Showing papers on "Basis function published in 1993"

Identification of nonlinear biological systems using Laguerre expansions of kernels.

Abstract: Identification of nonlinear dynamic systems using the Volterra-Wiener approach requires the estimation of system kernels from input-output data. A kernel estimation technique, originally proposed by Wiener (1958) and recently studied by Ogura (1986), employs Laguerre expansions of the kernels and estimates the unknown expansion coefficients via time-averaging of covariance samples. This paper presents another implementation of the technique which utilizes least-squares fitting instead of covariance time-averaging and provides for the proper selection of the intrinsic Laguerre parameter “α”. Results from simulation examples demonstrate that this implementation can yield accurate kernel estimates up to 3rd-order from short input-output data records. Furthermore, it is shown that this implementation remains effective in the presence of noise and when the spectral characteristics of the input signal deviate somewhat from the theoretical requirements of whiteness. the computational requirements and the overall performance of this technique compare favorably to existing methods, especially in cases where the system kernels can be represented with a relatively small number of Laguerre basis functions.

On the use of wavelet expansions in the method of moments (EM scattering)

Abstract: An approach which incorporates the theory of wavelet transforms in method-of-moments solutions for electromagnetic wave interaction problems is presented. The unknown field or response is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function, which is often referred to as the mother wavelet. The wavelet expansion can adaptively fit itself to the various length scales associated with the scatterer by distributing the localized functions near the discontinuities and the more spatially diffused ones over the smooth expanses of the scatterer. The approach is thus best suited for the analysis of scatterers which contain a broad spectrum of length scales ranging from a subwavelength to several wavelengths. Using a Galerkin method and subsequently applying a threshold procedure, the moment-method matrix is rendered sparsely populated. The structure of the matrix reveals the localized scale-fitting distribution long before the matrix equation is solved. The performance of the proposed discretization scheme is illustrated by a numerical study of electromagnetic coupling through a double-slot aperture. >

A posteriori error estimates based on hierarchical bases

Abstract: The authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions. The authors analyze nonlinear, nonselfadjoint, and indefinite problems as well as the selfadjoint, positive-definite case. Because both the analysis and the estimator itself are quite simple, it is easy to see how various approximations affect the quality of the estimator. As examples, the authors apply the theory to some scalar elliptic equations and the Stokes system of equations.

Wave‐net: a multiresolution, hierarchical neural network with localized learning

Abstract: A Wave-Net is an artificial neural network with one hidden layer of nodes, whose basis functions are drawn from a family of orthonormal wavelets. The good localization characteristics of the basis functions, both in the input and frequency domains, allow hierarchical, multiresolution learning of input-output maps from experimental data. Furthermore, Wave-Nets allow explicit estimation for global and local prediction error-bounds, and thus lend themselves to a rigorous and explicit design of the network. This article presents the mathematical framework for the development of Wave-Nets and discusses the various aspects of their practical implementation. Computational complexity arguments prove that the training and adaptation efficiency of Wave-Nets is at least an order of magnitude better than other networks. In addition, it presents two examples on the application of Wave-Nets; (a) the prediction of a chaotic time-series, representing population dynamics, and (b) the classification of experimental data for process fault diagnosis.

Electronic-structure calculations in adaptive coordinates.

Abstract: The plane-wave method for electronic-structure calculations is reformulated in generalized curvilinear coordinates. This introduces a new set of basis functions that depend continuously on a coordinate transformation, and can adapt themselves to represent optimally the solutions of the Schrodinger equation. As a consequence, the effective plane-wave energy cutoff is allowed to vary in the unit cell in an unbiased way. The efficiency of this method is demonstrated in the calculation of the equilibrium structures of the CO and H 2 O molecules using the local-density approximation of density-functional theory, and norm-conserving, nonlocal pseudopotentials. The easy evaluation of forces on all degrees of freedom makes the method suitable for ab initio molecular-dynamics applications

Learning control using fuzzified self-organizing radial basis function network

Abstract: This note describes an approach to integrating fuzzy reasoning systems with radial basis function (RBF) networks and shows how the integrated network can be employed as a multivariable self-organizing and self-learning fuzzy controller. In particular, by drawing some equivalence between a simplified fuzzy control algorithm (SFCA) and a RBF network, we conclude that the RBF network can be interpreted in the context of fuzzy systems and can be naturally fuzzified into a class of more general networks, referred to as FBFN, with a variety of basis functions (not necessarily globally radial) synthesized from each dimension by fuzzy logical operators. On the other hand, as a result of natural generalization from RBF to SFCA, we claim that the fuzzy system like RBF is capable of universal approximation. Next, the FBFN is used as a multivariable rule-based controller but with an assumption that no rule-base exists, leading to a challenging problem of how to construct such a rule-base directly from the control environment. We propose a simple and systematic approach to performing this task by using a fuzzified competitive self-organizing scheme and incorporating an iterative learning control algorithm into the system. We have applied the approach to a problem of multivariable blood pressure control with a FBFN-based controller having six inputs and two outputs, representing a complicated control structure. >

Priors Stabilizers and Basis Functions: From Regularization to Radial, Tensor and Additive Splines

Abstract: We had previously shown that regularization principles lead to approximation schemes, as Radial Basis Functions, which are equivalent to networks with one layer of hidden units, called Regularization Networks. In this paper we show that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models, Breiman''s hinge functions and some forms of Projection Pursuit Regression. In the probabilistic interpretation of regularization, the different classes of basis functions correspond to different classes of prior probabilities on the approximating function spaces, and therefore to different types of smoothness assumptions. In the final part of the paper, we also show a relation between activation functions of the Gaussian and sigmoidal type.

A generalized orthonormal basis for linear dynamical systems

Abstract: In many areas of signal, system and control theory orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems, and that compose an orthonormal basis for the signal space l/sub 2//sup n/. To this end use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, e.g., the Laguerre functions and the pulse functions, related to the use of the delay operator, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e. to obtain a good approximation by retaining only a finite number of expansion coefficients. >

Shape from shading with a linear triangular element surface model

Abstract: The authors propose to combine a triangular element surface model with a linearized reflectance map to formulate the shape-from-shading problem. The main idea is to approximate a smooth surface by the union of triangular surface patches called triangular elements and express the approximating surface as a linear combination of a set of nodal basis functions. Since the surface normal of a triangular element is uniquely determined by the heights of its three vertices (or nodes), image brightness can be directly related to nodal heights using the linearized reflectance map. The surface height can then be determined by minimizing a quadratic cost functional corresponding to the squares of brightness errors and solved effectively with the multigrid computational technique. The proposed method does not require any integrability constraint or artificial assumptions on boundary conditions. Simulation results for synthetic and real images are presented to illustrate the performance and efficiency of the method. >

On linear independence for integer translates of a finite number of functions

Abstract: We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Cluster expansions and the configurational energy of alloys

Abstract: A formal description of configurational functions in alloys is presented. The approach introduces an infinity of orthogonal and complete basis sets in the configurational space of finite clusters, and leads naturally to generalized cluster expansions. In the present approach, the orthogonality of the basis functions is defined with respect to the scalar product given in terms of unrestricted sums over all possible configurations of the cluster. In the thermodynamic limit, this definition of the scalar product corresponds to sums in the canonical ensemble and, as such, the basis are well adapted to describe functions at any fixed concentration. A general relation between the expansion coefficients in different basis is derived.

Use of locally dense basis sets for nuclear magnetic resonance shielding calculations

Abstract: The locally dense basis set approach to the calculation of nuclear magnetic resonance shieldings is one in which a sufficiently large or dense set of basis functions is used for an atom or molecular fragment containing the resonant nucleus or nuclei of interest and fewer or attenuated sets of basis functions employed elsewhere. Provided the dense set is of sufficient size, this approach is capable of determining chemical shieldings nearly as well as a calculation with a balanced basis set of quality equal to the locally dense set, but with considerable savings of CPU time

Linear color representations for full speed spectral rendering

Abstract: We present a general linear transform method for handling full spectral information in computer graphics rendering. In this framework, any spectral power distribution in a scene is described with respect to a set of fixed orthonormal basis functions. The lighting computations follow simply from this decision, and they can be viewed as a generalization of point sampling. Because any basis functions can be chosen, they can be tailored to the scenes that are to be rendered. We discuss efficient point sampling for scenes with smoothly varying spectra, and we present the use of characteristic vector analysis to select sets of basis functions that deal efficiently with irregular spectral power distributions. As an example of this latter method, we render a scene illuminated with fluorescent light.

Frequency-domain motion estimation using a complex lapped transform

Abstract: A frequency-domain algorithm for motion estimation based on overlapped transforms of the image data is developed as an alternative to block matching methods. The complex lapped transform (CLT) is first defined by extending the lapped orthogonal transform (LOT) to have complex basis functions. The CLT basis functions decay smoothly to zero at their end points, and overlap by 2:1 when a data sequence is transformed. A method for estimating cross-correlation functions in the CLT domain is developed. This forms the basis of a motion estimation algorithm that calculates vectors for overlapping, windowed regions of data. The overlapping data window used has no block edge discontinuities and results in smoother motion fields. Furthermore, when motion compensation is performed using similar overlapping regions, the algorithm gives comparable or smaller prediction errors than standard models using exhaustive search block matching, and computational load is lower for larger displacement ranges and block sizes. >

From regularization to radial, tensor and additive splines

Abstract: Poggio and Girosi showed that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called regularization networks. They summarize their results (1993) that show that regularization networks encompass a much broader range of approximation schemes, including many of the general additive models and some of the neural networks. In particular, additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. The same extension that extends radial basis functions to hyper basis functions leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions and some forms of projection pursuit regression. The authors propose to use the term generalized regularization networks for this broad class of approximation schemes that follow from an extension of regularization. >

Stability and linear independence associated with wavelet decompositions

Abstract: Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence, and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation that the basis function satisfies

Theoretical and numerical treatment of surface integrals involving the free-space Green's function

Abstract: This paper deals with the problem of the calculation of surface integrals for electromagnetic scattering in the case of the widely popular double-triangular basis functions first introduced by Rao, Wilton, and Glisson (1982). An entire set of formulas is obtained which overrides the difficulties inherent to the singularity of the integrands, and results showing the stability, accuracy, and efficiency of the methods developed are reported in an application of the method of moments to the case of perfectly conducting surfaces and computation of near field in domains including the surface itself. Furthermore, the authors provide insight as regards the capability of triangular basis functions to model near field patterns. >

Entire-domain Galerkin method for analysis of metallic antennas and scatterers

Abstract: The paper presents the general theory of an entire-domain Galerkin method for the analysis of metallic (i.e. perfectly conducting) antennas and scatterers. The antenna and scatterer surfaces are approximated by generalised quadrangles (i.e. curved curvilinear quadrangles defined by means of parametric equations, whose edges coincide with local coordinate lines). Surface currents are expanded in such local coordinate systems and the general form of the corresponding electric field integral equation is derived. A procedure is decribed for obtaining entire-domain basis functions which satisfy automatically the continuity equation along the surface element interconnections and free edges, and the expressions are derived for the impedance matrix elements in this case. Starting from the general theory, two new particular methods are presented. The first is intended for the analysis of general structures, and is based on application of truncated cones and bilinear surfaces for the approximation of geometry. The second is aimed for the analysis of spherical scatterer, and is based on the application of generalised rectangular elements which follow the sphere shape. Both methods use polynomials for approximation of currents. Very good agreement of the results obtained by the proposed method with available experimental and numerical results is achieved by using small number of unknowns per wavelength squared (about ten for large surfaces of simple form).

Repeatable generalized inverse control strategies for kinematically redundant manipulators

Abstract: The issue of generating a repeatable control strategy which possesses the desirable physical properties of a particular generalized inverse is addressed. The technique described is fully general and only requires a knowledge of the associated mill space of the desired inverse. While an analytical representation of the null vector is desirable, ultimately the calculations are done numerically so that a numerical knowledge of the associated full vector is sufficient. This method first characterizes repeatable strategies using a set of orthonormal basis functions to describe the null space of these transformations. The optimal repeatable inverse is then obtained by projecting the null space of the desired generalized inverse onto each of these basis functions. The resulting inverse is guaranteed to be the closest repeatable inverse to the desired inverse, in an integral norm sense, from the set of all inverses spanned by the selected basis functions. This technique is illustrated for a planar, three-degree-of-freedom manipulator and a seven-degree-of-freedom spatial manipulator. >

A novel approach for template matching by nonorthogonal image expansion

Abstract: A novel approach to template matching using nonorthogonal expansion with template-similar basis functions is described. The results show that this expansion is superior in comparison with the normalized-correlation (matched-filtering) approach. It is proved that the expansion can be efficiently implemented by restoration filtering and that this method yields optimal discriminative signal-to-noise ratio. It is further proved that the nonorthogonal expansion can be performed with an almost unlimited variety of basis functions. Results show that the expansion approach has definite advantages over correlation, especially in conditions of noise and severe occlusion. When recognition is required in audio, radar, or sonar signals, where the principle of occlusion is replaced by superposition, correlation techniques are even more ineffective and the expansion approach yields better results. >

Self generating radial basis function as neuro-fuzzy model and its application to nonlinear prediction of chaotic time series

Abstract: The authors propose a self-generating algorithm for radial basis functions to automatically determine the minimal number of basis functions to achieve the specified model error. This model is also regarded as a multilayered neural network or fuzzy model of class C/sup infinity /. The self-generating algorithm consists of two processes: model parameter tuning by the gradient method for a fixed number of rules, and a basis function generation procedure by which a new basis function is generated in such a way that the center is located at the point where maximal inference error takes place in the input space, when the effect of parameter tuning is diminished. A numerical example shows that the algorithm can achieve the specified model error with fewer basis functions than other methods by which only coefficients of the basis functions are tuned. The method is applied to the nonlinear prediction of optical chaotic time series. >

Interpolation with exponential B-splines in tension

Abstract: In [11, 13] we introduced a basis of B-splines for the exponential splines in tension considered by Schweikert already in 1966. For interpolation with these basis functions we give a necessary and sufficient condition for the existence of a unique interpolant. We consider bicubic interpolation of data on rectangular grids using this basis, and give several examples showing the usefulness of this scheme.

Quantum reactive scattering in three dimensions using hyperspherical (APH) coordinates. VI. Analytic basis method for surface functions

Abstract: We continue development of the theory of reactive (rearrangement) scattering using adiabatically adjusting principal axes hyperspherical (APH) coordinates. The surface functions, functions of the APH hyperangles covering the surface of the internal coordinate sphere, are expanded in analytic basis functions centered in each of the arrangement channels. The rotational functions are associated Legendre polynomials, and the vibrational functions are harmonic functions of an ‘‘anharmonic’’ variable which covers an infinite range, allows accurate Gauss–Hermite quadrature, and includes effects of anharmonicity. Example calculations show that these functions provide an efficient basis which can markedly decrease the computational effort required to generate accurate surface functions.

Active control of sound power using acoustic basis functions as surface velocity filters

Abstract: An improved method of active structural acoustics control is presented that is based on the minimization of the total power radiated from any structure expressed in terms of a truncated series sum. Each term of this sum is related to the coupling between the orthogonal eigenvectors of the radiation impedance matrix (referred to as ‘‘basis functions’’ in this paper) and the structural surface velocity vector. The basis functions act as surface velocity filters. These acoustic basis functions are found to be weak functions of frequency but their corresponding weighting coefficients increase monotonically with frequency. The minimization of the radiated power is shown to result in a structural surface velocity vector that couples poorly to those acoustic basis functions that account for high‐efficiency sound radiation. This strategy is demonstrated numerically for a clamped–clamped baffled beam in air. Point force primary and control actuators (shakers) are used to explore the control mechanisms. As expected...

Fast waveguide mode computation using wavelet-like basis functions

Abstract: The use of wavelet-like basis functions for solving electromagnetics problems is demonstrated. In particular, the modes of an arbitrarily shaped hollow metallic waveguide are found, using a surface-integral-equation/method-of-moments (MOM) formulation. A class of wavelet-like basis functions is used to produce a sparse MOM impedance matrix, allowing the use of sparse matrix methods for fast solution of the problem. The same method applies directly to the external scattering problem. For the examples considered, the wavelet-domain impedance matrix has about 20% nonzero elements, and the time required to compute its LU factorization is reduced by approximately a factor of 10 compared to that for the original full matrix. >

Memory based method and apparatus for computer graphics

Abstract: A memory-based computer graphic animation system generates desired images and image sequences from 2-D views. The 2-D views provide sparse data from which intermediate views are generated based on a generalization and interpolation technique of the invention. This technique is called a Hyper Basis function network and provides a smooth mapping between the given set of 2-D views and a resulting image sequence for animating a subject in a desired movement. A multilayer network provides learning of such mappings and is based on Hyper Basis Functions (HBF's). A special case of the HBFs) is the radial basis function technique used in a preferred embodiment. The invention generalization/integration technique involves establishing working axes along which different views of the subject are taken. Different points along the working axes define different positions (geometrical and/or graphical) of the subject. For each of these points, control points for defining a view of the subject are either given or calculated by interpolation/generalization of the present invention.

Spectral multidomain technique with local Fourier basis

Abstract: A novel domain decomposition method for spectrally accurate solutions of PDEs is presented. A Local Fourier Basis technique is adapted for the construction of the elemental solutions in subdomains.C 1 continuity is achieved on the interfaces by a matching procedure using the analytical homogeneous solutions of a one dimensional equation. The method can be applied to the solution of elliptic problems of the Poisson or Helmholtz type as well as to time discretized parabolic problems in one or more dimensions. The accuracy is tested for several stiff problems with steep solutions. The present domain decomposition approach is particularly suitable for parallel implementations, in particular, on MIMD type parallel machines.