Showing papers on "Basis function published in 1987"

Orthogonal Pyramid Transforms For Image Coding.

Abstract: We describe a set of pyramid transforms that decompose an image into a set of basis functions that are (a) spatial-frequency tuned, (b) orientation tuned, (c) spatially localized, and (d) self-similar. For computational reasons the set is also (e) orthogonal and lends itself to (f) rapid computation. The systems are derived from concepts in matrix algebra, but are closely connected to decompositions based on quadrature mirror filters. Our computations take place hierarchically, leading to a pyramid representation in which all of the basis functions have the same basic shape, and appear at many scales. By placing the high-pass and low-pass kernels on staggered grids, we can derive odd-tap QMF kernels that are quite compact. We have developed pyramids using separable, quincunx, and hexagonal kernels. Image data compression with the pyramids gives excellent results, both in terms of MSE and visual appearance. A non-orthogonal variant allows good performance with 3-tap basis kernels and the appropriate inverse sampling kernels.

A new basis set method for quantum scattering calculations

Abstract: A new basis set approach for quantum scattering calculations is described and tested on model problems of elastic and inelastic collisions. The approach is essentially the Kohn variational method, but applied to the S or T matrix directly rather than to the K matrix as is normally done; it is seen that the result of the present approach is not equivalent to the usual Kohn method (i.e., for the K matrix) and is indeed preferable to it. The present approach is seen to have the same structure as the complex scaling/coordinate rotation expressions for the T matrix (but with some added features). Its potential advantage over the Schwinger variational method, another useful basis set technique, is that matrix elements of the Green’s function for some reference Hamiltonian are not required; the present method requires only matrix elements of the Hamiltonian itself between the basis functions. The essential reason for all of these desirable features is that the basis set which is used incorporates the correct sca...

Solution of the Hartree–Fock equations for polyatomic molecules by a pseudospectral method

Abstract: A pseudospectral code for general polyatomic molecules has been developed using Gaussian basis functions As an example, the water molecule is studied using a 6‐31G** basis set Quantitative agreement with conventional calculations is obtained for the equilibrium geometry, total energy, first ionization potential, and vibrational force constants Timing results for a vectorized version of the code (run on a Cray X‐MP) indicate that for large molecules, rate enhancements of Hartree–Fock self‐consistent field calculations of order 103 can be achieved

A method for computing spectral reflectance

Abstract: Psychophysical experiments show that the perceived colour of an object is relatively independent of the spectrum of the incident illumination and mainly depends on the surface spectral reflectance. We first demonstrate a possible solution to this undetermined problem for a Mondrian world of flat rectangular patches. We expand the illumination and surface reflectances in terms of a finite number of basis functions. We assume that the number of colour receptors is greater than the number of basis functions. This yields a set of nonlinear equations for each colour patch. Number counting arguments show that, given a sufficient number of surface patches with the same illumination, there are enough equations to determine the surface reflectances up to an overall scaling factor. This theory is similar to previous and independent work by Maloney and Wandell (Maloney 1985). We demonstrate a simple method of solving these non-linear equations. We generalize to situations where the illumination varies in space and the objects are three dimensional shapes. To do this we define a method for detecting material changes, a colour edge detector, and illustrate a way of detecting the colour of a material at its boundaries and propagating it inwards.

Dispersion Characteristics of Open Microstrip Lines

Abstract: A method based on spectral-domain analysis is derived to compute effectively and automatically the effective relative permittivity of an open microstrip line. Simple but accurate closed-form expressions are used as the basis functions of the longitudinal and transverse current distributions. The numerical results are shown in tables and figures for various cases and compared WMI other available results. The results presented here are seen as having a high degree of accuracy and may be used as reference standards,

Wave solutions under complex space–time shifts

Abstract: The idea of analytic continuation in both time and spatial source variables is adopted here to seek new space–time solutions of the wave equation. The solutions obtained depend not only on the complex space–time-shift parameters but on the specific temporal source behavior, which, in principle, can be selected at will. Three choices are investigated explicitly: The time-harmonic excitation leads to the well-known and widely applied time-harmonic Gaussian beams; the temporal impulse and the Gaussian excitations yield new solutions and are of particular interest, as they possess several desirable properties discussed in the text. A major application for the obtained solutions is their use as basis functions for generalized space–time-field representations. This leads to certain requirements. First, the solutions must form a complete basis. Second, they must be simple. Simplicity, here, implies the capability to calculate the expansion coefficients at an acceptable computational cost. This requirement is nontrivial, as the available solutions do not constitute an orthogonal basis. The field response to the Gaussian excitation is shown to be simple in this sense.

Complex rays for radiation from discretized aperture distributions

Abstract: Many electromagnetic propagation problems require tracking of fields radiated by large actual or induced aperture distributions through complicated environments before reaching the observer. For a systematic approach to this problem area, it is desirable to represent the aperture field in terms of basis functions which are physically informative and well adapted to traversing the propagation path. At high frequencies, Ganssian beam-type basis functions meet these requirements. After referring to a rigorous aperture discretization scheme, various quasi-Gaussian basis field profiles are examined, with a special view toward expressing their radiation properties in terms of complex rays; complex ray tracing is promising for field tracking in complicated surroundings. By comparing reference solutions from numerical integration of radiation integrals with complex ray asymptotics, it is concluded that the true Gaussian has the most favorable attributes for matching aperture discretization, propagation requirements, and complex ray tracing. Thus, the analysis here may point the way toward systematic treatment of the above-noted class of propagation problems.

Optimization of a distributed Gaussian basis set using simulated annealing: Application to the solvated electron

Abstract: A distributed Gaussian basis set is employed to calculate the energy eigenvalues and eigenstates of an electron solvated in a molecular fluid. A combination of two techniques is proposed to optimize the basis functions. The first consists of treating the centers of the Gaussians as dynamical variables whose optimal distribution for a given solvent configuration is obtained by the technique of simulated annealing. In addition, the Gaussian basis functions are multiplied by Jastrow factors which suppress excessive overlap within the repulsive cores of the solvent molecules. The method is applied to a series of rigid solvent configurations sampled from a path integral Monte Carlo simulation of an electron solvated in ammonia. A careful comparison is made with averages derived from the path integral calculations. The two methods are found to agree within 10% for the ground state energy. From the sample of different solvent configurations the adiabatic fluctuation in the energy eigenvalues and the density of states is analyzed and compared to experimental data.

Chebyshev Spectral Methods for Limited-Area Models. Part I: Model Problem Analysis

Abstract: This study considers how spectral methods can be applied to limited-area models using Chebyshev polynomials as basis functions. We review the convergence of Sturm–Liouville series to motivate the use of the Chebyshev polynomials, and describe the tau and collocation projections which allow the use of general (nonperiodic) boundary conditions. These methods are illustrated for a simple model problem, the linear advection equation in one dimension, and numerical results confirm their high accuracy. Time differencing and efficiency are considered in detail using both asymptotic analysis and numerical result from the model problem. The stability condition for Chebyshev methods with explicit time differencing, often thought to be severe, is shown to be less severe than that for finite difference methods when high accuracy is desired. Fourth-order Runge-Kutta time differencing is the most efficient of the many schemes considered. When the accuracy desired is high enough, Chebyshev spectral methods are ...

Numerical implementation of the Sinc‐Galerkin method for second‐order hyperbolic equations

Abstract: A fully Galerkin method in both space and time is developed for the second-order, linear hyperbolic problem. Sinc basis functions are used and error bounds are given which show the exponential convergence rate of the method. The matrices necessary for the formulation of the discrete system are easily assembled. They require no numerical integrations (merely point evaluations) to be filled. The discrete problem is formulated in two different ways and solution techniques for each are described. Consideration of the two formulations is motivated by the computational architecture available. Each has advantages for the appropriate hardware. Numerical results reported show that if 2N + 1 basis functions are used then the exponential convergence rate , κ > 0, is attained for both analytic and singular problems.

On locally supported basis functions for the representation of geometrically continuous curves

Abstract: We say that a curve has geometric continuity if the curve, its unit tangent vector and its curvatures relative to a Frenet frame are continuous. Here we study locally supported basis functions in spaces of piecewise polynomials which can be used for the Bezier representation of geometrically continuous curves. For spaces with identical connection conditions at each knot, we study the dependence of the basis functions on the Parameters defining the connection conditions, and give a method for the computation of these functions. We also investigate the effect of the various parameters on the shape of the Bezier curve. For connection conditions prescribed by a totally positive matrix, we obtain conditions for the solvability of the cardinal Interpolation problem and give a new proof of the uniqueness and non-negativity of the basis functions. AMS 1980 Classification number: 41

A new local basis for designing with tensioned splines

Abstract: Recently there has been a great deal of interest in the use of “tension” parameters to augment control mesh vertices as design handles for piecewise polynomials. A particular local cubic basis called B-splines, which has been termed a “generalization of B-splines, v has been proposed as an appropriate basis. These functions are defined only for floating knot sequences. This paper uses the known property of B-splines that with appropriate knot vectors span what are called here spaces of tensioned splines, and that particular combinations of them, called LT-splines, form bases for the spaces of tensioned splines. In addition, this paper shows that these new proposed bases have the variation diminishing property, the convex hull property, and straightforward knot insertion algorithms, and that both curves and individual basis functions can be easily computed. Sometimes it is desirable to interpolate points and also use these tension parameters, so interpolation methods using the LT-spline bases are presented. Finally, the above properties are established for uniform and nonuniform knot vectors, open and floating end conditions, and homogeneous and nonhomogeneous tension parameter pairs.

Boundary element method for dielectric resonators and waveguides

Abstract: A general mode-matching or boundary element method is described for formulating the eigenvalue problem associated with dielectric resonators and waveguides. The method is based on the use of dyadic Green's functions for which various different forms can be chosen. These different choices and the resultant computational advantages are discussed. The method is applied to the problems of a cylindrical dielectric resonator and a rectangular dielectric waveguide. It is shown that very accurate values of the dominant mode eigenvalue can be obtained using only a few global basis functions.

Four-Index Transformations

Abstract: Atomic and molecular electronic structure calculations are most frequently performed by employing basis set expansion techniques; that is, by invoking the algebraic approximation (for recent reviews see Refs. 1 and 2). In electronic structure calculations which go beyond an independent electron or orbital model and take account of electron correlation effects, it is necessary to perform, either explicitly or implicitly, a transformation. Specifically, it is necessary to carry out a transformation of integrals involving the components of the Schrodinger operator over the chosen basis functions, usually either exponential-type functions or Gaussian-type functions, to integrals over the orbitals resulting from the independent electron model calculation, usually a self-consistent-field calculation.

Application of Pocklington's equation to analysis of dipole frequency-selective surfaces of finite size

Abstract: Pocklington's equation is applied to arrays of linear dipoles. An integral equation for the currents induced in each dipole is derived for plane-wave illumination. The equation is solved by the method of moments. The current mode amplitudes for a 5 × 5 array and a 20 × 5 array are compared with those given by the standard modal analysis of an infinite periodic array using the same set of basis functions. In the latter case predicted transmission responses in the frequency range 18-40 GHz link the calculations to experiment. The effect of the edge region on the induced current phasors and on the scattered fields is discussed. Near the resonance frequency for TE incidence there is a pronounced amplitude variation across the 20 × 5 array. The analysis described in the paper is not restricted to periodic lattices.

Convergence phenomenon in the solution of dichroic scattering problems by Galerkin's method

Abstract: The convergence problem that arises when the scattering problem of dichroic surfaces is solved numerically by means of Galerkin's method and Floquet's theorem is investigated. A theoretical discussion is outlined showing that a truncation of the infinite Floquet spectrum is actually equivalent to modifying the used basis functions. Examples of modified basis functions due to different Floquet-mode truncations and the effect on the numerical results are demonstrated for the particular cases of arrays of thin dipoles and crossed dipoles. Some useful truncation rules are also presented, and the theories are verified by comparing predicted values with experimental results obtained by waveguide simulator measurements

Multidimensional quantum eigenstates from the semiclassical dynamical basis set

Abstract: A new method for obtaining molecular vibrational eigenstates using an efficient basis set made up of semiclassical eigenstates is presented. Basis functions are constructed from a ‘‘primitive’’ basis of Gaussian wave packets distributed uniformly on the phase space manifold defined by a single quasiperiodic classical trajectory (an invariant N‐torus). A uniform distribution is constructed by mapping a grid of points in the Hamilton–Jacobi angle variables, which parametrize the surface of the N‐torus, onto phase space by means of a careful Fourier analysis of the classical dynamics. These primitive Gaussians are contracted to form the semiclassical eigenstates via Fourier transform in a manner similar to that introduced by De Leon and Heller [J. Chem. Phys. 81, 5957 (1984)]. Since the semiclassical eigenstates represent an extremely good approximation of the quantum eigenstates, small matrix diagonalizations are sufficient to obtain eigenvalues ‘‘converged’’ to 4–5 significant figures. Such small diagonali...

Variational method for computing surface acoustic pressure on vibrating bodies, applied to transversely oscillating disks

Abstract: A variational principle derived from the Kirchhoff-Helmholtz integral relation can be applied to acoustic radiation and diffraction problems. An illustrative example discussed here is that of sound radiation from a flat rigid circular disk in transverse oscillation. The variational formulation has the surface pressure as the unknown variable, with the normal velocity of the surface taken as given. The Rayleigh-Ritz method used in determining approximate solutions in terms of truncated expansions of basis functions encounters some numerical problems in the evaluation of integrals with singular integrands. The integrands are nevertheless integrable and techniques are described for handling the singularities. Another potential source of difficulty is that the tangential derivative of the surface pressure for the exact solution must be infinite at the edge of the disk. One makes use of prior knowledge of such a fact by using basis functions with the correct dependence on radial distance near the disk edge. Because basis functions in the Rayleigh-Ritz procedure have been selected with the aid of prior insight into the nature of the true solution, accurate results are obtained with a relatively small number of basis functions. The numerical solutions agree well with results calculated by Leitner in 1949.

Real-space formulation of the mixed-basis pseudopotential method: Bulk structural properties of elemental copper

Abstract: We present a formulation of the mixed-basis expansion for electronic structure calculations, which allows calculations on systems in which there is a strongly localized component of the valence charge density. As in the conventional mixed-basis expansion, a small plane-wave basis is augmented with a set of auxiliary functions to describe the localized component of the wave functions. Unlike the conventional mixed-basis scheme, however, a fixed set of optimized nonoverlapping auxiliary functions are employed, so that matrix elements involving this set are calculable by very fast and accurate one-dimensional k-independent quadrature. The method is applied to study the electronic structure and bulk structural properties of Cu. The electronic structure, based on the pseudopotential of Bachelet, Hamann, and Schl\"uter, compares well with that obtained from other self-consistent state-of-the-art all-electron methods. The total energies for Cu in the fcc and bcc crystal structures are calculated and compared. We find that the fcc structure is favored at all densities, although the bcc exhibits an unusually stable high-density structure. A number of technical points relating to the use of these optimized local basis functions in band-structure calculations are discussed.

Composite modeling of nonstationary signals

Abstract: Most temporal signals of practical interest are nonstationary and need to be modeled using time-varying systems. In this paper a composite model for these signals is proposed which accounts for nonstationarities in the mean and the autocorrelation functions. Unbiased and consistent time-varying estimators for the mean and the variance functions are studied and used to produce zero-mean, constant-variance signals that can be modeled using autoregressive systems with time-varying coefficients. The identification of the coefficients is implemented recursively using the parameterization of the coefficients by a set of basis functions. We illustrate the application of the composite model in the analysis and synthesis of speech and in the estimation of instantaneous frequencies in radar return signals.

Multilevel adaptive technique for quantum reactive scattering

Abstract: Discretization (with localized basis functions or grid points) of the coupled integral equations for molecular collisions leads to a very large system of linear algebraic equations. New methods, which are well adapted to vector supercomputers and parallel architectures, are developed for solving this large system. The multilevel adaptive technique (MLAT) is combined with recursive and iterative techniques. First, a multichannel solution is obtained on a low level grid. The basis is then adapted to this solution and the coarse solution is projected or interpolated onto the adapted basis. The scattering amplitudes (K‐matrix elements) on the high level are then developed through use of either the recursion method (for single amplitudes, or a small batch of them) or the iterative technique (for all transitions from a specified initial state). In both of these methods, the original large system of algebraic equations is projected into a much smaller subspace (an orthonormalized Krylov space) spanned by a few b...

Resistivity and induced polarization responses of arbitrarily shaped 3‐d bodies in a two‐layered earth*

Abstract: Numerical computations using the integral equation method are presented for resistivity and IP responses due to arbitrarily shaped 3-dimensional bodies in a layered earth. The unknown surface charge density distribution is expressed as the solution of Fredholm's integral equation of the second kind. Use of moment method (with pulse basis function and point-collocation) yields the matrix equations for the unknowns. The contributions to Green's function are solved (a) analytically for the primary and (b) by convolution for the secondary contributions resulting in a fast algorithm. The further step of computing potential, apparent resistivity, chargeability etc., for any electrode system, is straightforward. Our results show a good agreement with those from finite difference methods and physical tank experiments. The CPU time is only 138 s on a super-minicomputer for an apparent resistivity pseudo-section, even with 96 elementary cells as used for discretization. A large number of models for different geological situations were studied; some are presented here.

The Hylleraas‐CI method in molecular calculations: Two‐electron integrals

Abstract: In the Hylleraas-CI method, first proposed by Sims and Hagstrom, correlation factors of the type r are included into the configurations of a CI expansion. The computation of the matrix elements requires the evaluation of different two-, three-, and four-electron integrals. In this article we present formulas for the two-electron integrals over Cartesian Gaussian functions, the most used basis functions in molecular calculations. Most of the integrals have been calculated analytically in closed form (some of them in terms of the incomplete Gamma function), but in one case a numerical integration is required, although the interval for the integration is finite and the integrand well-behaved. We have also reported on partial and preliminary computations for the H2 molecule using our four-center general formulas; a basis set of s- and p-type functions yielded at R = 1.4001 A an energy of - 1.174380 a.u. to be compared with Kolos and Wolniewicz value of - 1.174475.

A generalization of Van den Berg's integral-square error iterative computational technique for scattering

Abstract: Van den Berg shows a way of using spectral iteration (SI) within a scheme that shows good convergence. His method can be reinterpreted as a global expansion of the field quantities in terms of a set of basis functions. It is shown here that the orthogonalization of these basis functions can considerably improve convergence rates. This involves extra storage requirements, but only a negligible decrease in speed of computation.

Uniform Convergence of Galerkin Solutions to Non-compact Integral Operator Equations

Abstract: We consider a class of second-kind integral equations in which the operator is not compact. These may be solved numerically by a Galerkin method using piecewise polynomials as basis functions. Here we improve the existing stability analysis for such methods, and establish optimal rates of convergence for the Galerkin solution in the uniform norm. Also, superconvergence is established for the iterated Galerkin solution and for a suitably averaged Galerkin solution.

Numerical treatment of boundary conditions at points connecting more than two electrically distinct regions

Abstract: Points of intersection of three or more lossy dielectric regions (i.e. tissues) often occur in anatomical cross-sections of the human body obtained with CT-scans. The boundary conditions for the electric and magnetic fields require special attention at these points, in contrast to points of connectin of only two tissues, thus posing a problems for numerical methods involved with calculating the electromagnetic fields in tissue. In this paper, a simple theoretical analysis, which exposes the dominant analytic behaviour near such points, is followed. The derived analytic behaviour is numerically enforced via a special basis function which is compatible with the finite element method. Results are shown which indicate that the special basis preserves the analytic behaviour while maintaining grid resolution that is typical of the overall mesh.

Rigorous finite-basis-set approach to the calculation of the angular dependence of photoionization

Abstract: A practical solution is provided for the problem of using finite-basis-set methods to calculate the asymmetry parameter, ..beta..(E), which determines the angular dependence of photoejection in the dipole approximation. This problem has been an outstanding limitation of basis-set approaches to photoionization in general, which have heretofore been able only to produce the total cross section. The procedure proposed here is given in two forms: an evaluation of the differential cross section in terms of the outgoing flux, and an alternative prescription involving matching to the outgoing wave function. In both cases the method of complex basis functions is used to compute the necessary matrix elements of the electronic Green's function. Numerical results are reported for ..beta..(E) for ionization of the 2p state of atomic hydrogen.

The calculation of transient eddy-current fields using null-field integral techniques

Abstract: The transient eddy-current problem is characteristically computationally intensive. The motivation for this research was to realize an efficient accurate solution technique involving small matrices via an eigenvalue approach. Such a technique is indeed realized and tested using the null-field integral technique. Using smart (i.e., efficient, global) basis functions to represent unknowns in terms of a minimum number of unknowns, homogeneous eigenvectors and eigenvalues are first determined. The general excitatory response is then represented in terms of these eigenvalues and eigenvectors. Excellent results are obtained for the Argonne FELIX cylinder experiments using a 4 × 4 matrix. Extension to the three-dimensional problem (short cylinder) is set-up in terms of an 8 × 8 matrix.