Set Of Orthonormal Basis Functions Research Articles

On the method of functional equations and the performance of desingularized boundary element methods

A correspondence is made between the reciprocal relation for linear elliptic partial differential equations and the Riesz integral representation. The former relates the boundary distributions and appropriate normal fluxes of two arbitrary solutions, and the latter expresses a continuous linear functional in terms of an integral involving a representing function. When sufficient regularity conditions are met, the representing function is identified with the unknown boundary distribution. In principle, the representing function may be expressed in terms of the images of a complete set of orthonormal basis functions with known normal fluxes, as suggested by Kupradze [Kupradze VD. On the approximate solution of problems in mathematical physics. Russ Math Surv 1967; 22: 59–107]; in practice, the representing function is computed by solving integral equations using boundary element methods. The basic procedure involves expressing the representing function in terms of finite-element or other basis functions, and requiring the satisfaction of the reciprocal relationship with a suitable set of test functions such as Green's functions and their dipoles. When the singular points are placed at the boundary, we obtain the standard boundary integral equation method. When the singular points are placed outside the domain of solution, we obtain a system of functional equations and associated class of desingularized boundary integral methods. When sufficient regularity conditions are met and the test functions comprise a complete set, then in the limit of infinite discretization the numerical solution converges to the unknown boundary distribution. An overview of formulations is presented with reference to Laplace's equation in two dimensions. Numerical experimentation shows that, in general, the solution obtained by desingularized methods becomes increasingly less accurate as the singular points of Green's functions move farther away from the boundary, but the loss of accuracy is significant only when the exact solution shows pronounced variations. Exceptions occur when the integral equation does not have a unique solution. In contrast, and in agreement with previous findings, the condition number of the linear system increases rapidly with the distance of the singular points from the boundary, to the extent that a dependable solution cannot be obtained when the singularities are located even a moderate distance away from the boundary. The desingularized formulation based on Green's function dipoles is superior in accuracy and reliability to the one that uses Green functions. The implementation of the method to the equations of elastostatics and Stokes flow are also discussed.

Fuzzy segmented image coding using orthonormal bases and derivative chain coding

A framework for fuzzy segmentation-based image coding (FSIC) is presented. The segmentation is carried out on the basis of a region-growing algorithm which uses a fuzzy rule-based system for the evaluation of the homogeneity criterion. This approach allows to incorporate complex systems of fuzzy rules and to find an appropriate solution even for contradicting image features. The specific features and rules have been chosen such that image segments with smooth contours result and very small region are avoided. Coding the contours of these regions by a derivative chain code leads to an efficient image compression scheme that can be utilized, where medium and high compression ratios together with acceptable reconstruction quality and great flexibility are needed. Segmentation results are compared with standard segmentation-based image coding (SIC) and with JPEG-compression. Afterwards, a method is presented to improve the reconstruction quality which is based on least-squares approximation. To efficiently calculate the estimate for each segmented region an orthonormal set of basis functions is constructed from a linearly independent starting base using the Gram–Schmidt method. With this approximation the reconstruction quality is improved significantly and false contours are reduced. Besides, it offers the possibility of progressive image reconstruction. This characteristic makes the approximation with orthonormal bases interesting for many state-of-the-art audio-visual applications. Segmentation results are shown as well as their improvement using orthonormal bases.

Errors in estimating spherical harmonic coefficients from partially sampled GCM output

This study describes a method of calculating the mean squared error (MSE) incurred when estimating the spherical harmonic coefficients of a climatological field that is sampled at a small network of points. The method can also be applied to the coefficients of any other set of orthonormal basis functions that are defined on the sphere. It, therefore, provides a formalism that can be applied in a variety of contexts, such as in climate change detection, where inferences are attempted about fingerprint coefficients that are imperfectly estimated from observational data. By incorporating the fingerprint as part of a set of basis functions, the methodology can be used to estimate the sampling error in the fingerprint coefficient. The MSE is expressed in terms of the spherical harmonics (or other orthonormal expansion) of the empirical orthogonal functions (EOFs), the locations of the points in the network and a set of weights that are applied at these points. The weights are optimised by minimising the expected MSE. The method is applied to a number of network configurations using monthly-mean screen temperature and 500 mb height simulated by the Canadian Climate Centre 2nd generation general circulation model in an ensemble of six 10-year simulations. In comparison with uniform weighting, optimal weighting can reduce the MSE by an order of magnitude or more for some spherical harmonic coefficients and some network configurations. Also, the MSEs vary seasonally for each network. In particular, the relative MSE of low order spherical harmonic coefficients is found to be larger in DJF than in JJA. We demonstrate how MSEs improve with increasing network density and identify graphically, the coefficients that can be estimated reliably with each network configuration.

Nonlinear time-domain modeling by multiresolution time domain (MRTD)

A multiresolution time-domain (MRTD) scheme based on the expansion in scaling functions is applied to the modeling of nonlinear pulse propagation. Appropriate absorbers for the MRTD scheme are presented and their performance is discussed. The differences using pulse functions and nonlocalized basis functions like the Battle-Lemarie scaling functions are demonstrated by deriving time-domain schemes for both sets of orthonormal basis functions.

Thermally activated escape processes in a double well coupled to a slow harmonic mode

We present accurate calculations of thermally activated rates for a symmetric double well system coupled to a dissipative harmonic mode. Diffusive barrier crossing is treated by solving the time-independent two-dimensional Smoluchowski equation as a function of a coupling and a diffusion anisotropy parameter. The original problem is transformed to a Schrödinger equation with a Hamiltonian describing a reactive system coupled to a one-dimensional harmonic bath. The calculations are performed using a matrix representation of the Hamiltonian operator in a set of orthonormal basis functions. An effective system-specific basis is introduced which consists of adiabatically displaced eigenfunctions of the coupled harmonic oscillator and those of the uncoupled reactive subsystem. This representation provides a very rapid convergence rate. Just a few basis functions are sufficient to obtain highly accurate eigenvalues with a small computational effort. The presented results demonstrate the applicability of the method in all regimes of interest, reaching from inter-well thermal activation (fast harmonic mode) to deep intra-well relaxation (slow harmonic mode). Our calculations reveal the inapplicability of the Kramers–Langer theory in certain regions of parameter space not only when the anisotropy parameter is exponentially small, but even in the isotropic diffusion case when the coupling is weak. The calculations show also that even for large barrier heights there is a region in the parameter space with multiexponential relaxation towards equilibrium. An asymptotic theory of barrier crossing in the strongly anisotropic case is presented, which agrees well with the numerically exact results.

Simplification of dynamic NMR spectroscopy by wavelet transforms

Wavelet transforms considerably simplify the analysis of FIDs influenced by dynamic processes. The projection of data onto an orthonormal set of basis functions which represent well-defined points on a time-frequency plane allows a more natural description of time-dependent frequencies than Fourier transforms. The properties are illustrated by application to simulated NMR signals showing dynamics in the intermediate to slow exchange regime. Construction of highly sophisticated pulse sequences from wavelets is briefly discussed. This very general method will be useful in many areas of magnetic resonance, and theory may benefit as well as imaging.

Using an image database to constrain the acquisition and reconstruction of MR images of the human head

A training set of MR images of normal and abnormal heads was used to derive a complete set of orthonormal basis functions which converged to head-like images more rapidly than Fourier basis functions. The new image representation was used to reconstruct MR images of other heads from a relatively small number of phase-encoded signal measurements. The training images also determined exactly which phase-encoded signals should be measured to minimize image reconstruction error. These signals were nonuniformly scattered throughout k-space. Experiments showed that head images reconstructed with the new method had less serious truncation artifacts than conventional Fourier images reconstructed from the same number of signals. The resulting images were characterized by spatially variable spatial resolution and were particularly well-resolved in regions where the training images had structural detail.

Comparison of geometries for ionospheric tomography

The distribution of electron density in the ionosphere can be imaged using computerized tomography techniques. Because of the limited view angle, resolution in the vertical direction is poor but can be improved by using a priori information in the reconstruction algorithm. The orthogonal decomposition algorithm provides a theoretical framework that unifies several traditional tomographic reconstruction algorithms, depending upon the choice of basis functions for the image domain. In this algorithm, the image is reconstructed as the weighted sum of a set of orthonormal basis functions. For ionospheric tomography, a priori information on the radial distribution of electron density can be used to guide the selection of basis functions. The resolution performance of the orthogonal decomposition algorithm depends upon the geometry used for the reconstruction. This paper analyzes three possible geometries. The factors that influence the selection of a particular geometry are (1) the ability to incorporate a priori information into the algorithm and (2) the ability to solve for the radial distribution. An analysis of the radial resolution in ionospheric tomography is also presented.

Inversion of light-scattering measurements for particle size and optical constants: theoretical study

We invert the Fredholm equation representing the light scattered by a single spherical particle or a distribution of spherical particles to obtain the particle size distribution function and refractive index. We obtain the solution by expanding the distribution function as a linear combination of a set of orthonormal basis functions. The set of orthonormal basis functions is composed of Schmidt-Hilbert eigenfunctions and a set of supplemental basis functions, which have been orthogonalized with respect to the Schmidt-Hilbert eigenfunctions by using the Gram-Schmidt orthogonalization procedure. We use the orthogonality properties of the basis functions and of the eigenvectors of the kernel covariance matrix to obtain the solution that minimizes the residual errors subject to a trial function constraint. The inversion process is described, and results from the inversion of several simulated data sets are presented.

A New Set of Orthonormal Modes for Linearized Meteorological Problems

Abstract A new orthogonal decomposition based on the Schmidt decomposition approach has been applied to the barotropic equation linearized around the January 300-mb climatological flow. The Schmidt decomposition can be computed numerically performing a singular value decomposition of the numerical representation of the equation. The decomposition provides a set of positive real numbers whose minimum is linked to the singularity of the linear equation. A nonzero minimum singular value guarantees nonsingularity. Within the limits of the numerical precision and resolution used (R15 and R30) the nondivergent, global, barotropic equation linearized around the winter climatology is not singular, but it is very badly conditioned. The Schmidt decomposition gives two sets of orthonormal basis functions, and a possible interpretation is offered by expressing the covariance matrix of forced responses in terms of Schmidt modes. An interpretation of the basis is obtained by showing that one set corresponds to the EOF ...

Spinless Salpeter equation as a simple matrix eigenvalue problem.

We propose a new method for solving the spinless Salpeter equation. Choosing a special set of orthonormal basis functions we are able to calculate analytically the integral over the corresponding kernel as well as the matrix elements of a class of potentials. In this way the problem can be reduced to the solution of a simple matrix eigenvalue problem with explicitly known matrices, which can be solved very fast numerically. The method is demonstrated in detail for $S$ waves using a typical interquark potential as an example.

Acoustic imaging of ribbed plates

In recent years, a number of measurement techniques have been developed for structural intensity in various systems such as beams and plates. These techniques are based on motion measurements by either contact sensors (e.g., accelerometers) or noncontact methods (e.g., near‐field acoustic holography). It has been shown that it is possible to calculate the two‐dimensional structural intensity vector in plates on the basis of acoustic imaging. However, the method is very sensitive to errors occurring in the image of the reconstructed surface motion. Thus the goal of this paper is to investigate the accuracy of the surface velocity when calculated from two different acoustic imaging methods: near‐field acoustic holography and a new acoustic imaging method based on surface motion expansion in a set of orthonormal basis functions. The imaging is performed by using a singular value decomposition method along with additional constraints to calculate appropriate coefficients in this expansion. Comparisons between imaging of thin homogeneous plates and ribbed plates will be discussed. [Work supported by ONR.]

Three-dimensional reconstruction from electron micrographs of disordered specimens I. Method

A method is presented for three-dimensional reconstruction from electron micrographs of a specimen containing a disordered collection of identical objects with unknown orientations. All data from all the images are simultaneously used to obtain an approximately maximum likelihood estimate of the three-dimensional electron density, which is represented as a truncated expansion in a complete orthonormal set of basis functions. Anomalous objects can nevertheless be detected and eliminated. The method remains under statistical control, and a hypothesis test is used to choose the lowest resolution reconstruction that is consistent with the data. Error propagation is quantitatively traced from the micrograph to the reconstructed electron density. Random orientation is not necessary, and prior knowledge of preffered orientation can be used to advantage. Similarly, symmetry in the object is not necessary, but it can be imposed and exploited, if appropriate. Evidence is presented that useful reconstructions can be obtained with only one or two extra tilts from highly disordered specimens, even if the objects are asymmetric. The companion paper discusses in detail the implementation and verification of the method.

Identification and Control of Linear Distributed Parameter Systems Through the Use of Experimentally Determined Singular Functions

A new method is presented for the identification of linear Distributed Parameter Systems (DPS) given only limited a priori information about the physical process. Knowledge of the locations of a finite number of spatially varying control actuators and the locations of point measurements is assumed. Basic information from the system inputs and outputs is used to identify approximate spatial characteristics of the DPS. The identification procedure results in a pseudo-modal input/output model for the system, Identification of the spatial character of the process is accomplished by assuming the general model of a linear time-invariant compact unsymmetric integral operator. The kernel of this integral operator is expressed in terms of two sets of orthonormal basis functions, known as singular functions, whose existence is given by singular value theory. Approximations to these singular functions are determined numerically from input/output data using Galerkin's method and the Singular Value Decomposition. Application of this procedure to DPS described by both a self-adjoint and a non-self-adjoint parabolic PDE model is demonstrated. The usefulness of the identified model for an example control system design is shown

Adiabatic separations of stretching and bending vibrations: Application to H2O

A detailed investigation is made into the use of adiabatic approximations for describing excited stretching and bending vibrations of the water molecule. The goal is to determine precisely how effective this approach can be in a fully quantum mechanical triatomic calculation which incorporates anharmonicities to all orders in each of the modes. Great care is taken to avoid introducing unnecessary limitations or approximations: (i) Curvilinear coordinates are used rather than the Cartesian coordinates which form the starting point for normal mode calculations; (ii) the exact quantum kinetic energy operator in these coordinates is used as the basis for both the adiabatic and full three-dimensional calculations; (iii) a Sorbie–Murrell-type potential energy surface is used, giving a reasonable representation of the ground electronic surface for large excursions from the equilibrium configuration. In addition to the bond and bond-angle variables of earlier local mode investigations, a slightly different set of fully curvilinear coordinates is also investigated. These coordinates are shown to provide a more nearly separable description in both the exact and adiabatic treatments of this specific problem. The conventional adiabatic approach, in which the slower bending mode experiences an effective force due to averaging over the faster stretching modes, is reaffirmed to be accurate for excited stretching states. For states with any appreciable bending excitation, however, it turns out that the adiabatic calculations quickly erode in reliability. In answer to this problem, the reverse adiabatic procedure (with the bend treated first) is also implemented here. While counterintuitive, this latter method is found to yield a significant improvement for the calculated bending overtones, as well as many of the combination bands. Thus, by thorough consideration of both the coordinates and order of averaging employed, the adiabatic method is shown to be very effective for either bending or stretching overtones in a realistic, fully anharmonic, triatomic vibrational problem. In addition, introduction of a new orthonormal set of basis functions for the bending angle overcomes some of the problems associated with use of the less flexible Legendre basis.

Iterative inverse scattering method employing Gram-Schmidt orthogonalization

An inverse scattering method is tested which appears to work very well. It is an iterative procedure based on the distorted wave Born approximation. The resulting Fredholm equation of the first kind is solved by a projection method that requires the construction of an orthonormal set of basis functions from the set of kernel functions. This can be done by the spectral expansion method which entails matrix diagonalization or by Gram-Schmidt orthogonalization. The method is tested on a simple one-dimensional optical wave inverse scattering problem. Both orthonormalization methods lead to a rapidly convergent and stable iteration. The spectral expansion method gives somewhat better results, but the Gram-Schmidt orthogonalization method, which is the novel aspect of our approach, is much less time-consuming. We show how the projection method can be generalized to the case of acoustic wave scattering where the target must be characterized by more than a single independent function of position.

Iterative inverse scattering method employing Gram‐Schmidt orthogonalization

An inverse scattering method is tested which appears to work very well. It is an iterative procedure based on the distorted wave Born approximation. The resulting Fredholm equation of the first kind is solved by a projection method that requires the construction of an orthonormal set of basis functions from the set of kernel functions. This can be done by the spectral expansion method which entails matrix diagonalization or by Gram‐Schmidt orthogonalization. The method is tested on a simple one‐dimensional optical wave inverse scattering problem. Both orthonormalization methods lead to a rapidly convergent and stable iteration. The spectral expansion method gives somewhat better results, but the Gram‐Schmidt orthogonalization method, which is the novel aspect of our approach, is much less time consuming. We show how the projection method can be generalized to the case of acoustic wave scattering where the target must be characterized by more than a single independent function of position.

Redundancy reduction for improved display and analysis of body surface potential maps. II. Temporal compression.

This paper describes use of the Karhunen-Loeve expansion to identify and reduce temporal redundancy in electrocardiographic body surface potential maps (192 body surface leads recorded simultaneously at 1 kHz/channel for approximately 600 msec). Temporal data compression of about 20 to 1 was obtained with accurate representation of the original data. Use of separate sets of orthonormal basis functions for QRS and ST-T provided a more accurate representation than the basis derived from QRST. Combined with the spatial compression described in the preceding paper, overall map data compression of about 320 to 1 was obtained without significant loss of accuracy of representation or map appearance. With both spatial and temporal compression the 100,000 numbers which typically comprise a single cardiac complex were accurately represented by 216 coefficients. Using basis functions derived from a single cardiac complex were accurately represented by 216 coefficients. Using basis functions derived from a training set of 221 maps, the estimated average rms error of representation was 60 microV during the ST-T. For 34 test maps which were not part of the training set, measured average errors were 64 microV during the QRS and 23 microV during the ST-T. This technique provides a basis for quantification of the diagnostic content of maps and automated classification of maps.

Generalized N-body harmonic oscillator basis functions for the hyperspherical approach

The concept of harmonic oscillator functions is generalized to the multidimensional space of the N-body system. Explicit expressions are found for these functions. They are shown to constitute a complete orthonormal set of basis functions in which one can expand the radial functions of the hyperspherical approach. Because of the wrong asymptotic behavior of the harmonic oscillator basis a method for correcting the expansion is also given.

Generalized Wannier Functions and Effective Hamiltonians

An effective Hamiltonian for an electron in a periodic lattice in a uniform magnetic field can be formulated rigorously in terms of an orthonormal set of basis functions. These functions are obtained from one another by application of magnetic translation operators. The Wannier functions turn out to be a special case of this more general set, applicable in the absence of magnetic fields. Not only are these functions not uniquely defined, but the form of the effective Hamiltonian will depend on their manner of selection, with the exception that at zero field the effective Hamiltonian is unique. A multiband effective Hamiltonian suitable for breakdown calculations is described.