Spline-based Method Research Articles

Effect of incomplete datasets on the calculation of continuous relaxation spectra from dynamic-mechanical data

In this article, the effect of an incomplete frequency range on relaxation spectra calculated with the new spline-based method (Stadler and Bailly, Rheol Acta 48(1):33–49, 2009) presented before is discussed. The range, in which the spectrum can be determined, is limited by the range of the input data, but not directly by the inverse frequency. The actual limits depend on the range of input data. Depending on the shape of the spectrum the relaxation spectrum can be determined from the input data in a range up to three decades larger than the input data. This can be explained by the influence of the modes outside the inverse frequency range. For this purpose, a new concept, the relevance factor analysis, was introduced, which allows for a determination of the limits of spectrum calculation. The characteristic relaxation times are discussed in comparison for to the calculation of $J_e^{\rm 0}$ and η 0 from the spectrum.

Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints

We study a spline-based likelihood method for the partly linear model with monotonicity constraints. We use monotone B -splines to approximate the monotone nonparametric function and apply the generalized Rosen algorithm to compute the estimators jointly. We show that the spline estimator of the nonparametric component achieves the possible optimal rate of convergence under the smooth assumption and that the estimator of the regression parameter is asymptotically normal and efficient. Moreover, a spline-based semiparametric likelihood ratio test is established to make inference of the regression parameter. Also an observed profile information method to consistently estimate the standard error of the spline estimator of the regression parameter is proposed. A simulation study is conducted to evaluate the finite sample performance of the proposed method. The method is illustrated by an air pollution study.

Semiparametric specification of the utility function in a travel mode choice model

We propose a semiparametric approach that can capture the nonlinearity of deterministic components of the utility functions in discrete choice models and demonstrate it by analyzing travel mode choice behaviour for an interregional trip. The proposed smoothing spline-based specification method can be used to make ex ante evaluations regarding the parametric specifications of the deterministic utility functions in discrete choice models.

The Individual Virtual Eye: a Computer Model for Advanced Intraocular Lens Calculation

Purpose To describe the individual virtual eye, a computer model of a human eye with respect to its optical properties. It is based on measurements of an individual person and one of its major application is calculating intraocular lenses (IOLs) for cataract surgery. Methods The model is constructed from an eye's geometry, including axial length and topographic measurements of the anterior corneal surface. All optical components of a pseudophakic eye are modeled with computer scientific methods. A spline-based interpolation method efficiently includes data from corneal topographic measurements. The geometrical optical properties, such as the wavefront aberration, are simulated with real ray-tracing using Snell's law. Optical components can be calculated using computer scientific optimization procedures. The geometry of customized aspheric IOLs was calculated for 32 eyes and the resulting wavefront aberration was investigated. Results The more complex the calculated IOL is, the lower the residual wavefront error is. Spherical IOLs are only able to correct for the defocus, while toric IOLs also eliminate astigmatism. Spherical aberration is additionally reduced by aspheric and toric aspheric IOLs. The efficient implementation of time-critical numerical ray-tracing and optimization procedures allows for short calculation times, which may lead to a practicable method integrated in some device. Conclusions The individual virtual eye allows for simulations and calculations regarding geometrical optics for individual persons. This leads to clinical applications like IOL calculation, with the potential to overcome the limitations of those current calculation methods that are based on paraxial optics, exemplary shown by calculating customized aspheric IOLs.

Efficient accurate numerical treatment of the modified Burgers’ equation

In this article, we propose a non-polynomial spline-based method to obtain numerical solutions of the non-linear modified Burgers’ equation. Applying the Von Neumann stability analysis, the developed method is shown to be conditionally stable. A numerical example is given to illustrate the applicability, and the accuracy of the proposed method compared with other existing recent method.

A smoothing spline-based regularization of the initial inverse problem in a two-dimensional heat equation

A smoothing spline-based method and a hyperbolic heat conduction model is applied to regularize the recovery of the initial profile from a parabolic heat conduction model in two-dimensions. An ill-posed inverse problem involving recovery of the initial temperature distribution from measurements of the final temperature distribution is investigated. A hyperbolic heat conduction model is considered instead of a parabolic model and smoothing splines are applied to regularize the recovered initial profile. The comparison of the proposed procedure and parabolic model is presented graphically by examples.

Selecting optimal weighting factors in iPDA for parameter estimation in continuous-time dynamic models

Iteratively refined principal differential analysis (iPDA) is a spline-based method for estimating parameters in ordinary differential equation (ODE) models. In this article we extend iPDA for use in differential equation models with stochastic disturbances and we demonstrate the probabilistic basis for the iPDA objective function using a maximum likelihood argument. This development naturally leads to a method for selecting the optimal weighting factor in the iPDA objective function. We demonstrate the effectiveness of iPDA using a simple two-output continuous-stirred-tank-reactor example, and we use Monte Carlo simulations to show that iPDA parameter estimates are superior to those obtained using traditional nonlinear least squares techniques, which do not account for stochastic disturbances.

B-spline-based complex-rotation method with spin-dependent interaction

We present a $B$-spline-based complex-rotation (BSCR) method with spin-dependent interaction for the study of atomic photoionization leading to multiple ionization channels dominated by doubly excited resonances for two-electron and divalent atoms. The degree of mixing between different spin states and between the bound and continuum components of the state function of the resonance state can be easily identified in the BSCR method. Its application to Mg photoionization gives good agreement with observed singlet-triplet mixed Mg spectra.

Higher-Order Nonlinear Vibration Analysis of Timoshenko Beams by the Spline-Based Differential Quadrature Method

Higher-order nonlinear vibrations of Timoshenko beams with immovable ends are studied. The nonlinear effects of axial deformation, bending curvature and transverse shear strains are considered. The nonlinear governing differential equations are solved using a spline-based differential quadrature method (SDQM), which is constructed based on quartic B-splines. Ratios of the nonlinear to the linear frequencies are extracted and their variations with the ratio of amplitude to radius of gyration are examined. In contrast to the well-recognized finding for the nonlinear fundamental frequency of beams, some higher-order nonlinear frequencies decrease with the increase of ratio of amplitude to radius of gyration.

Adaptive feedrate scheduling for NC machining along curvilinear paths with improved kinematic and geometric properties

In precision machining of free-form surfaces at high speed, it is very important to ensure part accuracy and machining properties. To achieve this, this paper develops a guide spline-based feedrate scheduling method for machining along curvilinear paths with simultaneous constraints of chord errors and ac/deceleration. Based on two metrics about feedrate scheduling from the view of geometric and kinematic properties, the nonlinear relationships between the arc-length parameter and path parameter are subsequently established with a parameter correction spline. Then a guide spline associated with the modified curvature radius of paths is constructed to the schedule feedrate. It is shown that determining the largest safe feedrate with constant chord error can be reduced to the issue of changing the feedrate proportionally with the square root of path curvature radii and ensuring the safe feed acceleration can be converted into the issue of presetting the largest slope of guide spline. The simulation results prove that the proposed feedrate scheme has potential applications in the field of finish machining.

On the analysis of cubic smoothing spline-based stem curve prediction for forest harvesters

In the cut-to-length (CTL) harvesting system the felling, delimbing, and bucking processes take place at the harvesting site. The optimal cutting points along the stem can be determined if the whole stem curve is known. In practice, however, it is not economically feasible to measure the whole stem first before crosscutting, and hence the first cutting decisions are usually made when only a short part of the stem is known. Predictions are used to determine the cutting pattern to compensate for the unknown part of the stem. In this paper our interest focuses on stem curve prediction in a harvesting situation and we study a modified version of a cubic smoothing spline-based prediction method devised by Nummi and Mottonen (T. Nummi and J. Mottonen. 2004. J. Appl. Stat. 31: 105–114). The method's performance was assessed in five different final felling stands of spruce and pine, collected by harvesters in southern Finland. The results for the spline approach are very promising and show the superiority of the method over the linear mixed-model-based approach of Liski and Nummi (E. Liski and T. Nummi. 1995. Scand. J. Stat. 22: 255–269) and also over the approach based on the variable-exponent taper equation of Kozak (A. Kozak. 1988. Can. J. For. Res. 18: 1363–1368).

Continuous delay estimation with polynomial splines

Delay estimation is used in ultrasonic imaging to estimate blood flow, determine phase aberration corrections, and to calculate elastographic images. Several algorithms have been developed to determine these delays. The accuracy of these methods depends in differing ways on noise, bandwidth, and delay range. In most cases relevant to delay estimation in ultrasonics, a subsample estimate of the delay is required. We introduce two new delay algorithms that use cubic polynomial splines to continuously represent the delay. These algorithms are compared to conventional delay estimators, such as normalized cross correlation and autocorrelation, and to another spline-based method. We present simulations that compare the algorithms' performance for varying amounts of noise, delay, and bandwidth. The proposed algorithms have better performance, in terms of bias and jitter, in a realistic ultrasonic imaging environment. The computational requirements of the new algorithms also are considered.

Solution of nonlinear initial-value problems by the spline-based differential quadrature method

The spline-based differential quadrature method (SDQM) is applied to the solution of nonlinear initial-value problems. Explicit expressions of weighting coefficients for approximation of derivatives are presented. Dynamic systems with Duffing-type nonlinearity are solved to demonstrate the effectiveness of the method. Numerical results of three examples show that the spline-based differential quadrature method is versatile and stable in the solution of nonlinear initial-value problems. It can be counted on to achieve satisfactory accuracy for long-term integration.

Symbolic computing in spline-based differential quadrature method

The differential quadrature method based on spline functions (SDQM), considered as a piecewise polynomials of any degree, is proposed in the paper. The conditions for the determination of the interpolation function are presented. The symbolic computing has been employed to determine the weighting coefficients for the SDQM. The method has been examined in the vibration analysis of a plate. Effects of the spline degree, the number of nodes, the grid point distribution and two types of end conditions on the accuracy and convergence have been investigated in the example. Copyright © 2006 John Wiley & Sons, Ltd.

Three-body scattering at intermediate energies

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial-wave decomposition. In its simplest form, the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Pad\'e summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention is given to the selection of independent variables and the treatment of three-body breakup singularities with a spline-based method. The elastic differential cross section, semiexclusive $d(N,{N}^{'}$) cross sections, and total cross sections of both elastic and breakup processes in the intermediate-energy range up to about 1 GeV are calculated and the convergence of the multiple-scattering series is investigated in every case. In general, a truncation in the first or second order in the two-body t matrix is quite insufficient.

Spline-Based Airfoil Curvature Smoothing and Its Applications

The performance of a transonic airfoil is directly related to the airfoil curvature profile and its smoothness. Whereas univariate data smoothing has been studied extensively, very little research has been conducted on curvature smoothing. Consequently, airfoil smoothing in design environments is largely based on heuristic methods, and there is an art to the generation of an unbiased smooth fit of the airfoil’s curvature profile by the modification of its geometry. In this paper, the sum of squares of the third derivative jumps is used as a curvature smoothness measure for the development of a spline-based airfoil smoothing method, called constrained fitting for airfoil curvature smoothing (CFACS). CFACS can take out dramatic curvature oscillations with extremely small geometry changes and smooth an airfoil segment without creating curvature oscillations near the endpoints. Visually, CFACS generates an unbiased smooth fit of the curvature profile. Examples demonstrating the utility of CFACS show how the smoothing can be tailored to promote desirable characteristics in performance trade studies. Nomenclature A = matrix representing smoothness measure c = chord length of airfoil cd = drag coefficient cl = lift coefficient c p = pressure coefficient f (t) = function of t for 0 ≤ t ≤ 1 f (r) (t) = r th derivative of f (t) f¯(t) = cubic spline interpolation of vector ¯ f ( 3) ¯ (t) = third derivative of f¯(t) M = freestream Mach number n = number of data points S(¯) = smoothness measure of vector ¯ t = column vector of t1, t2 ,..., tn ti = knot location of spline functions wi = positive weight in smoothness measure x = column vector of x coordinates of airfoil data xi, yi = coordinates of point on the plane y, ¯ = column vectors of y coordinates of airfoil data β, δ = parameters for choosing leading-edge segment � = percentage error tolerance for smoothing θ = control parameter (0 ≤ θ ≤ 1) for smoothing τi = positive weight for least-squares fitting �� = 2-norm of any vector

SU-FF-J-12: Computed 4D Patient Models for Motion Compensation in Radiotherapy

Purpose: We designed a system that can track tumors moving due to respiratory motion, allowing more specific and effective irradiation of the tumor. Method and Materials: First two CT scans are taken during maximal inhalation and exhalation. Then several synthetic intermediate scans are computed by using morphing methods, yielding a 3D motion picture. Before treatment x-ray images are taken periodically and compared to the 4D model. After registration of the x-ray with the model we know the best matching stack and therefore tumor position and respiratory state. Thus we acquire correlation from respiratory state to tumor position. Due to a registration time of 10 seconds for each stack, we only get intermittent information about the target location. The target may already have moved. Therefore we use an infrared tracking system, with emitters attached to significant positions of the patient's body, to report information on the current state of respiration in real-time. The information of the sensor is correlated to the target location computed by the comparison between the live shot and the model. Results: To create the model we use a thin-plate spline-based method. 47 corresponding control points were manually selected for deforming a lung. To evaluate the results transversal snap shots were compared with the model, yielding the corresponding respiratory state. We also tested the 2D/4D registration process by generating two mutually orthogonal DRRs, which we matched afterwards to the model. The best match was the stack containing the DRRs, furthermore the neighboring stacks led to the next best results. Conclusion: We have shown that it is possible to determine the respiratory state by matching a synthetic x-ray to a generated 4D model of an internal organ. Next steps will be to improve our method of generating the model and to test the registration with real x-rays.

Prognostic factors for the long-term development of ocular lesions in 327 children with congenital toxoplasmosis.

The aim of this study was to identify the high-risk factors associated with the development of ocular lesions in a large cohort of children with congenital toxoplasmosis (CT), irrespective of their gestational age at the time of maternal infection. Children were managed according to a standardized protocol and monitored for up to 14 years at the Croix-Rousse Hospital, Lyon, France. Cox model and a flexible regression, spline-based method were used for the analysis. During a median follow-up time of 6 years, 79 of the 327 children (24%) had at least one retinochoroidal lesion. No bilateral impairment of visual acuity was observed. The risk of a child developing ocular disease was higher not only when mothers were infected early during pregnancy, which was expected, but also when CT was diagnosed prior to or at the time of birth, when non-ocular manifestations were present at baseline and when birth was premature.

Non-linear vibration analysis of beams by a spline-based differential quadrature method

Non-linear vibration analysis of beams by a spline-based differential quadrature method

Image Registration Using Wavelet-Based Motion Model

An image registration algorithm is developed to estimate dense motion vectors between two images using the coarse-to-fine wavelet-based motion model. This motion model is described by a linear combination of hierarchical basis functions proposed by Cai and Wang (SIAM Numer. Anal., 33(3):937–970, 1996). The coarser-scale basis function has larger support while the finer-scale basis function has smaller support. With these variable supports in full resolution, the basis functions serve as large-to-small windows so that the global and local information can be incorporated concurrently for image matching, especially for recovering motion vectors containing large displacements. To evaluate the accuracy of the wavelet-based method, two sets of test images were experimented using both the wavelet-based method and a leading pyramid spline-based method by Szeliski et al. (International Journal of Computer Vision, 22(3):199–218, 1996). One set of test images, taken from Barron et al. (International Journal of Computer Vision, 12:43–77, 1994), contains small displacements. The other set exhibits low texture or spatial aliasing after image blurring and contains large displacements. The experimental results showed that our wavelet-based method produced better motion estimates with error distributions having a smaller mean and smaller standard deviation.