Showing papers on "Biorthogonal system published in 2013"
TL;DR: This paper relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that are k-hop localized with compact spectral spread and still satisfy the perfect reconstruction conditions.
Abstract: This paper extends previous results on wavelet filterbanks for data defined on graphs from the case of orthogonal transforms to more general and flexible biorthogonal transforms. As in the recent work, the construction proceeds in two steps: first we design “one-dimensional” two-channel filterbanks on bipartite graphs, and then extend them to “multi-dimensional” separable two-channel filterbanks for arbitrary graphs via a bipartite subgraph decomposition. We specifically design wavelet filters based on the spectral decomposition of the graph, and state sufficient conditions for the filterbanks to be perfect reconstruction and orthogonal. While our previous designs, referred to as graph-QMF filterbanks, are perfect reconstruction and orthogonal, they are not exactly k-hop localized, i.e., the computation at each node is not localized to a small k-hop neighborhood around the node. In this paper, we relax the condition of orthogonality to design a biorthogonal pair of graph-wavelets that are k-hop localized with compact spectral spread and still satisfy the perfect reconstruction conditions. The design is analogous to the standard Cohen-Daubechies-Feauveau's (CDF) construction of factorizing a maximally-flat Daubechies half-band filter. Preliminary results demonstrate that the proposed filterbanks can be useful for both standard signal processing applications as well as for signals defined on arbitrary graphs.
235 citations
TL;DR: In this article, the Hermiticity condition in quantum mechanics required for the characterisation of physical observables and generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and their eigenstates are complete.
Abstract: The Hermiticity condition in quantum mechanics required for the characterisation of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert space dimensionality is finite. Specifically, characterisations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.
173 citations
TL;DR: In this article, a biorthogonal system of functions consisting of two Riesz bases of the space L2[(0, 1) × ( 0, 1), obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse source problem.
Abstract: We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L2[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.
106 citations
TL;DR: In this article, an alternative definition for pseudo-bosons is proposed, which simplifies the mathematical structure, minimizing the required assumptions, and some physical examples are discussed, as well as some mathematical results related to the biorthogonal sets arising out of their framework.
Abstract: We propose an alternative definition for pseudo-bosons. This simplifies the mathematical structure, minimizing the required assumptions. Some physical examples are discussed, as well as some mathematical results related to the biorthogonal sets arising out of our framework. We also briefly extend the results to the so-called nonlinear pseudo-bosons.
55 citations
TL;DR: In this paper, the authors show that even a small deviation from strict self-adjointness of the hamiltonian $H$ produces two deep consequences, not well understood in the literature: first, the original orthonormal basis splits into two families of biorthogonal vectors.
Abstract: Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the hamiltonian $H$ changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of $H$ produces two deep consequences, not well understood in the literature: first of all, the original orthonormal basis of $H$ splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space $\Hil$. Secondly, the so-called metric operator is unbounded, as well as its inverse.
In the second part of the paper, after an extension of some previous results on the so-called $\D$ pseudo-bosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.
48 citations
TL;DR: In this article, the authors consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin Morrey spaces and derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline), and more general r-regular wavelets.
Abstract: We consider local means with bounded smoothness for Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r-regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov-Triebel-Lizorkin spaces.
46 citations
TL;DR: In this article, the authors solved a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations.
Abstract: We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral problems for those equations, this one is of a 'discrete cubic string' type -- a nonselfadjoint generalization of a classical inhomogeneous string -- but presents some interesting novel features: there are two Lax pairs, both of which contribute to the correct complete spectral data, and the solution to the inverse problem can be expressed using quantities related to Cauchy biorthogonal polynomials with two different spectral measures. The latter extends the range of previous applications of Cauchy biorthogonal polynomials to peakons, which featured either two identical, or two closely related, measures. The method used to solve the spectral problem hinges on the hidden presence of oscillatory kernels of Gantmacher-Krein type implying that the spectrum of the boundary value problem is positive and simple. The inverse spectral problem is solved by a method which generalizes, to a nonselfadjoint case, M. G. Krein's solution of the inverse problem for the Stieltjes string.
43 citations
TL;DR: In this paper, it was shown that a non-self-adjoint Hamiltonian of harmonic-oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudobosonic operators.
Abstract: We demonstrate that a non-self-adjoint Hamiltonian of harmonic-oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudobosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT-symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that these sets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space L2(R2), but instead only D quasibases. As recently proved by one of us, this is sufficient to deduce several interesting consequences.
36 citations
TL;DR: In this article, an approximate method for solving a system of nonlinear Volterra integro-differential equations is proposed, which involves the use of biorthogonal systems in adequate spaces of continuous functions associated with such a system.
31 citations
TL;DR: The proposed admissible diffusion wavelets (ADW) are constructed in a bottom-up manner that starts from a local operator in a high frequency, and dilates by its dyadic powers to low frequencies, hence facilitating data analysis and geometry processing.
Abstract: As signal processing tools, diffusion wavelets and biorthogonal diffusion wavelets have been propelled by recent research in mathematics. They employ diffusion as a smoothing and scaling process to empower multiscale analysis. However, their applications in graphics and visualization are overshadowed by nonadmissible wavelets and their expensive computation. In this paper, our motivation is to broaden the application scope to space-frequency processing of shape geometry and scalar fields. We propose the admissible diffusion wavelets (ADW) on meshed surfaces and point clouds. The ADW are constructed in a bottom-up manner that starts from a local operator in a high frequency, and dilates by its dyadic powers to low frequencies. By relieving the orthogonality and enforcing normalization, the wavelets are locally supported and admissible, hence facilitating data analysis and geometry processing. We define the novel rapid reconstruction, which recovers the signal from multiple bands of high frequencies and a low-frequency base in full resolution. It enables operations localized in both space and frequency by manipulating wavelet coefficients through space-frequency filters. This paper aims to build a common theoretic foundation for a host of applications, including saliency visualization, multiscale feature extraction, spectral geometry processing, etc.
24 citations
20 Sep 2013
TL;DR: In this paper, it was shown that every Banach space containing an isomorphic copy of a given copy of the given copy has an infinite equilateral set and also that if a bounded biorthogonal system of size α can be renormed so that it admits equilateral sets of equal size.
Abstract: We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $\alpha$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.
Journal Article•
TL;DR: In this article, the concept of biorthogonal wavelets was generalized to a local field of positive characteristic, and it was shown that wavelets generate Riesz bases for local fields with positive characteristic.
Abstract: We generalize the concept of biorthogonal wavelets to a local field $K$ of positive characteristic We show that if the translates of the scaling functions of two multiresolution analyses are biorthogonal, then the associated wavelet families are also biorthogonal Under mild assumptions on the scaling functions and the wavelets, we also show that the wavelets generate Riesz bases for $L^2(K)$
26 Jun 2013
TL;DR: The coefficients of the DOST and its variations are shown to exhibit a directed graph structure as opposed to the tree-like structure demonstrated by wavelet coefficients.
Abstract: We examine the so-called Discrete Orthonormal Stockwell Transform (DOST) and show that a number of quite simple modifications can be made to obtain various desired properties. For example, we introduce a real-valued Discrete Cosine-based DOST (DCST). The coefficients of the DOST and its variations are shown to exhibit a directed graph structure as opposed to the tree-like structure demonstrated by wavelet coefficients. Finally, we employ the DOST and DCST in a series of simple compression experiments and compare the results to those obtained with biorthogonal wavelets and the DCT.
TL;DR: The solution of the first order nonlinear Fredholm–Volterra integro-differential equation is approximate by means of a new method for its numerical resolution, using the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.
TL;DR: In this article, it was shown that the relevant biorthogonal polynomials are actually special cases of multiple orthogonal ensembles associated with modified Bessel functions, which was first introduced by Van Assche and Yakubovich.
Abstract: The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg, and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M = 2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.
TL;DR: In this article, the authors apply the nonlinear steepest descent method to a class of 3 × 3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model.
Abstract: We apply the nonlinear steepest descent method to a class of 3 × 3 Riemann-Hilbert problems introduced in connection with the Cauchy two-matrix random model. The general case of two equilibrium measures supported on an arbitrary number of intervals is considered. In this case, we solve the Riemann-Hilbert problem for the outer parametrix in terms of sections of a spinorial line bundle on a three-sheeted Riemann surface of arbitrary genus and establish strong asymptotic results for the Cauchy biorthogonal polynomials.
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TL;DR: In this article, the authors studied the central limit theorem for biorthogonal ensembles and showed that whenever the right limit is a Laurent matrix, a Central Limit Theorem holds.
Abstract: We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a Central Limit Theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a Central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai Class of an interval. Our results also extend previous results on Unitary Ensembles in the one-cut case. Finally, we will illustrate our results by deriving Central Limit Theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.
TL;DR: The key point of the theory is that the derivative of a 1D BMRA is also a BMRA, as established by Jouini and Lemarie-Rieusset (1993) [16].
Patent•
20 Nov 2013
TL;DR: In this article, an integral coding/decoding method based on MIMO radar communication is proposed, which guarantees the orthogonality of bipolar phase spread spectrum code sequence based on WalsH matrix.
Abstract: The invention provides an integral coding/decoding method based on MIMO radar communication. The orthogonal spread spectrum code sequence guarantees the orthogonality of bipolar phase spread spectrum code sequence based on WalsH matrix. The requirement of the radar to the detection of signal self-correlation peak value, mutual correlation peak value low side-lobe can be satisfied through the genetic algorithm. The signal encoding is based on soft spread spectrum biorthogonal encoding. To satisfy the requirement of the MIMO radar to detection signal, different spreading coding sequences can be used on different encode element positions; special sending channel can be reserved as the time and frequency synchronous standard, the first encode element position of each sending channel is reserved as the phase position standard, and special synchronous spreading codes are used on the positions. According to the method provided by the invention, conventional carrier frequency offset extraction technology, carrier phase partial extraction technology and soft spread spectrum biorthogonal decoding method are improved specifically according to the requirement of the radar/radio frequency integrate realization, accompanied by the frequency offset tracking technology with Alpha -Beta as the core, accuracy of data deviation correction is guaranteed.
01 Jan 2013
TL;DR: In this article, the performance of DWT-OFDM systems using Fast Fourier Transforms (FFT) and Discrete Wavelet Transform (DWT) is analyzed by various parameters such as BER, eye diagram and constellation diagram.
Abstract: Performance of Orthogonal Frequency Division Multiplexing (OFDM) systems using Fast Fourier Transforms (FFT) and Discrete Wavelet Transform (DWT) are analyzed in this paper. The performance of DWT-OFDM is assessed by various parameters such as BER, eye diagram and constellation diagram. BER is the ratio of the no. of bits with error to the total no. of bits transmitted through the channel. From the Eye diagram, we can find the eye height through which the interference can be calculated. By varying the different wavelets and the different modulation schemes the BER is calculated and the eye diagram and the constellation diagram are plotted with the aid of MATLAB- SIMULINK. In this work, performance of most widely used wavelet based OFDM systems Haar, Daubechies, Biorthogonal, and Reverse Biorthogonal wavelets are studied and wavelet basis suitable for optimum OFDM system is investigated. Bit Error Rate (BER) versus Signal to Noise Ratio (SNR) is used as system parameter.
TL;DR: In this article, the classical L2 or L2 projection theory is extended to H(grad), H(curl) and H(div) to obtain increased projection accuracy for the distributional derivatives.
Abstract: Coupled problems are made up of subproblems of which the physical nature differs. Using indirect coupling models, the subproblems are calculated separately on their own meshes to ensure precision. To obtain a precise solution for the total problem, it is important to ensure the transmission of information between the subproblems. In this paper, we present field projection methods on overlapping domains. In comparison to earlier works, the classical L2 or L2 projection theory is extended to H(grad), H(curl) and H(div) to obtain increased projection accuracy for the distributional derivatives. A Petrov-Galerkin method is then presented to fill the test space using a biorthogonal basis, without losing the optimality of the result in comparison to the L2 or L2 Ritz-Galerkin method. Using the Petrov-Galerkin method and biorthogonal test functions, the projection is presented using a diagonal matrix. However, in the standard Ritz-Galerkin projections, a linear system must be solved.
TL;DR: The coset sum shares many essential features of the tensor product that make it attractive in practice and suggest that it is worthwhile to develop and practice alternative methods to the Tensor product for constructing multivariate wavelet systems.
Abstract: A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in multiresolution analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: 1) it preserves the biorthogonality of univariate refinement masks, 2) it preserves the accuracy number of the univariate refinement mask, and 3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.
TL;DR: In this article, the authors consider some modifications of the canonical commutation relations, leading to non commutative bosons and show how biorthogonal bases of the Hilbert space of the system can be obtained out of them.
Abstract: We consider some modifications of the two dimensional canonical commutation relations, leading to non commutative bosons and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (F.B.), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.
TL;DR: In this paper, a mixed finite element method based on simplicial triangulations for a three-field formulation of linear elasticity is considered and a uniform convergence of the finite element approximation in the incompressible limit is shown.
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TL;DR: In this article, a toplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomial which allow of a combinatorial interpretation in terms of Schroder paths.
Abstract: A Toplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schroder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schroder paths developed by Eu and Fu.
Patent•
22 May 2013
TL;DR: In this article, a second generation small-wave support vector machine assessment method for damage and remaining life of a metal structure is presented, where an intrinsic mode function is obtained through decomposition of an experience mode, a time-frequency domain statistic characteristic of the intrinsic mode is extracted, a most sensitive characteristic is chosen according to a distance accessment principle to construct a best characteristic set, a minimum quantization error indicator which has an obvious performance degradation trend along with time changes is constructed by means of self-organization neural network characteristic fusion techniques, a biorthogonal small
Abstract: The invention discloses a second generation small-wave support vector machine assessment method for damage and remaining life of a metal structure. According to the method, an intrinsic mode function is obtained through decomposition of an experience mode, a time-frequency domain statistic characteristic of the intrinsic mode function is extracted, a most sensitive characteristic is chosen according to a distance accessment principle to construct a best characteristic set, a minimum quantization error indicator which has an obvious performance degradation trend along with time changes is constructed by means of self-organization neural network characteristic fusion techniques, a biorthogonal small-wave support vector machine kernel function of second generation small-wave transform is provided, a service life prediction model of the second generation small-wave support vector machine is constructed, the minimum quantization error indicator serves as a prediction characteristic, and quantitative assessment for the damage and the remainig service life of a metal structural component of mechanical equipment under a small subsample is achieved.
DOI•
01 Jan 2013
TL;DR: Compared with the DCT-based JPEG image compression algorithm, this paper uses the all phase biorthogonal transform matrix to reduce interpixel redundancy and proposes uniform quantization to the APBT coefficients aiming at reducing the computation complexity.
Abstract: In this paper, the properties of the all phase biorthogonal transform (APBT) matrix are deduced based on the concepts of all phase biorthogonal transform. A novel algorithm is presented to compress color image using the all phase Walsh biorthogonal transform (APWBT), the all phase discrete cosine biorthogonal transform (APDCBT) and the all phase inverse discrete cosine biorthogona transform (APIDCBT), instead of the conventional discrete cosine transform (DCT). Compared with the DCT-based JPEG image compression algorithm, we use the all phase biorthogonal transform matrix to reduce interpixel redundancy and propose uniform quantization to the APBT coefficients aiming at reducing the computation complexity. Experimental results show that the CPSNR of the proposed algorithm performs close to the DCT and outperforms the DCT at low bit rates especially.
01 Dec 2013
TL;DR: This paper proposes to use a biorthogonal 5/3 spline filter which provides linear phase response and solves the data expansion problem and reduces the operation count and yield a better compression ratio.
Abstract: Although several methods are used to measure various power system quantities such as voltage, current and frequency, it is still inadequate for post processing of power quality issues due to the possibility of missing significant events relevant to power quality. Therefore, IEC 61000-4-30 recommends storage of the power system raw waveform samples over a certain period. However, due to the large amount data, it is mandatory to compress the data for transmission and storage. In an earlier work, an approach for compression is based on separating the stationary and nonstationary components of the waveform. The stationary components are represented by Fourier coefficients and the nonstationary components are represented by discrete wavelet transform coefficients. However, the filter used has nonlinear phase response and data expansion problem. Therefore, the data is subjected to phase distortion. In this paper, it is proposed to use a biorthogonal 5/3 spline filter which provides linear phase response and solves the data expansion problem. Further, the filter reduces the operation count and yield a better compression ratio. Simulation results confirm the validity of the proposed approach.
TL;DR: A novel approach for the construction of symmetric compactly supported biorthogonal multiwavelets with multiplicity 2 is proposed.
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TL;DR: In this article, the authors considered some modifications of the canonical commutation relations, leading to non commutative bosons and showed how biorthogonal bases of the Hilbert space of the system can be obtained out of them.
Abstract: We consider some modifications of the two dimensional canonical commutation relations, leading to {\em non commutative bosons} and we show how biorthogonal bases of the Hilbert space of the system can be obtained out of them. Our construction extends those recently introduced by one of us (FB), modifying the canonical anticommutation relations. We also briefly discuss how bicoherent states, producing a resolution of the identity, can be defined.