Model-order selection in Zernike polynomial expansion of corneal surfaces using the efficient detection criterion

Corneal-height data are typically measured with videokeratoscopes and modeled using a set of orthogonal Zernike polynomials. We address the estimation of the number of Zernike polynomials, which is formalized as a model-order selection problem in linear regression. Classical information-theoretic criteria tend to overestimate the corneal surface due to the weakness of their penalty functions, while bootstrap-based techniques tend to underestimate the surface or require extensive processing. In this paper, we propose to use the efficient detection criterion (EDC), which has the same general form of information-theoretic-based criteria, as an alternative to estimating the optimal number of Zernike polynomials. We first show, via simulations, that the EDC outperforms a large number of information-theoretic criteria and resampling-based techniques. We then illustrate that using the EDC for real corneas results in models that are in closer agreement with clinical expectations and provides means for distinguishing normal corneal surfaces from astigmatic and keratoconic surfaces.

Modeling videokeratoscopic height data with spherical harmonics

Purpose: All currently considered parametric models used for decomposing videokeratoscopy height data are viewercentered and hence describe what the operator sees rather than what the surface is. The purpose of this study was to ascertain the applicability of an object-centered representation to modeling of corneal surfaces. Methods: A three-dimensional surface decomposition into a series of spherical harmonics is considered and compared with the traditional Zernike polynomial expansion for a range of videokeratoscopic height data. Results: Spherical harmonic decomposition led to significantly better fits to corneal surfaces (in terms of the root mean square error values) than the corresponding Zernike polynomial expansions with the same number of coefficients, for all considered corneal surfaces, corneal diameters, and model orders. Conclusions: Spherical harmonic decomposition is a viable alternative to Zernike polynomial decomposition. It achieves better fits to videokeratoscopic height data and has the advantage of an object-centered representation that could be particularly suited to the analysis of multiple corneal measurements.

Modeling corneal surfaces with three-dimensional basis functions

Purpose: To ascertain the effectiveness of object-centered three-dimensional representations for the modeling of corneal surfaces. Methods: Three-dimensional (3D) surface decomposition into series of basis functions including: (i) spherical harmonics, (ii) hemispherical harmonics, and (iii) 3D Zernike polynomials were considered and compared to the traditional viewer-centered representation of two-dimensional (2D) Zernike polynomial expansion for a range of retrospective videokeratoscopic height data from three clinical groups. The data were collected using the Medmont E300 videokeratoscope. The groups included 10 normal corneas with corneal astigmatism less than −0.75 D, 10 astigmatic corneas with corneal astigmatism between −1.07 D and 3.34 D (Mean = −1.83 D, SD = ±0.75 D), and 10 keratoconic corneas. Only data from the right eyes of the subjects were considered. Results: All object-centered decompositions led to significantly better fits to corneal surfaces (in terms of the RMS error values) than the corresponding 2D Zernike polynomial expansions with the same number of coefficients, for all considered corneal surfaces, corneal diameters (2, 4, 6, and 8 mm), and model orders (4th to 10th radial orders) The best results (smallest RMS fit error) were obtained with spherical harmonics decomposition which lead to about 22% reduction in the RMS fit error, as compared to the traditional 2D Zernike polynomials. Hemispherical harmonics and the 3D Zernike polynomials reduced the RMS fit error by about 15% and 12%, respectively. Larger reduction in RMS fit error was achieved for smaller corneral diameters and lower order fits. Conclusions: Object-centered 3D decompositions provide viable alternatives to traditional viewer-centered 2D Zernike polynomial expansion of a corneal surface. They achieve better fits to videokeratoscopic height data and could be particularly suited to the analysis of multiple corneal measurements, where there can be slight variations in the position of the cornea from one map acquisition to the next.

Stability of corneal topography and wavefront aberrations in young Singaporeans

Purpose To investigate the differences between and variations across time in corneal topography and ocular wavefront aberrations in young Singaporean myopes and emmetropes. Methods We used a videokeratoscope and wavefront sensor to measure the ocular surface topography and wavefront aberrations of the total eye optics in the morning, mid-day and late afternoon on two separate days. Topography data were used to derive the corneal surface wavefront aberrations. Both the corneal and total wavefronts were analysed up to the 4th radial order of the Zernike polynomial expansion, and were centred on the entrance pupil (5 mm). The participants included 12 young progressing myopes, 13 young stable myopes and 15 young age-matched emmetropes. Results For all subjects considered together there were significant changes in some of the aberrations terms across the day, such as spherical aberration ( ) and vertical coma ( ) (repeated measures ANOVA, p<0.05). The magnitude of positive spherical aberration ( ) was significantly lower in the progressing myope group than that of the stable myopes (p=0.04) and emmetrope group (p=0.02). There were also significant interactions between refractive group and time of day for with/against-the-rule astigmatism ( ). Significantly lower 4th order RMS of ocular wavefront aberrations were found in the progressing myope group compared with the stable myopes and emmetropes (p<0.01). Conclusions These differences and variations in the corneal and total aberrations may have significance for our understanding of refractive error development and for clinical applications requiring accurate wavefront measurements.

Anomalous behavior of hydrodynamic modes in the two dimensional shear flow of a granular material

The growth rates of the hydrodynamic modes in the homogeneous sheared state of a granular material are determined by solving the Boltzmann equation. The steady velocity distribution is considered to be the product of the Maxwell Boltzmann distribution and a Hermite polynomial expansion in the velocity components; this form is inserted into them Boltzmann equation and solved to obtain the coeificients of the terms in the expansion. The solution is obtained using an expansion in the parameter epsilon =(1 - e)(1/2), and terms correct to epsilon(4) are retained to obtain an approximate solution; the error due to the neglect of higher terms is estimated at about 5% for e = 0.7. A small perturbation is placed on the distribution function in the form of a Hermite polynomial expansion for the velocity variations and a Fourier expansion in the spatial coordinates: this is inserted into the Boltzmann equation and the growth rate of the Fourier modes is determined. It is found that in the hydrodynamic limit, the growth rates of the hydrodynamic modes in the flow direction have unusual characteristics. The growth rate of the momentum diffusion mode is positive, indicating that density variations are unstable in the limit k--> 0, and the growth rate increases proportional to kslash} k kslash}(2/3) in the limit k --> 0 (in contrast to the k(2) increase in elastic systems), where k is the wave vector in the flow direction. The real and imaginary parts of the growth rate corresponding to the propagating also increase proportional to kslash k kslash(2/3) (in contrast to the k(2) and k increase in elastic systems). The energy mode is damped due to inelastic collisions between particles. The scaling of the growth rates of the hydrodynamic modes with the wave vector I in the gradient direction is similar to that in elastic systems. (C) 2000 Elsevier Science B.V. All rights reserved.

The deuteron-deuteron reaction and the stopping cross section of D_2O ice below 600 kev

The energy loss of protons and deuterons in D_2O ice has been measured over the energy range, E_p 18 - 541 kev. The double focusing magnetic spectrometer was used to measure the energy of the particles after they had traversed a known thickness of the ice target. One method of measurement is used to determine relative values of the stopping cross section as a function of energy; another method measures absolute values. The results are in very good agreement with the values calculated from Bethe’s semi-empirical formula. Possible sources of error are considered and the accuracy of the measurements is estimated to be ± 4%.

The D(dp)H^3 cross section has been measured by two methods. For E_D = 200 - 500 kev the spectrometer was used to obtain the momentum spectrum of the protons and tritons. From the yield and stopping cross section the reaction cross section at 90° has been obtained.

For E_D = 35 – 550 kev the proton yield from a thick target was differentiated to obtain the cross section. Both thin and thick target methods were used to measure the yield at each of ten angles. The angular distribution is expressed in terms of a Legendre polynomial expansion. The various sources of experimental error are considered in detail, and the probable error of the cross section measurements is estimated to be ± 5%.

复杂条件地层滤波预测的理论与关键技术研究

The theory researches of prediction about stratigraphic filtering in complex condition are carried out, and three key techniques are put forward in this dissertation. Theoretical aspects: The prediction equations for both slant incidence in horizontally layered medium and that in laterally variant velocity medium are expressed appropriately. Solving the equations, the linear prediction operator of overlaid layers, then corresponding reflection/transmission operators, can be obtained. The properties of linear prediction operator are elucidated followed by putting forward the event model for generalized Goupillaud layers. Key technique 1: Spectral factorization is introduced to solve the prediction equations in complex condition and numerical results are illustrated. Key technique 2: So-called large-step wavefield extrapolation of one-way wave under laterally variant velocity circumstance is studied. Based on Lie algebraic integral and structure preserving algorithm, large-step wavefield depth extrapolation scheme is set forth. In this method, the complex phase of wavefield extrapolation operator’s symbol is expressed as a linear combination of wavenumbers with the coefficients of this linear combination in the form of the integral of interval velocity and its derivatives over depth. The exponential transform of the complex phase is implemented through phase shifting, BCH splitting and orthogonal polynomial expansion. The results of numerical test show that large-step scheme takes on a great number of advantages as low accumulating error, cheapness, well adaptability to laterally variant velocity, small dispersive, etc. Key technique 3: Utilizing large-step wavefield extrapolation scheme and based on the idea of local harmonic decomposition, the technique generating angle gathers for 2D case is generalized to 3D case so as to solve the problems generating and storing 3D prestack angle gathers. Shot domain parallel scheme is adopted by which main duty for servant-nodes is to compute trigonometric expansion coefficients, while that for host-node is to reclaim them with which object-oriented angle gathers yield. In theoretical research, many efforts have been made in probing into the traits of uncertainties within macro-dynamic procedures.

Thickness-induced stabilization of ferroelectricity in SrRuO3/Ba0.5Sr0.5TiO3/Au thin film capacitors

Pulsed-laser deposition has been used to fabricate Au/Ba0.5Sr0.5TiO3/SrRuO3/MgO thin film capacitor structures. Crystallographic and microstructural investigations indicated that the Ba0.5Sr0.5TiO3 (BST) had grown epitaxially onto the SrRuO3 lower electrode, inducing in-plane compressive and out- of-plane tensile strain in the BST. The magnitude of strain developed increased systematically as film thickness decreased. At room temperature this composition of BST is paraelectric in bulk. However, polarization measurements suggested that strain had stabilized the ferroelectric state, and that the decrease in film thickness caused an increase in remanent polarization. An increase in the paraelectric-ferroelectric transition temperature upon a decrease in thickness was confirmed by dielectric measurements. Polarization loops were fitted to Landau-Ginzburg-Devonshire (LGD) polynomial expansion, from which a second order paraelectric-ferroelectric transition in the films was suggested at a thickness of similar to500 nm. Further, the LGD analysis showed that the observed changes in room temperature polarization were entirely consistent with strain coupling in the system. (C) 2002 American Institute of Physics.

Developing a numerical inverse-theory-based extraction of orientation-dependent relaxation rates from partially- relaxed spectra

Second-rank tensor interactions, such as quadrupolar interactions between the spin- 1 deuterium nuclei and the electric field gradients created by chemical bonds, are affected by rapid random molecular motions that modulate the orientation of the molecule with respect to the external magnetic field. In biological and model membrane systems, where a distribution of dynamically averaged anisotropies (quadrupolar splittings, chemical shift anisotropies, etc.) is present and where, in addition, various parts of the sample may undergo a partial magnetic alignment, the numerical analysis of the resulting Nuclear Magnetic Resonance (NMR) spectra is a mathematically ill-posed problem. However, numerical methods (de-Pakeing, Tikhonov regularization) exist that allow for a simultaneous determination of both the anisotropy and orientational distributions. An additional complication arises when relaxation is taken into account. This work presents a method of obtaining the orientation dependence of the relaxation rates that can be used for the analysis of the molecular motions on a broad range of time scales. An arbitrary set of exponential decay rates is described by a three-term truncated Legendre polynomial expansion in the orientation dependence, as appropriate for a second-rank tensor interaction, and a linear approximation to the individual decay rates is made. Thus a severe numerical instability caused by the presence of noise in the experimental data is avoided. At the same time, enough flexibility in the inversion algorithm is retained to achieve a meaningful mapping from raw experimental data to a set of intermediate, model-free

Vibration–rotation variational calculations: precise results on HCN up to 25 000 cm−1

Variation calculations of the vibration–rotation energy levels of many isotopomers of HCN are reported, for J=0, 1, and 2, extending up to approximately 8 quanta of each of the stretching vibrations and 14 quanta of the bending mode. The force field, which is represented as a polynomial expansion in Morse coordinates for the bond stretches and even powers of the angle bend, has been refined by least squares to fit simultaneously all observed data on the Σ and Π state vibrational energies, and the Σ state rotational constants, for both HCN and DCN. The observed vibrational energies are fitted to roughly ±0.5 cm−1, and the rotational constants to roughly ±0.0001 cm−1. The force field has been used to predict the vibration rotation spectra of many isotopomers of HCN up to 25 000 cm−1. The results are consistent with the axis‐switching assignments of some weak overtone bands reported recently by Jonas, Yang, and Wodtke, and they also fit and provide the assignment for recent observations by Romanini and Lehmann of very weak absorption bands above 20 000 cm−1.

Variational calculations of rovibrational states: a precise high-energy potential surface for HCN [and discussion]

We report the results of variational calculations of the rovibrational energy levels of HCN for J = 0, 1 and 2, where we reproduce all the ca. 100 observed vibrational states for all observed isotopic species, with energies up to 18000 cm$^{-1}$, to about $\pm $1 cm$^{-1}$, and the corresponding rotational constants to about $\pm $0.001 cm$^{-1}$. We use a hamiltonian expressed in internal coordinates r$_{1}$, r$_{2}$ and $\theta $, using the exact expression for the kinetic energy operator T obtained by direct transformation from the cartesian representation. The potential energy V is expressed as a polynomial expansion in the Morse coordinates y$_{i}$ for the bond stretches and the interbond angle $\theta $. The basis functions are built as products of appropriately scaled Morse functions in the bond-stretches and Legendre or associated Legendre polynomials of cos $\theta $ in the angle bend, and we evaluate matrix elements by Gauss quadrature. The hamiltonian matripx is factorized using the full rovibrational symmetry, and the basis is contracted to an optimized form; the dimensions of the final hamiltonian matrix vary from 240 $\times $ 240 to 1000 $\times $ 1000.We believe that our calculation is converged to better than 1 cm$^{-1}$ at 18 000 cm$^{-1}$. Our potential surface is expressed in terms of 31 parameters, about half of which have been refined by least squares to optimize the fit to the experimental data. The advantages and disadvantages and the future potential of calculations of this type are discussed.

Studying nonlinear effects on the early stage of phase ordering using a decomposition method

Nonlinear effects on the early stage of phase ordering are studied using Adomian's decomposition method for the Ginzburg-Landau equation for a nonconserved order parameter. While the long-time regime and the linear behavior at short times of the theory are well understood, the onset of nonlinearities at short times and the breaking of the linear theory at different length scales are less understood. In the Adomians decomposition method, the solution is systematically calculated in the form of a polynomial expansion for the order parameter, with a time dependence given as a series expansion. The method is very accurate for short times, which allows to incorporate the short-time dynamics of the nonlinear terms in a analytical and controllable way. (c) 2005 Elsevier B.V. All rights reserved.

On a control of a non-ideal mono-rail system with periodics coefficients

In this work, the problem in the loads transport (in platforms or suspended by cables) it is considered. The system in subject is composed for mono-rail system and was modeled through the system: inverted pendulum, car and motor and the movement equations were obtained through the Lagrange equations. In the model, was considered the interaction among of the motor and system dynamics for several potencies motor, that is, the case studied is denominated a non-ideal periodic problem. The non-ideal periodic problem dynamics was analyzed, qualitatively, through the comparison of the stability diagrams, numerically obtained, for several motor torque constants. Furthermore, one was made it analyzes quantitative of the problem through the analysis of the Floquet multipliers. Finally, the non-ideal problem was controlled. The method that was used for analysis and control of non-ideal periodic systems is based on the Chebyshev polynomial expansion, in the Picard iterative method and in the Lyapunov-Floquet transformation (L-F trans formation). This method was presented recently in [3-9].

On non-linear dynamics and a periodic control design applied to the potential of membrane action

The Fitzhugh-Nagumo (fn) mathematical model characterizes the action potential of the membrane. The dynamics of the Fitzhugh-Nagumo model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods, which can be applied to more complex models. This paper deals with the dynamics in the (FH) model. Here, the dynamics are analyzed, qualitatively, through the stability diagrams to the action potential of the membrane. Furthermore, we also analyze quantitatively the problem through the evaluation of Floquet multipliers. Finally, the nonlinear periodic problem is controlled, based on the Chebyshev polynomial expansion, the Picard iterative method and on Lyapunov-Floquet transformation (L-F transformation).

Análise da estabilidade de sistemas dinâmicos periódicos usando Teoria de Sinha

Pós-graduação em Matemática - IBILCE