Showing papers on "Basis function published in 1988"
TL;DR: An approach for enforcing integrability, a particular implementation of the approach, an example of its application to extending an existing shape-from-shading algorithm, and experimental results showing the improvement that results from enforcingIntegrability are presented.
Abstract: An approach for enforcing integrability, a particular implementation of the approach, an example of its application to extending an existing shape-from-shading algorithm, and experimental results showing the improvement that results from enforcing integrability are presented. A possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions, and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes. The integrability projection constraint was applied to extending an iterative shape-from-shading algorithm of M.J. Brooks and B.K.P. Horn (1985). Experimental results show that the extended algorithm converges faster and with less error than the original version. Good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces. >
1,090 citations
TL;DR: In this paper, a new method for quantum mechanical calculations of cross sections for molecular energy transfer and chemical reactions is presented, and it is applied to inelastic and reactive collisions of I, H, and D with H2.
Abstract: A new method for quantum mechanical calculations of cross sections for molecular energy transfer and chemical reactions is presented, and it is applied to inelastic and reactive collisions of I, H, and D with H2. The method involves the expansion in a square‐integrable basis set of the amplitude density due to the difference between the true interaction potential and a distortion potential and the solution of a large set of coupled equations for the basis function coefficients. The transition probabilities, which correspond to integrals over the amplitude density, are related straightforwardly to these coefficients.
127 citations
TL;DR: The microstrip-patch resonant frequency problem is formulated in terms of an integral equation using vector Fourier transforms using Galerkin's method and the solutions obtained are interpolated with a three-variable polynomial.
Abstract: The microstrip-patch resonant frequency problem is formulated in terms of an integral equation using vector Fourier transforms. Using Galerkins's method in solving the integral equation, the resonant frequency of the microstrip patch is studied with both Chebyshev polynomials and sinusoidal functions as basis functions. In the case of the Chebyshev polynomials, the edge singularity is included, but it is not important for convergence. Furthermore, the resonant frequency of the microstrip patch is ascertained with a perturbation calculation. The results for Galerkin's method and experiments are in good agreement. The perturbation calculation agrees asymptotically with Galerkin's method. With the aim of developing a computer-aided design formula, the solutions obtained by Galerkin's method are interpolated with a three-variable polynomial. >
122 citations
06 Jun 1988
TL;DR: In this paper, the electric field integral equation is used with the moment method to develop an efficient numerical procedure for treating problems of arbitrary conducting wire/surface configurations, and three triangular-type basis functions are used to represent the currents on the bodies, wires, and wires/surface junctions.
Abstract: The electric field integral equation is used with the moment method to develop an efficient numerical procedure for treating problems of arbitrary conducting wire/surface configurations. Three triangular-type basis functions are used to represent the currents on the bodies, wires, and wire/surface junctions. A junction basis function is developed which is applicable to any junction configuration. >
96 citations
TL;DR: Spherical cap harmonic analysis as discussed by the authors is an analytical technique for modeling either a potential function and its spatial derivatives over and above a spherical cap, or a general function and their surface derivatives on the spherical cap surface.
91 citations
TL;DR: In this paper, the transfinite element-method is used in conjunction with the planar waveguide model to analyze MMIC devices, using analytic basis functions together with finite-element approximation functions in a variational technique.
Abstract: A numerical procedure called the transfinite element-method is used in conjunction with the planar waveguide model to analyze MMIC devices. By using analytic basis functions together with finite-element approximation functions in a variational technique, the transfinite-element method determines the fields and scattering parameters for a wide variety of stripline and microstrip devices. With minor modification, the method can be applied to waveguide junctions, treating singular points in the junctions very efficiently. Calculations are shown for a rectangular-waveguide two-slot 20-dB coupler, stripline band-elimination filter, and several microstrip discontinuity problems. Good agreement of the numerical results with published values demonstrates the validity of the procedure. >
89 citations
TL;DR: In the present paper the three-center nuclear attraction integrals and the two-electron multicenter integrals with B functions are evaluated in a unified way via the Fourier-transform method and Feynman's identity.
Abstract: The method of Bonham, Peacher, and Cox for computing molecular multicenter integrals for 1s Slater-type orbitals is generalized to include all states. This was possible by using B functions as basis functions which have the simplest structure under Fourier transformation, compared with other commonly used exponential-type orbitals (ETO's). Those ETO's which differ from B functions, like Slater-type orbitals (STO's), can be expressed by finite linear combinations of B functions. Therefore multicenter integrals occurring in molecular calculations with any of the commonly used ETO basis sets can be represented by integrals with B functions. In the present paper the three-center nuclear attraction integrals and the two-electron multicenter integrals with B functions are evaluated in a unified way via the Fourier-transform method and Feynman's identity. The resulting expressions require a two- or three-dimensional numerical integration, respectively. The numerical and computational properties of the resulting formulas are discussed and various test values are given. Comparison is made with some values of integrals with STO's which exist in the literature.
87 citations
TL;DR: In this paper, a molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many-body system, where the instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions with equal amplitude.
Abstract: A molecular dynamics technique is introduced for the simulation of the adiabatic dynamics of an excess electron coupled to a classical many‐body system. The instantaneous ground state wave function of the electron is represented by a superposition of distributed Gaussian basis functions, each with equal amplitude. We present generalized equations of motion for the coupled system, which optimize the positions and widths of the Gaussians by simulated annealing. The condition of equal amplitude ensures the aggregation of the Gaussians in regions of finite electron probability density and hence yields a particularly efficient representation of localized ground states. The method is applied to an electron solvated in liquid ammonia and results for equilibrium properties are compared to quantum path integral calculations. New results for the dynamics are discussed in the light of mobility measurements.
78 citations
TL;DR: In this paper, the quantum mechanical problem of reactive or nonreactive scattering of atoms and molecules is formulated in terms of square-integrable basis sets with variational expressions for the reactance matrix.
Abstract: The quantum mechanical problem of reactive or nonreactive scattering of atoms and molecules is formulated in terms of square-integrable basis sets with variational expressions for the reactance matrix. Several formulations involving expansions of the wave function (the Schwinger variational principle) or amplitude density (a generalization of the Newton variational principle), single-channel or multichannel distortion potentials, and primitive or contracted basis functions are presented and tested. The test results, for inelastic and reactive atom-diatom collisions, suggest that the methods may be useful for a variety of collision calculations and may allow the accurate quantal treatment of systems for which other available methods would be prohibitively expensive.
77 citations
TL;DR: The reliability of finite-element methods for modal analysis of two-and three-dimensional eddy-current problems is addressed in this article, where separation of variables is used to convert transient-eddy-current problem into an ordinary differential equation in time and linear combination of normal modes in space.
Abstract: The reliability of finite-element methods for modal analysis of two- and three-dimensional eddy-current problems is addressed. Separation of variables is used to convert transient-eddy-current problems into an ordinary differential equation in time and linear combination of normal modes in space. The eigensolution of the vector wave equation by the usual finite-element basis functions usually results in numerical instabilities that render the procedure worthless. It has been found that the root cause of these instabilities is the improper approximation of the null space of the curl operator. Three different methods that eliminate the instabilities completely have been developed. The first method uses C/sup 1/ or derivative continuous finite elements; the second uses tangential vector basis functions developed in a companion paper; and the third uses ordinary Lagrangian finite elements but places them in a special mesh pattern so that C/sup 1/ continuous polynomials are possible, although C/sup 1/ continuity is not imposed. >
72 citations
01 Jan 1988
TL;DR: The goal is to develop a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements and can be applied to more practical problems involving systems of PDE's, time dependence, and three spatial dimensions.
Abstract: Many elliptic partial differential equations can be solved numerically with near optimal efficiency through the uses of adaptive refinement and multigrid solution techniques It is our goal to develop a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements The basic step of the refinement process is the bisection of a pair of triangles, which corresponds to the addition of one or more basis functions to the approximation space An approximation of the resulting change in the solution is used as an error indicator to determine which triangles to divide The multigrid iteration uses a red-black Gauss-Seidel relaxation in which the black relaxations are used only locally The grid transfers use the change between the nodal and hierarchical bases This multigrid iteration requires only O(N) operations, even for highly nonuniform grids, and is defined for any finite element space The full multigrid method is an optimal blending of the processes of adaptive refinement and multigrid iteration So as to minimize the number of operations required, the duration of the refinement phase is based on increasing the dimension of the approximation space by some fixed factor which is determined to be the largest possible for the given error-reducing power of the multigrid iteration The result is an algorithm which (i) uses only O(N) operations with a reasonable constant of proportionality, (ii) solves the discrete system to the accuracy of the discretization error, (iii) is able to achieve the optimal order of convergence of the discretization error in the presence of singularities Numerical experiments confirm this for linear, quadratic and cubic elements It is believed that the method can also be applied to more practical problems involving systems of PDE's, time dependence, and three spatial dimensions
25 May 1988
TL;DR: In this paper, a numerical procedure called the transfinite-element method is used in conjunction with the planar waveguide model to analyze monolithic microwave integrated circuit (MMIC) devices.
Abstract: A numerical procedure called the transfinite-element method is used in conjunction with the planar waveguide model to analyze monolithic microwave integrated circuit (MMIC) devices. By using analytic basis functions together with finite-element approximation functions in a variational technique, the transfinite-element method can be used to determine the fields and scattering parameters for a wide variety of stripline and microstrip devices. >
TL;DR: In this article, the numerical solution of a class of second-kind integral equations in which the integral operator is not compact is discussed, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary.
Abstract: We discuss the numerical solution of a class of second-kind integral equations in which the integral operator is not compact Such equations arise, for example, when boundary integral methods are applied to potential problems in a two-dimensional domain with corners in the boundary We are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes, provided some simple modifications are made to the underlying basis functions These are sufficient to ensure stability, but do not damage the rate of convergence Numerical experiments show that such modifications are necessary in certain circumstances
11 Apr 1988
TL;DR: It is pointed out that in the analysis of transient signals such as those encounters in speech, or in certain kinds of image processing, standard Fourier analysis is often non satisfactory because the basic functions of the Fourier Analysis extend over infinite time, whereas the signals to be analyzed are short-time transients.
Abstract: It is pointed out that in the analysis of transient signals such as those encounters in speech, or in certain kinds of image processing, standard Fourier analysis is often non satisfactory because the basic functions of the Fourier analysis (sines, cosines, complex exponentials) extend over infinite time, whereas the signals to be analyzed are short-time transients. Reference is made to a method for dealing with transient signals which has recently appeared in the literature. The basis functions are referred to as wavelets, and they utilize time compression (or dilation) rather than a variation of frequency of the modulated sinusoid. Hence, all the wavelets have the same number of cycles. The analyzing wavelets must satisfy a few simple conditions, but are not otherwise specified. There is a wide latitude in the choice of these functions and they can be tailored to specific applications. The wavelets are founded on rigorous mathematical theory, and the expansions are robust. They are applied to detect ventricular delayed potentials (VLP) in the electrocardiogram. >
01 Jun 1988
TL;DR: This work presents a technique that permits constant-time filtering for space-variant kernels, and allows the use of arbitrary filters, and is useful to explore interesting mappings and special filtering techniques.
Abstract: Filtering is an essential but costly step in many computer graphics applications, most notably in texture mapping. Several techniques have been previously developed which allow prefiltering of a texture (or in general an image) in time that is independent of the number of texture elements under the filter kernel. These are limited, however, to space-invariant kernels whose shape in texture space is the same independently of their positions, and usually are also limited to a small range of filters.We present here a technique that permits constant-time filtering for space-variant kernels. The essential step is to approximate a filter surface in texture space by a sum of suitably-chosen basis functions. The convolution of a filter with a texture is replaced by the weighted sum of the convolution of the basis functions with the texture, which can be precomputed. To achieve constant time, convolutions with the basis functions are computed and stored in a pyramidal fashion, and the right level of the pyramid is selected so that only a constant number of points on the filter kernel need be evaluated.The technique allows the use of arbitrary filters, and as such is useful to explore interesting mappings and special filtering techniques. We give examples of applications to perspective and conformal mappings, and to the use of filters such as gaussians and sinc functions.
TL;DR: In this article, two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically, and it is shown that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number.
Abstract: Two different methods for the evaluation of overlap integrals of B functions with different scaling parameters are analyzed critically. The first method consists of an infinite series expansion in terms of overlap integrals with equal scaling parameters [14]. The second method consists of an integral representation for the overlap integral which has to be evaluated numerically. Bhattacharya and Dhabal [13] recommend the use of Gauss-Legendre quadrature for this purpose. However, we show that Gauss-Jacobi quadrature gives better results, in particular for larger quantum number. We also show that the convergence of the infinite series can be improved if suitable convergence accelerators are applied. Since an internal error analysis can be done quite easily in the case of an infinite series even if it is accelerated, whereas it is very costly in the case of Gauss quadratures, the infinite series is probably more efficient than the integral representation. Overlap integrals of all commonly occurring exponentially declining basis functions such as Slater-type functions, can be expressed by finite sums of overlap integrals of B functions, because these basis functions can be represented by linear combinations of B functions.
TL;DR: The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted as discussed by the authors.
Abstract: The Husimi function provides a phase‐space view of quantum systems. This paper considers a number of its properties, including ways in which it can be expanded in terms of basis functions. A spectral or ‘‘natural’’ expansion and an expansion analogous to the Carlson–Keller expansion in terms of coordinate‐density momentum‐density products are considered, as is a method for separating the angular dependence of the momentum. There is a set of functions in phase space having the same overlap properties as the initial orbital basis in terms of which the charge density matrix is expressed. A Husimi matrix is defined and a scalar product in the space containing such matrices as elements is introduced. The connection with the vector space of density matrices is examined. The Husimi function is related to the Husimi matrix in a way analogous to the relationship between a density matrix and a density, but there is a major difference in that this map can always be inverted. For a harmonic oscillator basis the phase space basis functions, in terms of which the Husimi function is expressed, span a linear space but do not provide a complete set; their products provide a linearly independent set that is complete. It is suggested that similar behavior can be expected for other basis sets.
TL;DR: In this article, the propagation characteristics of wave along a periodic array of parallel signal lines in a multilayered structure in the presence of a periodically perforated ground plane are studied.
Abstract: The propagation characteristics of waves along a periodic array of parallel signal lines in a multilayered structure in the presence of a periodically perforated ground plane are studied. To analyze this structure, the spectral-domain immittance approach is used to derive the Green's function. The surface current density on the conductors is expressed in terms of a set of rooftop subdomain basis functions. The boundary condition is then enforced in conjunction with the Galerkin procedure, leading to an eigenvalue problem for the propagation constant which is solved by the Newton-Raphson algorithm. The dispersion characteristics of these signal lines are studied for both balanced and unbalanced excitations with the relative permittivities of the various layers as parameters. Numerical results are presented and compared with available data. Extension of the present method to treat conductors with finite sheet resistances is also discussed. >
TL;DR: In this paper, a variational formulation involving the expansion of the transverse E field at the step in terms of suitable basis functions is presented. But it is only applied to a five-section low-pass filter.
Abstract: A rigorous analysis of boxed microstrip single-step discontinuities and cascades of strongly coupled discontinuities is presented. Use is made of a variational formulation involving the expansion of the transverse E field at the step in terms of suitable basis functions. Strongly coupled steps are analyzed using the concept of localized and accessible modes and making use of a network model. The method is applied to a five-section low-pass filter. >
14 Nov 1988
TL;DR: Rotation-invariant operators of order 0 and 2 are used first to detect and then to enhance lines, suitable for fingerprints, based on suppression of line responses with an orientation that creates a mismatch with the locally dominating direction of the fingerprint.
Abstract: Rotation-invariant operators of order 0 and 2 are used first to detect and then to enhance lines. The enhancement procedure, suitable for fingerprints, is based on suppression of line responses with an orientation that creates a mismatch with the locally dominating direction of the fingerprint. The rotation-invariant operators also serve as basis functions. By construction they are mutually orthogonal and by proper normalization they form an orthonormal set. At the origin any pattern a(r, phi ) can be expanded or at least approximated by such a set of basis functions. >
TL;DR: Two reconstruction methods that recognize smoothness to be a priori information and use numerically spacelimited basis functions have been developed and the efficacy has been demonstrated using a limited number of projections synthetically generated from a distribution phantom.
Abstract: Two reconstruction methods that recognize smoothness to be a priori information and use numerically space-limited basis functions have been developed. The first method is a modification of the well-known convolution method and uses such basis functions for the projections. The second method is a continuous algebraic reconstruction technique that employs consistent basis functions for the projections as well as the distribution and makes use of other a priori information like the constraints on the domain as well as the range of the distribution in a rigorous way. The efficacy of these methods has been demonstrated using a limited number of projections synthetically generated from a distribution phantom.
TL;DR: The equations of motion of an arbitrary piezoelectric plate are represented by a set of second-order differential equations involving only the transverse spatial coordinates, which provides a quite general basis for modeling devices such as trapped energy resonators.
Abstract: The equations of motion of an arbitrary piezoelectric plate are represented by a set of second-order differential equations involving only the transverse spatial coordinates. This is achieved by expanding the thickness dependence in a set of basis functions derived from the solutions to the plate problem at cutoff. Techniques are presented for constructing approximate plate equations using only chosen mode amplitudes; such equations predict the true cutoff frequencies and give dispersion curves which are rigorously correct up to terms of order k/sup 2/. The coefficients in these equations can be computed analytically, and techniques for doing this are presented. Comparisons with dispersion curves calculated by partial wave analysis are given both for quartz and for lithium niobate. The theory provides a quite general basis for modeling devices such as trapped energy resonators. >
TL;DR: In this article, a Timoshenko beam finite element is developed using trigonometric basis functions and the properties and performance of these new elements are explored through a series of illustrative problems that treat both straight and curved geometries.
Abstract: A Timoshenko beam finite element is developed using trigonometric basis functions. The properties and performance of these new elements are explored through a series of illustrative problems that treat both straight and curved geometries. Both linear and nonlinear strain-displacement relations are employed in the formulation and comparison is made to results obtained from full, completely reduced, and partially reduced integration of polynomial basis functions of degree one through three. The results obtained in this work indicate that the trigonometric basis functions are a competitive alternative to polynomial basis functions with regard to accuracy and convergence and most importantly they possess a freedom from shear and membrane locking. The trigonometric basis functions also allow the recovery of rigid-body motions in the case of straight and circular arc curved beams.
TL;DR: In this article, a specific numerical algorithm for solving the Schrodinger equation for a one-dimensional hydrogen atom in an oscillating electric field is described which uses these results for the dipole matrix elements along with a discrete representation of the continuum.
Abstract: Exact integral expressions and simple analytical estimates for the bound-bound, bound-continuum, and continuum-continuum dipole matrix elements are derived by use of the momentum-space eigenfunctions for a one-dimensional model of a hydrogen atom. These results provide the essential ingredients for the numerical study of the quantum mechanisms responsible for the chaotic ionization of highly excited hydrogen atoms in intense microwave fields. A specific numerical algorithm for solving the Schr\"odinger equation for a one-dimensional hydrogen atom in an oscillating electric field is described which uses these results for the dipole matrix elements along with a discrete representation of the continuum. In addition, the momentum-space representation of the Sturmian basis functions has been used to derive exact integral expressions and convenient analytical estimates for the projections of the Sturmian basis functions onto the hydrogenic bound and continuum states. These results are used to provide a direct comparison of numerical calculations for the ionization of one-dimensional hydrogen atoms using the hydrogenic and Sturmian bases.
TL;DR: These methods employing continuous basis functions are appropriate for reflecting the peculiar characteristics of interferometric tomography of fluid flow fields, namely, continuity, data sparsity, and nonuniform sampling.
Abstract: Series-expansion methods for interferometric tomography of continuous flow fields are discussed. The techniques are based on series expansion by orthogonal polynomials and circular harmonics multiplied by an envelope function. These methods employing continuous basis functions are appropriate for reflecting the peculiar characteristics of interferometric tomography of fluid flow fields, namely, continuity, data sparsity, and nonuniform sampling. The high approximating power of the methods allows accurate representation of fields with a small number of series terms. This generates enough redundancy for a given number of data points in setting up a system of linear algebraic equations. The data redundancy thus generated yields accurate reconstruction even under ill-posed conditions including limited view angle, incomplete projections, and high noise level.
TL;DR: A musical tone synthesis technique based on the Karhunen-Loeve (KL) transform is introduced, reducing the computational load considerably by replacing the sinusoids with a smaller set of basis functions derived from KL statistical techniques.
Abstract: A musical tone synthesis technique based on the Karhunen-Loeve (KL) transform is introduced. Currently, most additive synthesis methods are based on a truncated Fourier series representation. The computational load has been reduced considerably, compared to that of Fourier-based additive synthesis, by replacing the sinusoids with a smaller set of basis functions derived from KL statistical techniques. Algorithms are developed for the analysis of tones, yielding the basis functions and synthesis parameters. Energy compression in the KL basis functions is enhanced by a classification procedure applied to the tones to be synthesized. A high level of natural quality is achieved in tones synthesized with this method. >
TL;DR: In this paper, an algorithm for approximating an unknown function f(x) given many function values containing random noise is described, where the approximation constructed is a cubic spline with sufficient basis functions to represent the function accurately.
Abstract: An algorithm is described for approximating an unknown function f(x), given many function values containing random noise. The approximation constructed is a cubic spline g(x) with sufficient basis functions to represent f(x) accurately. The basis-function coefficients are determined by minimizing a combination of the infidelity E (the mean-square errorz between g(x) and the data,and the roughness T (which is a measure of the tortuosity of g(x)). The quantity minimized is E+pT, where p is a smoothing parameter. A suitable value of p is determined by cross validation.Results of numerical tests are reported which show that this algorithm is superior to least-squares cubic splines: in general the statistical errors are substantially less, and they are insensitive to the number of basis functions used.
TL;DR: In this paper, the Surface Integral Equation (SIE) method is applied to the solution of electromagnetic transmission boundary value problems, and the results generally compare well with analytical solutions, or solutions obtained numerically by other means.
Abstract: The theory by which the Surface Integral Equation method may be applied to the solution of electromagnetic transmission boundary value problems is presented. For a 3D target of arbitrary electrical property contrast with its host medium excited by an arbitrary time‐harmonic source, two integral equations are derived which need to be simultaneously solved for tangential electric and magnetic source density on the target's surface. If the target is 2D, though still excited by an arbitrary source (the 2½ D case), the problem is best solved in the transform domain for a number of different wavenumbers in the target's strike direction. Then a set of four simultaneous scalar integral equations needs to be solved for the components of the surface source density transforms in the target's strike direction and in the direction of the tangent vector to the target's cross‐sectional contour. Examples are presented in which the 2½D problem is solved numerically using the method of moments with piecewise linear basis functions. Although the results generally compare well with analytical solutions, or solutions obtained numerically by other means, errors appear in the calculation of the real response of these targets to excitation by a magnetic dipole source at low frequencies. This is attributed to ill‐conditioning of the system resulting from a non‐unique solution at zero frequency. Copyright
TL;DR: In this article, a detailed presentation of the recently introduced integration-free method, with applications to determine the energy levels of the generalised quantum anharmonic oscillators, are given.
Abstract: A detailed presentation of the recently introduced integration-free method, with applications to determine the energy levels of the generalised quantum anharmonic oscillators, are given. Numerical calculations are realised for the quartic and the sextic oscillators. Energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method. Certain remarks about the selection of the basis functions and a convergence discussion on the presented simple approximation scheme are also included in this paper.
TL;DR: Wavelets are well founded on rigorous mathematical theory, and the expansions are robust, and they are applied to detect ventricular delayed potentials in the electrocardiogram.