A class of C 2piecewise quintic polynomials

A new class of C ² piecewise quintic interpolatory polynomials is defined. It is shown that this new class contains a number of interpolatory functions which present practical advantages, when compared with the conventional cubic spline.     

Exponentially fitted cubic spline for two-parameter singularly perturbed boundary value problems

In this paper, we derive an exponentially fitted difference scheme using cubic splines for a singularly perturbed ordinary differential equation with two small parameters affecting the convection and diffusion terms. The solution of the problem exhibits a boundary layer on the left-hand side of the domain. Bounds on the derivatives of the solution are derived. A first-order numerical method is constructed. Numerical results are presented to establish the theoretical results and to show the efficiency of the method.     

Sextic C 1-spline collocation methods for solving delay differential equations

In this paper, we propose a global collocation method for the numerical solution of the delay differential equations (DDEs). The method presented is based on sextic C ¹-splines s(x) as an approximation to the exact solution y(x) of the DDEs. Convergence results shows that the error bounds ‖ s ^j ?y ^j ‖=O(h ⁶), j=0, 1, in the uniform norm. Moreover, the stability analysis properties of these methods have been studied. Numerical experiments will also be considered.     

A new point matching algorithm for non-rigid registration

Feature-based methods for non-rigid registration frequently encounter the correspondence problem. Regardless of whether points, lines, curves or surface parameterizations are used, feature-based non-rigid matching requires us to automatically solve for correspondences between two sets of features. In addition, there could be many features in either set that have no counterparts in the other. This outlier rejection problem further complicates an already difficult correspondence problem. We formulate feature-based non-rigid registration as a non-rigid point matching problem. After a careful review of the problem and an in-depth examination of two types of methods previously designed for rigid robust point matching (RPM), we propose a new general framework for non-rigid point matching. We consider it a general framework because it does not depend on any particular form of spatial mapping. We have also developed an algorithm—the TPS–RPM algorithm—with the thin-plate spline (TPS) as the parameterization of the non-rigid spatial mapping and the softassign for the correspondence. The performance of the TPS–RPM algorithm is demonstrated and validated in a series of carefully designed synthetic experiments. In each of these experiments, an empirical comparison with the popular iterated closest point (ICP) algorithm is also provided. Finally, we apply the algorithm to the problem of non-rigid registration of cortical anatomical structures which is required in brain mapping. While these results are somewhat preliminary, they clearly demonstrate the applicability of our approach to real world tasks involving feature-based non-rigid registration.     

Reevaluation of daylight spectral distributions

The spectral power distribution of colorimetric daylight illuminants was determined based on measurements1 done some 40 years ago. The data were averaged and standardized at every full 10 nm. Later, to follow colorimetric practice, these data have been interpolated (linearly) to every 5 nm, but the factors to calculate the values have not been changed. We show in this article that this is not correct and leads to some discrepancies, especially if data with even smaller step size are required. We suggest that when correcting this discrepancy one should at the same time go from the linear interpolation technique, a rather crude method for a spectral distribution with many minor minima and maxima in the spectrum, to a nonlinear one. We show that a third‐order spline interpolation can provide smooth functions resembling the original data to a high degree. © 2000 John Wiley & Sons, Inc. Col Res Appl, 25: 250–259, 2000     

Use of smoothing splines and rapid Fourier transformation for constructing periodic differentiable approximations of discrete signals

A method of thinning out voltage and current discrete signals for electromagnetic alternating current systems using smoothing splines followed by use of rapid Fourier transformation in order to obtain periodic differentiable approximations is used. It is shown that use of some splines may be regarded as a method of smoothing by frequency windows. __________ Translated from Izmeritel’naya Tekhnika, No. 7, pp. 26–29, July, 2006.     

Degree elevation of B-spline curves

An algorithm is presented which expresses any given linear combination of B-splines of order k as a linear combination of B-splines of order k + 1. This algorithm consists of two parts: (1) A simple representation of the curve by a sum of B-splines of order k + 1, and (2) a representation of these splines on a common knot vector by inserting new knots.     

Calculations of curvature continuous cubic splines

The connection between the recursion formula for B-splines and the de Boor algorithm is well-known. This connection can be transferred to the curvature continuous cubic case where the use of results of Goodman & Unsworth (for a recursion formula) and Boehm (for a de Boor-like algorithm) yields two different pairs of recursion formulas and de Boor-like algorithms. Some properties are discussed.     

Smoothing splines estimators in functional linear regression with errors-in-variables

The total least squares method is generalized in the context of the functional linear model. A smoothing splines estimator of the functional coefficient of the model is first proposed without noise in the covariates and an asymptotic result for this estimator is obtained. Then, this estimator is adapted to the case where the covariates are noisy and an upper bound for the convergence speed is also derived. The estimation procedure is evaluated by means of simulations.     

Smoothing of bicubic parametric surfaces

Bicubic parametric surfaces are often used to represent complex shapes in systems for computer-aided design and manufacture. Such as surface can be defined by a topologically rectangular mesh of cubic parametric splines, a curve which is an approximate mathematical model of the linear elastic beam.Smoothing a bicubic parametric surface can be done by smoothing the curve net that defines it. This paper describes a method for moving datapoints in a curve net to new ‘smoother’ positions. Different techniques to analyse the result of the smoothing are also discussed.