Hermite Basis Functions Research Articles

Detecting acute myocardial infarction in the 12-lead ECG using Hermite expansions and neural networks

We use artificial neural networks (ANNs) to detect signs of acute myocardial infarction (AMI) in ECGs. The 12-lead ECG is decomposed into Hermite basis functions, and the resulting coefficients are used as inputs to the ANNs. Furthermore, we present a case-based method that qualitatively explains the operation of the ANNs, by showing regions of each ECG critical for ANN response. Key ingredients in this method are: (i) a cost function used to find local ECG perturbations leading to the largest possible change in ANN output and (ii) a minimization scheme for this cost function using mean field annealing. Our approach was tested on 2238 ECGs recorded at an emergency department. The obtained ROC areas for ANNs trained with the Hermite representation and standard ECG measurements were 83.4 and 84.3% ( P=0.4), respectively. We believe that the proposed method has potential as a decision support system that can provide good advice for diagnosis, as well as providing the physician with insight into the reason underlying the advice.

Shape Reconstruction and Weak Lensing Measurement with Interferometers: A Shapelet Approach

We present a new approach for image reconstruction and weak lensing measurements with interferometers. Based on the shapelet formalism presented in Refregier (2001), object images are decomposed into orthonormal Hermite basis functions. The shapelet coefficients of a collection of sources are simultaneously fit on the uv plane, the Fourier transform of the sky brightness distribution observed by interferometers. The resulting chi-square fit is linear in its parameters and can thus be performed efficiently by simple matrix multiplications. We show how the complex effects of bandwidth smearing, time averaging and non-coplanarity of the array can be easily and fully corrected for in our method. Optimal image reconstruction, co-addition, astrometry, and photometry can all be achieved using weighted sums of the derived coefficients. As an example we consider the observing conditions of the FIRST radio survey (Becker et al. 1995; White et al. 1997). We find that our method accurately recovers the shapes of simulated images even for the sparse uv sampling of this snapshot survey. Using one of the FIRST pointings, we find our method compares well with CLEAN, the commonly used method for interferometric imaging. Our method has the advantage of being linear in the fit parameters, of fitting all sources simultaneously, and of providing the full covariance matrix of the coefficients, which allows us to quantify the errors and cross-talk in image shapes. It is therefore well-suited for quantitative shape measurements which require high-precision. In particular, we show how our method can be combined with the results of Refregier & Bacon (2001) to provide an accurate measurement of weak lensing from interferometric data.

Computational Mechanics of the Heart

Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

Hermite correction method in hyperspherical coordinates: Application to chemical reactions

The rotational product state distribution for the D+H 2(v=1, j=1) → HD( v′=1, j′ )+H reaction, has been calculated using the Hermite correction method in hyperspherical coordinates. We have also considered the geometric phase effect in our calculations either by introducing a vector potential in the system Hamiltonian or by multiplying the wavefunction by a phase factor. In the present application, we have investigated the performance of the simplest approach where only one time-dependent Gauss–Hermite basis function in the hyperradius is included.

Vlasov Simulations Using Velocity-Scaled Hermite Representations

The efficiency, accuracy, and stability of two different pseudo-spectral methods using scaled Hermite basis and weight functions, applied to the nonlinear Vlasov–Poisson equations in one dimension (1d-1v), are explored and compared. A variable velocity scaleUis introduced into the Hermite basis and is shown to yield orders of magnitude reduction in errors, as compared to linear kinetic theory, with no increase in workload. A set of Fourier–Hermite coefficients, representing a periodic Gaussian distribution function, are advanced through time with anO(Δt2)-accurate splitting method. Within this splitting scheme, the advection and acceleration terms of the Vlasov equation are solved separately using anO(Δt4)-accurate Runge–Kutta method. The asymmetrically weighted (AW) Hermite basis, which has been used previously by many authors, conserves particles and momentum exactly and total energy toO(Δt3); however, the AW Hermite method doesnotconserve the square integral of the distribution and is, in fact, numerically unstable. The symmetrically weighted (SW) Hermite algorithm, applied here to the Vlasov system for the first time, can either conserve particles and energy (forNueven) or momentum (forNuodd) as Δ t → 0, whereNuis the largest Hermite mode number. The SW Hermite method conserves the square integral of the distribution and, therefore, remains numerically stable. In addition, careful velocity scaling improves the conservation properties of the SW Hermite method. Damping and growth rates, oscillation frequencies, E-field saturation levels, and phase-space evolution are seen to be qualitatively correct during simulations. Relative errors with respect to linear Landau damping and linear bump-on-tail instability are shown to be less than 1% using only 64 velocity-scaled Hermite functions. Comparisons to particle-in-cell (PIC) simulations show that as the number of particles increases to more than 106, the PIC solutions converge to scaled SW Hermite solutions that were found in only 1/10 of the run-time. The SW Hermite method with velocity scaling is well-suited to kinetic simulations of warm plasmas.

A Fourier–Hermite Pseudospectral Method for Penetrative Convection

A Fourier–Hermite pseudospectral method is developed to study numerically the three-dimensional penetrative convection problem under the Boussinesq approximation. An S-shaped temperature profile in the absence of motion is prescribed in the vertical direction. All variables are expanded in terms of Fourier–Hermite basis functions. The Hermite functions are scaled to adjust the length of the computational domain in the vertical. A semi-implicit scheme is used for time marching with the third-order Adam–Bashforth and Crank–Nicholson scheme for the nonlinear and linear terms, respectively. An implementation of the numerical method on a parallel computer is also described. Numerical simulation results of resolution 643are presented for low-to-moderate Rayleigh numbers with a Prandtl number of unity. The highly stable outer regions are seen to act as effective lids and all penetrative flow are contained within the computational box. Variances, heat fluxes, and their budgets are reported for several Rayleigh numbers to demonstrate the efficacy of the numerical method.

Laminar structure of the heart: a mathematical model.

A mathematical description of cardiac anatomy is presented for use with finite element models of the electrical activation and mechanical function of the heart. The geometry of the heart is given in terms of prolate spheroidal coordinates defined at the nodes of a finite element mesh and interpolated within elements by a combination of linear Lagrange and cubic Hermite basis functions. Cardiac microstructure is assumed to have three axes of symmetry: one aligned with the muscle fiber orientation (the fiber axis); a second set orthogonal to the fiber direction and lying in the newly identified myocardial sheet plane (the sheet axis); and a third set orthogonal to the first two, in the sheet-normal direction. The geometry, fiber-axis direction, and sheet-axis direction of a dog heart are fitted with parameters defined at the nodes of the finite element mesh. The fiber and sheet orientation parameters are defined with respect to the ventricular geometry such that 1) they can be applied to any heart of known dimensions, and 2) they can be used for the same heart at various states of deformation, as is needed, for example, in continuum models of ventricular contraction.

Modelling 1-D signals using Hermite basis functions

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

Hermite expansions of compact support waveforms: applications to myoelectric signals.

Nonstationary signals with finite time support are frequently encountered in electrophysiology and other fields of biomedical research. It is often desirable to have a compact description of their shape and of their time evolution. For this purpose, Fourier analysis is not necessarily the best tool. The Hermite-Rodriguez and Associated Hermite basis functions are applied in this work. Both are based on the product of Hermite polynomials and Gaussian functions. Their general properties relevant to biomedical signal processing are reviewed. Preliminary applications are described concerning the analysis and description of: a) test signals such as a square pulse and a single cycle of a sinewave, b) electrically evoked myoelectric signals, and c) power spectra of either voluntary or evoked signals. It is shown that expansions with only five to ten terms provide an excellent description of the computer simulated and real signals. It is shown that these two families of Hermite functions are well suited for the analysis of nonstationary biological evoked potentials with compact time support. An application to the estimation of scaling factors of electrically evoked myoelectric signals is described. The Hermite functions show advantages with respect to the more traditional spectral analysis, especially in the case of signal truncation due to stimulation with interpulse intervals smaller than the duration of the evoked response. Finally, the Hermite approach is found to be suitable for classification of spectral shapes and compression of spectral information of either voluntary or evoked signals. The approach is very promising for neuromuscular diagnosis and assessment because of its capability for information compression and waveform classification.

Mathematical model of geometry and fibrous structure of the heart

We developed a mathematical representation of ventricular geometry and muscle fiber organization using three-dimensional finite elements referred to a prolate spheroid coordinate system. Within elements, fields are approximated using basis functions with associated parameters defined at the element nodes. Four parameters per node are used to describe ventricular geometry. The radial coordinate is interpolated using cubic Hermite basis functions that preserve slope continuity, while the angular coordinates are interpolated linearly. Two further nodal parameters describe the orientation of myocardial fibers. The orientation of fibers within coordinate planes bounded by epicardial and endocardial surfaces is interpolated linearly, with transmural variation given by cubic Hermite basis functions. Left and right ventricular geometry and myocardial fiber orientations were characterized for a canine heart arrested in diastole and fixed at zero transmural pressure. The geometry was represented by a 24-element ensemble with 41 nodes. Nodal parameters fitted using least squares provided a realistic description of ventricular epicardial [root mean square (RMS) error less than 0.9 mm] and endocardial (RMS error less than 2.6 mm) surfaces. Measured fiber fields were also fitted (RMS error less than 17 degrees) with a 60-element, 99-node mesh obtained by subdividing the 24-element mesh. These methods provide a compact and accurate anatomic description of the ventricles suitable for use in finite element stress analysis, simulation of cardiac electrical activation, and other cardiac field modeling problems.

An approximate inverse based multigrid approach to the biharmonic problem

The approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations is examined. This algorithm has proven successful in the numerical solution of several second order boundary value problems. Here we are concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using bicubic Hermite basis functions. Central to our development is the concept of an approximate inverse of a matrix. In particular, we use a least squares approximate inverse found by solving a Frobenius matrix norm minimization problem. This approximate inverse is used in the multigrid smoothers of our algorithm FAPIN. The algorithms presented are well suited for implementation on hypercube multiprocessors.

Solution of the advective‐dispersive transport equation using a least squares collocation, Eulerian‐Lagrangian method

AbstractNumerical solution of the advective‐dispersive transport equation is difficult when advection dominates. Difficulties arise because of the first‐order spatial derivatives which can be elminated by a local coordinate transformation to the characteristic lines of the first order hyperbolic portion of the equation. The resulting differential equation is discretized using a finite difference in time and finite elements in space employing cubic Hermite basis functions. The residuals at individual collocation points are then computed. The sum of the squares of the residuals is minimized to form the necessary set of algebraic equations. The method has performed well in one‐dimensional test problems.

Application of the frontal algorithm in the collocation method

The author reports on a numerical experimentation with the collocation finite element procedure using Hermite basis functions and the frontal elimination technique to solve some large-scale problems where up to 1000 linear equations are involved. Several test cases, including some applications to engineering problems, are presented. The implementation of the frontal technique applied to collocation is discussed to some extent.

Continuous source in tidal flow: A numerical study of the transport equation

A detailed comparative numerical study of a continuous source discharging in a tidal flow has been performed. The advective term in the one-dimensional transport equation consists of a constant freshwater velocity and an isotropic oscillating component. Various finite element solutions are investigated. Compared to the analytical solution of the dynamic steady state concentration, the numerical results for typical estuarine conditions clearly indicate the superiority of the collocation method with Hermite basis functions. An extended Fourier series analysis that accounts for the source condition is developed to explain the numerical behaviour of the different schemes.

A Collocation Finite Element Method for Integration of the Boundary Layer Equations

The author proposes a collocation finite element procedure using the Hermite basis functions. The particular choice of the trial functions gives a physical interpretation to all the nodal values. In order to demonstrate the versatility of collocation, three classical problems from boundary layer theory are solved numerically. The calculations verify that the method can produce accurate information at low cost.

Exponential and polynomial methods for applying tension to an interpolating spline curve

Two approaches are explained for the application of tension to a cubic interpolating spline curve. This was first accomplished by Schweikertusing an exponential-based formulation, the spline under tension, and later Nielson developed a polynomial alternative called the v-spline. A generalized form of the spline under tension and the v-spline are juxtaposed in a comparative, unified fashion, deriving each one from a variational principle. The v-spline is derived using Hermite basis functions, to emphasize its relation to the conventional cubic interpolatory spline. The derivation of each formulation is completed for two different choices of end conditions and the matrix corresponding to each of these conditions is analyzed. Various desirable properties are shown to be possessed by the matrices for the spline under tension for all tension values, and by the matrices for the v-spline for certain ranges of tension values, which are derived.

Solution of the neutron group-diffusion equations by orthogonal collocation with cubic hermite interpolants

Abstract The method of orthogonal collocation (OC) is used to find approximate solutions for the neutron group flux and precursor concentration equations of space-time nuclear reactor dynamics. In the OC method, the unknown solution is approximated by a finite linear combination U of local basis functions. The coefficients of this linear combination are determined by forcing the “defect”, LU - f, to zero at the Gaussian quadrature abscissae selected as collocation points for maximum accuracy. After discretizing the space variables by collocation using C1 cubic Hermite basis functions, the resulting system of ordinary differential equations is integrated in time by a version of the “θ-method”. A general group diffusion computer code COLFEM was developed and was applied in particular to the solution of a onedimensional two-energy group “benchmark” problem. Numerical results show excellent agreement with those obtained by the finite difference (FD) and the conventional finite element method (FEM) with an app...