Showing papers on "Biorthogonal system published in 2009"

The research of a class of biorthogonal compactly supported vector-valued wavelets

Abstract: In this paper, we introduce the biorthogonal vector-valued wavelets. We prove that, like in the scalar wavelet case, the existence of a pair of biorthogonal compactly supported vector-valued scaling functions guarantees the existence of a pair of biorthogonal compactly supported vector-valued wavelet functions. An algorithm for constructing a pair of biorthogonal compactly supported vector-valued wavelet functions is presented by means of vector-valued multiresolution analysis and matrix theory. The notion of biorthogonal vector-valued wavelet packets is introduced, and their properties are investigated by virtue of time–frequency analysis and algebra theory. Three biorthogonality formulas concerning the wavelet packets are established. Relation to some physical theories such as E -infinity Cantorian space–time theory is also discussed.

All phase biorthogonal transform and its application in JPEG-like image compression

Abstract: This paper proposes new concepts of the all phase biorthogonal transform (APBT) and the dual biorthogonal basis vectors. In the light of all phase digital filtering theory, three kinds of all phase biorthogonal transforms based on the Walsh transform (WT), the discrete cosine transform (DCT) and the inverse discrete cosine transform (IDCT) are proposed. The matrices of APBT based on WT, DCT and IDCT are deduced, which can be used in image compression instead of the conventional DCT. Compared with DCT-based JPEG (DCT-JPEG) image compression algorithm at the same bit rates, the PSNR and visual quality of the reconstructed images using these transforms are approximate to DCT, outgoing DCT-JPEG at low bit rates especially. But the advantage is that the quantization table is simplified and the transform coefficients can be quantized uniformly. Therefore, the computing time becomes shorter and the hardware implementation easier.

The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets

Abstract: Multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing, due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks. In order to process and represent vector-valued discrete data efficiently and sparsely by a multiwavelet transform, a multiwavelet has to be prefiltered or balanced. Balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature. Dual multiframelets include (bi)orthogonal multiwavelets as special cases, but their fundamental prefiltering and balancing property has not yet been investigated in the literature. In this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform. This approach, to our best knowledge, has not been considered so far in the literature even for multiwavelets, but it reveals the essential structure of prefiltering and the balancing property of multiwavelets and multiframelets. We prove that every biorthogonal multiwavelet can be prefiltered with the balancing order matching the order of its vanishing moments; that is, from every given compactly supported multivariate biorthogonal multiwavelet, one can always build another (essentially equivalent) compactly supported biorthogonal multiwavelets with its balancing order matching the order of the vanishing moments of the original one. More generally, we show that if a dual multiframelet can be prefiltered, then it can be equivalently transformed into a balanced dual multiframelet with the same balancing order. However, we notice that most available dual multiframelets in the literature cannot be simply prefiltered with its balancing order matching its order of vanishing moments and they must be designed to possess high balancing orders. The key ingredient of our approach is based on investigating some properties of the subdivision and transition operators acting on discrete vector polynomial sequences, which play a central role in a discrete multiframelet transform and are of interest in their own right. We also establish a new canonical form of a matrix mask, which greatly facilitates the investigation and construction of multiwavelets and multiframelets. In this paper, we obtain a complete criterion and the essential structure for balanced or prefiltered dual multiframelets in the most general setting. Our investigation of the balancing property of a multiframelet deepens our understanding of the multiframelet transform in signal processing and scientific computation.

Review of biorthogonal coupled cluster representations for electronic excitation

Abstract: Single reference coupled-cluster (CC) methods for electronic excitation are based on a biorthogonal representation (bCC) of the (shifted) Hamiltonian in terms of excited CC states, also referred to as correlated excited (CE) states, and an associated set of states biorthogonal to the CE states, the latter being essentially configuration interaction (CI) configurations. The bCC representation generates a non-hermitian secular matrix, the eigenvalues representing excitation energies, while the corresponding spectral intensities are to be derived from both the left and right eigenvectors. Using the perspective of the bCC representation, a systematic and comprehensive analysis of the excited-state CC methods is given, extending and generalizing previous such studies. Here, the essential topics are the truncation error characteristics and the separability properties, the latter being crucial for designing size-consistent approximation schemes. Based on the general order relations for the bCC secular matrix and the (left and right) eigenvector matrices, formulas for the perturbation-theoretical (PT) order of the truncation errors (TEO) are derived for energies, transition moments, and property matrix elements of arbitrary excitation classes and truncation levels. In the analysis of the separability properties of the transition moments, the decisive role of the so-called dual ground state is revealed. Due to the use of CE states the bCC approach can be compared to so-called intermediate state representation (ISR) methods based exclusively on suitably orthonormalized CE states. As the present analysis shows, the bCC approach has decisive advantages over the conventional CI treatment, but also distinctly weaker TEO and separability properties in comparison with a full (and hermitian) ISR method.

Cauchy Biorthogonal Polynomials

Abstract: The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in term of a Riemann-Hilbert problem.

Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz–Ladik hierarchy

Abstract: Adler and van Moerbeke (2001 Commun. Pure Appl. Math. 54 153-205) described a reduction of the 2D-Toda hierarchy called the Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik (1975 J. Math. Phys. 16 598-603) using semidiscrete zero-curvature equations. In this paper, we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computations to the matrix case. This generalization leads us to the semidiscrete zero-curvature equations for the non-Abelian (or multicomponent) version of the Ablowitz-Ladik equations (Gerdzhikov and Ivanov 1982 Theor. Math. Phys. 52 676-85). In this way, we extend the link between biorthogonal polynomials on the unit circle and the Ablowitz-Ladik hierarchy to the matrix case.

New biorthogonal potential-density basis functions

Abstract: We use the weighted integral form of spherical Bessel functions, and introduce a new analytical set of complete and biorthogonal potential–density basis functions. The potential and density functions of the new set have finite central values and they fall off, respectively, similar to r (1+l) and r (4+l) at large radii where l is the latitudinal quantum number of spherical harmonics. The lowest order term associated with l = 0 is the perfect sphere of de Zeeuw. Our basis functions are intrinsically suitable for the modeling of three dimensional, soft-centred stellar systems and they complement the basis sets of Clutton-Brock, Hernquist & Ostriker and Zhao. We test the performance of our functions by expanding the density and potential profiles of some spherical and oblate galaxy models.

Daubechies Versus Biorthogonal Wavelets for Moving Object Detection in Traffic Monitoring Systems

Abstract: Moving object detection is a fundamental task for a variety of traffic applications. In this paper the Daubechies and biorthogonal wavelet families are exploited for extracting the relevant movement information in moving image sequences in a 3D wavelet-based segmentation algorithm. The proposed algorithm is applied for traffic monitoring systems. The objective and subjective experimental results obtained by applying both wavelet types are compared and interpreted in terms of the different wavelet properties and the characteristics of the image sequences. The comparisons show the superior performance of the symmetric biorthogonal wavelets in the presence of noisy images and changing lighting conditions when compared to the application of high order Daubechies wavelets. The algorithm is evaluated using simulated images in the Matlab environment.

Biorthogonal wavelets on Vilenkin groups

Abstract: We describe algorithms for constructing biorthogonal wavelet systems and refinable functions whose masks are generalized Walsh polynomials. We give new examples of biorthogonal compactly supported wavelets on Vilenkin groups.

Bases for the description of monochromatic, strongly focused, scalar fields

Abstract: Two bases (one biorthogonal and one orthonormal) are proposed for the expansion of strongly focused (high numerical aperture) scalar monochromatic fields. The performance of these bases is tested and compared, both with each other and with a similar basis proposed by Alonso et al. [Opt. Express14, 6894 (2006)]. It is found that the orthonormal basis proposed herein exhibits the lowest truncation error of these three bases for the same truncation order for the examples considered. Additionally, this basis is advantageous because it allows for the expansion of fields without rotational symmetry.

Triangular wavelets: an isotropic image representation with hexagonal symmetry

Abstract: This paper introduces triangular wavelets, which are two-dimensional nonseparable biorthogonal wavelets defined on the regular triangular lattice. The construction that we propose is a simple nonseparable extension of one-dimensional interpolating wavelets followed by a straightforward generalization. The resulting three oriented high-pass filters are symmetrically arranged on the lattice, while low-pass filters have hexagonal symmetry, thereby allowing an isotropic image processing in the sense that three detail components are distributed uniformly. Applying the triangular filter to images, we explore applications that truly benefit from the triangular wavelets in comparison with the conventional tensor product transforms.

A New Image Fusion Algorithm Based on Biorthogonal Wavelet

Abstract: A new image fusion method based on biorthogonal wavelet that different frequency component on each decomposition level uses different operator is proposed. First, images involved in fusion are decomposed by biorthogonal multi-resolution wavelet. Then different fusion operators are used to high frequency images and low frequency images according to correlation coefficient between texture information function and edge luminance function. The simulation result indicated that the algorithm is able to obtain fused images of higher clarity and lower deviation index compared with traditional methods.

Cubic string boundary value problems and Cauchy biorthogonal polynomials

Abstract: Cauchy biorthogonal polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis?Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third-order ODE boundary value problem ?f ''' = zgf which is a generalization of the inhomogeneous string problem studied by Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated with these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L2g.

How to construct log-Gabor Filters?

Abstract: Orthogonal and biorthogonal wavelets became very popular image processing tools but exhibit major drawbacks, namely a poor resolution in orientation and the lack of translation invariance due to aliasing between subbands. We propose here the construction of log-Gabor wavelet transforms which allow exact reconstruction and strengthen the excellent mathematical properties of the Gabor filters. Two major improvements on the previous Gabor wavelet schemes are proposed: first the highest frequency bands are covered by narrowly localized oriented filters. Secondly, the set of filters cover uniformly the Fourier domain including the highest and lowest frequencies and thus exact reconstruction is achieved using the same filters in both the direct and the inverse transforms (which means that the transform is selfinvertible). In this paper the procedure for constructing log-Gabor filters is described and a Matlab toolbox that implements such multiresolution scheme is available from the authors.

Block diagonalization of four-dimensional metrics

Abstract: It is shown that, in four dimensions, it is possible to introduce coordinates so that an analytic metric locally takes block diagonal form, i.e. one can find coordinates such that gαβ = 0 for (α, β) S where S = {(1, 3), (1, 4), (2, 3), (2, 4)}. We call a coordinate system in which the metric takes this form a 'doubly biorthogonal coordinate system'. We show that all such coordinate systems are determined by a pair of coupled second-order partial differential equations.

Existence and design of biorthogonal matrix-valued wavelets

Abstract: In this paper, we study biorthogonal matrix-valued wavelets for analyzing matrix-valued signals based on matrix multiresolution analysis. Firstly, sufficient conditions for the existence of biorthogonal matrix-valued scaling function are established in terms of two-scale matrix symbols. Then we focus on the construction of biorthogonal matrix-valued wavelet. Two designs based on factorizations of biorthogonal two-scale matrix symbol are presented. In particular, explicit constructing formulations for biorthogonal matrix-valued wavelets are given. With these formulations, highpass filters {Gk} and {Gk} of biorthogonal matrix-valued wavelets can be given explicitly by lowpass filters {Hk} and {Hk} of their corresponding biorthogonal scaling functions. Finally, according to our designs, two examples of two-scale matrix filters are given.

Biorthogonal Cubic Hermite Spline Multiwavelets on the Interval with Complementary Boundary Conditions

Abstract: In this article, a new biorthogonal multiwavelet basis on the interval with complementary homogeneous Dirichlet boundary conditions of second order is presented. This construction is based on the multiresolution analysis on \({\mathbb{R}}\) introduced in [5] which consists of cubic Hermite splines. Numerical results for the Riesz constants and a discretization of the biharmonic equation, both non-adaptive and adaptive, are given, showing the superiority over other known boundary-adapted interval wavelet bases.

Construction and decomposition of biorthogonal vector-valued wavelets with compact support

Abstract: In this article, we introduce vector-valued multiresolution analysis and the biorthogonal vector-valued wavelets with four-scale The existence of a class of biorthogonal vector-valued wavelets with compact support associated with a pair of biorthogonal vector-valued scaling functions with compact support is discussed A method for designing a class of biorthogonal compactly supported vector-valued wavelets with four-scale is proposed by virtue of multiresolution analysis and matrix theory The biorthogonality properties concerning vector-valued wavelet packets are characterized with the aid of time–frequency analysis method and operator theory Three biorthogonality formulas regarding them are presented

Construction of Arbitrary Dimensional Biorthogonal Multiwavelet Using Lifting Scheme

Abstract: This paper focuses on the construction of multidimensional biorthogonal multiwavelets and the perfect reconstruction multifilter banks. Based on the Hermite-Neville filter, two lifting structures have been proposed and systematically investigated, and a general design framework has been developed for building biorthogonal multiwavelets and Hermite interpolation filter banks with any multiplicity for any lattice in any dimension with any number of primal and dual vanishing moments. The construction is an important generalization of the Neville-based lifting scheme and inherits all of the advantages of lifting schemes such as fast transform, in-place computation and integer-to-integer transforms. Our multi wavelet systems preserve most of the desirable properties for applications, such as interpolating, short support, symmetry, and high vanishing moments.

On wavelet decomposition of spaces of first order splines

Abstract: We consider approximate relations in the form of a system of linear algebraic equations that yield B φ -splines. We construct Lagrange type splines of the first order and give examples of polynomial, trigonometric, hyperbolic, and exponential B φ -splines. We also construct a system of linear functionals biorthogonal to the B φ -splines and resolve an interpolation problem generated by this system. For refined nonuniform grids we establish an embedding of spaces of B φ -splines. The decomposition and reconstruction formulas are obtained. Bibliography: 20 titles.

Generalization of the New Resonance Theory: Second Quantization Operator, Localization Scheme, and Basis Set

Abstract: We have recently proposed a method to evaluate the weights of resonance structures embedded in a molecular orbital by utilizing singlet-coupling scheme of an electron pair [J. Phys. Chem. A 2006, 110, 9028]. The method was formulated on the basis of the second quantization, in which a biorthogonal operator related to Mulliken population (MP) was used together with the Boys−Foster (BF) localization scheme. Our method is very easy to use; only a standard localization procedure is required to obtain the resonance weights. In addition, obtained results agreed well with our chemical intuition. In the present Article, the restrictions, namely MP and BF, were removed, and an operator related to Lowdin population (LP) and other various types of localization schemes were employed to examine the generality of the method. We found that computed resonance weights were virtually independent not only on the choice of these combinations but also on basis set. This new finding, the invariant nature in terms of resonance,...

Wavelet Bi-frames with Uniform Symmetry for Curve

Abstract: This paper is about the construction of wavelet bi-frames with each framelets being symmetric. When fllter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms, including the algorithms for boundary vertices, have high symmetry which makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices. When the multiresolution algorithms derived from univariate wavelet bi-frames are used as the boundary algorithms for surface multiresolution processing, it is required that not only the scaling functions but also all framelets are symmetric. In addition, for curve/surface multiresolution processing, it is also desirable that the algorithms should be given by templates so that the algorithms can be easily implemented. In this paper, flrst, by associating appropriately the lowpass and highpass outputs to the nodes of Z, we show that both biorthogonal wavelet multiresolution algorithms and bi-frame multiresolution algorithms can be represented by templates. Then, using the idea of lifting scheme, we provide frame algorithms given by several iterative steps with each step represented by a symmetric template. Finally, with the given iterative algorithms, we construct bi-frames based on their smoothness and vanishing moments. Two types of symmetric bi-frames are studied. In order to provide a clearer picture on the procedure for bi-frame construction, in this paper we also consider template-based construction of biorthogonal wavelets. The approach of the template-based bi-frame construction introduced in this paper can easily be extended to the construction of bivariate bi-frames with high symmetry for surface multiresolution processing.

Cubic String Boundary Value Problems and Cauchy Biorthogonal Polynomials

Abstract: Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.

Biorthogonal Systems on the Unit Circle, Regular Semiclassical Weights, and the Discrete Garnier Equations

Abstract: We demonstrate that a system of biorthogonal polynomials and their associated functions corresponding to a regular semiclassical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation, and showing that it coincides with that of a Garnier system. Such systems can also be characterized by recurrence relations of the discrete Painleve type, for example, in the case with one free deformation variable, the system was found to be characterized by a solution to the discrete fifth Painleve equation. Here we derive the canonical forms of the multivariable analogs of the discrete fifth Painleve for the Garnier systems, i.e. for arbitrary numbers of deformation variables.

Method for generating and receiving multi-scale dual-quadrature straight-extend frequency-hopping mixing signal

Abstract: A method for producing and receiving multi-level biorthogonal direct-sequence and frequency hopping (DS/FH) mixed signals relates to a method for producing and receiving DS/FH mixed signals, aiming at solving the problem that the transmission speed of the existing mixed DS/FH signals has more limitation of bandwidth. Serial-parallel conversion of information data is carried out at the transmitting terminal, and multi-level biorthogonal keying modulation of the produced parallel data is carried out; the signals after modulation have stronger anti-interference ability and multi-access ability by frequency hopping processing and then are transmitted by an antenna; the received signals are firstly dehopped and demodulated at the receiving terminal and then corresponding processing is carried out by a local biorthogonal code and the demodulated received signals; the transmitted biorthogonal code is judged by calculating correlation values and positive and negative relation thereof, and then original data is recovered by the serial-parallel conversion. The method can improve the data transmission speed without changing occupied bandwidth of system so that the system can simultaneously support more users.

Orthogonal and Biorthogonal $\sqrt 3$ -Refinement Wavelets for Hexagonal Data Processing

Abstract: The hexagonal lattice was proposed as an alternative method for image sampling. The hexagonal sampling has certain advantages over the conventionally used square sampling. Hence, the hexagonal lattice has been used in many areas. A hexagonal lattice allows radic3, dyadic and radic7 refinements, which makes it possible to use the multiresolution (multiscale) analysis method to process hexagonally sampled data. The radic3-refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale, and hence, it provides more resolution levels from which one can choose. This fact is the main motivation for the study of radic3-refinement surface subdivision, and it is also the main reason for the recommendation to use the radic3-refinement for discrete global grid systems. However, there is little work on compactly supported radic3 -refinement wavelets. In this paper, we study the construction of compactly supported orthogonal and biorthogonal radic3-refinement wavelets. In particular, we present a block structure of orthogonal FIR filter banks with twofold symmetry and construct the associated orthogonal radic3-refinement wavelets. We study the sixfold axial symmetry of perfect reconstruction (biorthogonal) FIR filter banks. In addition, we obtain a block structure of sixfold symmetric radic3-refinement filter banks and construct the associated biorthogonal wavelets.

Fermionic approach for evaluating integrals of rational symmetric functions

Abstract: We use the fermionic construction of two-matrix model partition functions to evaluate integrals over rational symmetric functions. This approach is complementary to the one used in our previous paper, where these integrals were evaluated by a direct method. Using Wick’s theorem, we obtain the same determinantal expressions in terms of biorthogonal polynomials.