of the template-based bi-frame construction introduced in this paper can be extended easily to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing.

Journal ArticleDOI
TL;DR: In this paper, the authors used the discrete Walsh transform to construct orthogonal and biorthogonal wavelets for complex periodic sequences similar to those studied earlier for the Cantor group.
Abstract: In this paper, using the discreteWalsh transform, we construct orthogonal and biorthogonal wavelets for complex periodic sequences similar to those studied earlier for the Cantor group. Results of numerical experiments demonstrate the effectiveness of the use of constructed discrete wavelets in image processing.

01 Jan 2011
TL;DR: Wavelet is used as denoising algorithm and performance of haar and biorthogonal wavelets are experimentally evaluated.
Abstract: Clear speech sometimes will be polluted by noise. Reduction of noise aims at reducing noise from noisy speech signal and extracting the clean speech. As speech is transmitted and received using various media it introduces distortions and have bandwidth constraints. These degradations lower intelligibility of speech message causing severe problems in downstream processing and user perception of speech signal. There has been a lot of research in speech denoising so far but there always remains room for improvement. The motivation to use wavelet as a possible alternative is to explore new ways to reduce computational complexity and to achieve better noise reduction performance. Wavelet denoising technique is simple and efficient. In this paper wavelet is used as denoising algorithm. Performance of haar and biorthogonal wavelets are experimentally evaluated.

01 Jan 2011
TL;DR: In this article, the authors consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of sub-spaces, Lindelof degree, irredundant families of clopen sets and others.
Abstract: We consider topological invariants on compact spaces related to the sizes of discrete subspaces (spread), densities of subspaces, Lindelof degree of subspaces, irredundant families of clopen sets and others and look at the following associations between compact topological spaces and Banach spaces: a compact K induces a Banach space C(K) of real valued continuous functions on K with the supremum norm; a Banach space X induces a compact space BX , the dual ball with the weak topology. We inquire on how topological invariants on K and BX are linked to the sizes of biorthogonal systems and their versions in C(K) and X respectively. We gather folkloric facts and survey recent results like that of Abad-Lopez and Todorcevic that it is consistent that there is a Banach space X without uncountable biorthogonal systems such that the spread of BX is uncountable or that of Brech and Koszmider that it is consistent that there is a compact space where spread of K 2 is countable but C(K) has uncountable biorthogonal systems.