Showing papers on "Orthonormal basis published in 1997"

A unifying construction of orthonormal bases for system identification

Abstract: This paper develops a general and very simple construction for complete orthonormal bases for system identification. This construction provides a unifying formulation of many previously studied orthonormal bases since the common FIR and recently popular Laguerre and two-parameter Kautz model structures are restrictive special cases of the construction presented here. However, in contrast to these special cases, the basis vectors in the unifying construction of this paper can have arbitrary placement of pole position according to the prior information the user wishes to inject. Results characterizing the completeness of the bases and the accuracy properties of models estimated using the bases are provided.

On the geometry and shape of brain sub-manifolds

Abstract: This paper develops mathematical representations for neuro-anatomically significant substructures of the brain and their variability in a population. The focus of the paper is on the neuro-anatomical variation of the geometry and the "shape" of two-dimensional surfaces in the brain. As examples, we focus on the cortical and hippocampal surfaces in an ensemble of Macaque monkeys and human MRI brains. The "shapes" of the substructures are quantified via the construction of templates; the variations are represented by defining probabilistic deformations of the template. Methods for empirically estimating probability measures on these deformations are developed by representing the deformations as Gaussian random vector fields on the embedded sub-manifolds. The Gaussian random vector fields are constructed as quadratic mean limits using complete orthonormal bases on the sub-manifolds. The complete orthonormal bases are generated using modes of vibrations of the geometries of the brain sub-manifolds. The covari...

Fractional Fourier-Kravchuk transform

Abstract: We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.

Regularity of multiwavelets

Abstract: The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Holder regularity in arbitrary L p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Holder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.

Multiwavelets for Second-Kind Integral Equations

Abstract: We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N2 to O(N log N) nonzero entries and still obtain (up to log N terms) the same convergence rates as for the exact Galerkin method.

Stability Theorems for Fourier Frames and Wavelet Riesz Bases.

Abstract: In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis, our estimate reduces to Kadec’ optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.

VLSI architectures for lattice structure based orthonormal discrete wavelet transforms

Abstract: We present efficient single-rate architectures for the one-dimensional orthonormal discrete wavelet transform (DWT). In the paper we make two contributions. First, we show that architectures that are based on the quadrature mirror filter (QMF) lattice structure require approximately half the number of multipliers and adders than corresponding direct-form structures. Second, we present techniques for mapping the 1-D orthonormal DWT to folded and digit-serial architectures which are based on the QMF lattice structure. For folded architectures, we discuss two techniques for mapping the QMF lattice structure to hardware. For digit-serial architectures, we show that any two-channel subband system can be implemented using digit-serial processing techniques by utilizing the polyphase decomposition. Using this result, we describe an orthonormal DWT architecture which uses the QMF lattice structure and digit-serial processing techniques. The proposed folded and digit-serial QMF lattice structures are attractive choices for implementations of the orthonormal DWT which require low area and low power dissipation.

Method and apparatus for transmission and reception

Abstract: In a method for transmission and reception over N transmission channels, the transmitting side performs orthonormal transform processing of N-channel modulated signals so that the resulting transformed output signals have lower cross-correlation, and the transformed N-channel signals are multiplexed and transmitted over the N-channels. At the receiving side, the N-channel received signals are subjected to inverse orthonormal transform processing to obtain N-channel modulated signals, which are demodulated to obtain a digital signal.

Nonlinear approximation of random functions

Abstract: Given an orthonormal basis and a certain class X of vectors in a Hilbert space H, consider the following nonlinear approximation process: approach a vector $x\in X$ by keeping only its N largest coordinates, and let N go to infinity. In this paper, we study the accuracy of this process in the case where $H=L^2(I)$, and we use either the trigonometric system or a wavelet basis to expand this space. The class of function that we are interested in is described by a stochastic process. We focus on the case of "piecewise stationary processes" that describe functions which are smooth except at isolated points. We show that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method, i.e., keeping the N first coordinates. The main motivation of this work is the search for a suitable mathematical model to study the compression of images and of certain types of signals.

Positivity conditions for bihomogeneous polynomials

Abstract: In this paper we continue our study of a complex variables version of Hilbert's seventeenth problem by generalizing some of the results from [CD]. Given a bihomogeneous polynomial $f$ of several complex variables that is positive away from the origin, we proved that there is an integer $d$ so that $||z||^{2d} f(z,{\overline z})$ is the squared norm of a holomorphic mapping. Thus, although $f$ may not itself be a squared norm, it must be the quotient of squared norms of holomorphic homogeneous polynomial mappings. The proof required some operator theory on the unit ball. In the present paper we prove that we can replace the squared Euclidean norm by squared norms arising from an orthonormal basis for the space of homogeneous polynomials on any bounded circled pseudoconvex domain of finite type. To do so we prove a compactness result for an integral operator on such domains related to the Bergman kernel function.

Orthonormal Wavelet Bases for Quantum Molecular Dynamics

Abstract: We report on the use of compactly supported, orthonormal wavelet bases for quantum molecular-dynamics (Car-Parrinello) algorithms. A wavelet selection scheme is developed and tested for prototypical problems, such as the three-dimensional harmonic oscillator, the hydrogen atom, and the local density approximation to atomic and molecular systems. Our method shows systematic convergence with increased grid size, along with improvement on compression rates, thereby yielding an optimal grid for self-consistent electronic structure calculations. {copyright} {ital 1997} {ital The American Physical Society}

Improved transient signal detection using a wavepacket-based detector with an extended translation-invariant wavelet transform

Abstract: This paper presents the theory of M-band, extended translation-invariant (ETI) wavelet transforms. The ETI generalizes the translation-invariant wavelet transform of Weiss (1933). It is shown that iteration of the ETI, in a tree structure, provides a signal decomposition into an orthonormal wavepacket basis. Other properties such as translation invariance and invertibility of the transform are proven. The theory is then applied to transient signal detection through development of a family of translation-invariant wavepacket-based detectors. This family of detectors provides improved the performance over previously defined wavepacket-based detectors. A performance analysis is conducted. ROC curves generated by Monte-Carlo simulation are presented, indicating the detector performance. The detector performance is demonstrated to be independent of the signal translation.

Gravity and Signature Change

Abstract: The use of proper “time” to describe classical “spacetimes” which contain both Euclidean and Lorentzian regions permits the introduction of smooth (generalized) orthonormal frames. This remarkable fact permits one to describe both a variational treatment of Einstein's equations and distribution theory using straightforward generalizations of the standard treatments for constant signature.

System identification using an over-parametrized model class-improving the optimization algorithm

Abstract: The use of an over-parametrized state-space model for system identification has some clear advantages: A single model structure covers the entire class of multivariable systems up to a given order. The over-parametrization also leads to the possibility to choose a numerically stable parametrization. During the parametric optimization the gradient calculations constitute the main computational part of the algorithm. Consequently using more than the minimal number of parameters required slows down the algorithm. However, we show that for any chosen (over)-parametrization it is possible to reduce the gradient calculations to the minimal amount by constructing the parameter subspace which is orthonormal to the tangent space of the manifold representing equivalent models.

Modelling 1-D signals using Hermite basis functions

Abstract: The paper discusses a method for estimating the Hermite coefficients of a discrete-time one-dimensional signal. To estimate the Hermite coefficients a solution based on Gaussian quadratures is used. The paper also looks at various least mean squared (LMS) estimation methods to estimate two further parameters which are incorporated into the orthonormal Hermite basis function; a spread term and a shift term. In addition, the effects of scaling, dilation and translates of a signal on its Hermite coefficients, spread and shift terms are presented. The paper concludes with a brief discussion on the potential application of the Hermite parameters as features for use in problems requiring shape discrimination within a one-dimensional signal. It also mentions those applications where this was found to be appropriate.

Positivity conditions for bihomogeneous polynomials

Abstract: In this paper we continue our study of a complex variables version of Hilbert's seventeenth problem by generalizing some of the results from [CD]. Given a bihomogeneous polynomial $f$ of several complex variables that is positive away from the origin, we proved that there is an integer $d$ so that $||z||^{2d} f(z,{\overline z})$ is the squared norm of a holomorphic mapping. Thus, although $f$ may not itself be a squared norm, it must be the quotient of squared norms of holomorphic homogeneous polynomial mappings. The proof required some operator theory on the unit ball. In the present paper we prove that we can replace the squared Euclidean norm by squared norms arising from an orthonormal basis for the space of homogeneous polynomials on any bounded circled pseudoconvex domain of finite type. To do so we prove a compactness result for an integral operator on such domains related to the Bergman kernel function.

Von Neumann Algebras and Wavelets

Abstract: Orthonormal wavelets can be regarded as complete wandering vectors for a system of bilateral shifts acting on a separable infinite dimensional Hilbert space. The local (or “point”) commutant of a system at a vector ψ is the set of all bounded linear operators which commute with each element of the system locally at ψ. In the theory we shall develop, we will show that in the standard one-dimensional dyadic orthonormal wavelet theory the local commutant at certain (perhaps all) wavelets ψ contains non-commutative von Neumann algebras. The unitary group of such a locally-commuting von Neumann algebra parameterizes in a natural way a connected family of orthonormal wavelets. We will outline, as the simplest nontrivial special case, how Meyer’s classical class of dyadic orthonormal wavelets with compactly supported Fourier transform can be derived in this way beginning with two wavelets (an interpolation pair) of a much more elementary nature. From this pair one computes an interpolation von Neumann algebra. Wavelets in Meyer’s class then correspond to elements of its unitary group. Extensions of these results and ideas are also discussed.

A new approach for subset 2-D AR model identification for describing textures

Abstract: This paper addresses the problem of identification of appropriate autoregressive (AR) components to describe textural regions of digital images by a general class of two-dimensional (2-D) AR models. In analogy with univariate time series, the proposed technique first selects a neighborhood set of 2-D lag variables corresponding to the significant multiple partial auto-correlation coefficients. A matrix is then suitably formed from these 2-D lag variables. Using singular value decomposition (SVD) and orthonormal with column pivoting factorization (QRcp) techniques, the prime information of this matrix corresponding to different pseudoranks is obtained. Schwarz's (1978) information criterion (SIG) is then used to obtain the optimum set of 2-D lag variables, which are the appropriate autoregressive components of the model for a given textural image. A four-class texture classification scheme is illustrated with such models and a comparison of the technique with the work of Chellappa and Chatterjee (1985) is provided.

Hyperbolic Equations for Vacuum Gravity Using Special Orthonormal Frames

Abstract: By adopting Nester's four-dimensional special orthonormal frames, the tetrad equations for vacuum gravity are put into explicitly causal and symmetric hyperbolic form, independent of any time slicing or other gauge or coordinate specialization.

Algorithm 776: SRRIT: a Fortran subroutine to calculate the dominant invariant subspace of a nonsymmetric matrix

Abstract: SRRT is a Fortran program to calculate an approximate orthonomral basis fr a dominant invariant subspace of a real matrix A by the method of simultaneous iteration. Specifically, given an integer m, SRRIT computes a matrix Q with m orthonormal columns and real quasi-triangular matrix T or order m such that the equation AQ = QT is satisfied up to a tolerance specified by the user. The eigenvalues of T are approximations to the m eigenvalues of largest absolute magnitude of A and the columns of Q span the invariant subspace corresponding to those eigenvalues. SRRIT references A only through a user-provided subroutine to form the product AQ; hence it is suitable for large sparse problems.

Time-stepping and preserving orthonormality

Abstract: Certain applications produce initial value ODEs whose solutions, regarded as time-dependent matrices, preserve orthonormality. Such systems arise in the computation of Lyapunov exponents and the construction of smooth singular value decompositions of parametrized matrices. For some special problem classes, there exist time-stepping methods that automatically inherit the orthonormality preservation. However, a more widely applicable approach is to apply a standard integrator and regularly replace the approximate solution by an orthonormal matrix. Typically, the approximate solution is replaced by the factorQ from its QR decomposition (computed, for example, by the modified Gram-Schmidt method). However, the optimal replacement—the one that is closest in the Frobenius norm—is given by the orthonormal polar factor. Quadratically convergent iteration schemes can be used to compute this factor. In particular, there is a matrix multiplication based iteration that is ideally suited to modern computer architectures. Hence, we argue that perturbing towards the orthonormal polar factor is an attractive choice, and we consider performing a fixed number of iterations. Using the optimality property we show that the perturbations improve the departure from orthonormality without significantly degrading the finite-time global error bound for the ODE solution. Our analysis allows for adaptive time-stepping, where a local error control process is driven by a user-supplied tolerance. Finally, using a recent result of Sun, we show how the global error bound carries through to the case where the orthonormal QR factor is used instead of the orthonormal polar factor.

Global behaviours of circles in a complex hyperbolic space

Abstract: for some positive constant k and a field of unit vectors Ys along y. Here Vs denotes the covariant differentiation along y with respect to the Riemannian connection V of M. The positive constant k is called the curvature of y. For given a positive k and an orthonormal pair of vectors u, v e TXM at a given point x e M, we have a unique circle y defined for ―oo ) we have an

Fast rotations: low-cost arithmetic methods for orthonormal rotation

Abstract: Introduces a new type of arithmetic operations that we have called "fast rotations" or "orthonormal /spl mu/-rotations". These are methods for orthonormal rotation over a set of fixed angles, with a very low cost in implementation-basically a few shift and add operations, as opposed to lengthy CORDIC operations. We also present the underlying theory for the construction of such fast rotation methods. Furthermore, we give examples where fast rotations have been applied successfully in a wide variety of applications. These include the low-cost and robust implementation of an FIR filter bank for image coding, the generation of spherical sample rays in 3D graphics, and the computation of the eigenvalue decomposition (EVD) and singular value decomposition (SVD).

Tomographic reconstruction and estimation based on multiscale natural-pixel bases

Abstract: We use a natural pixel-type representation of an object, originally developed for incomplete data tomography problems, to construct nearly orthonormal multiscale basis functions. The nearly orthonormal behavior of the multiscale basis functions results in a system matrix, relating the input (the object coefficients) and the output (the projection data), which is extremely sparse. In addition, the coarsest scale elements of this matrix capture any ill conditioning in the system matrix arising from the geometry of the imaging system. We exploit this feature to partition the system matrix by scales and obtain a reconstruction procedure that requires inversion of only a well-conditioned and sparse matrix. This enables us to formulate a tomographic reconstruction technique from incomplete data wherein the object is reconstructed at multiple scales or resolutions. In case of noisy projection data we extend our multiscale reconstruction technique to explicitly account for noise by calculating maximum a posteriori probability (MAP) multiscale reconstruction estimates based on a certain self-similar prior on the multiscale object coefficients. The framework for multiscale reconstruction presented can find application in regularization of imaging problems where the projection data are incomplete, irregular, and noisy, and in object feature recognition directly from projection data.