Showing papers on "Monotone cubic interpolation published in 2002"
TL;DR: It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternION unknowns, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family.
Abstract: The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.
127 citations
TL;DR: In this article, the authors describe the use of cubic splines for interpolating monotonic data sets and present an energy minimization framework to yield linear and nonlinear optimization-based methods.
89 citations
TL;DR: In this article, a new, faster iterative root finder for cubic polynomials is proposed, and the computational accuracy and computing time requirements of the analytical root finding method (Cardano's formula) are investigated.
Abstract: Some cubic equations of state can eventually have unphysical solutions for the molar volume. The conditions for this phenomenon are discussed. The computational accuracy and computing time requirements of the analytical root finding method (Cardano's formula) are investigated. A new, faster iterative root finder for cubic polynomials is proposed.
38 citations
TL;DR: A curve interpolation scheme for the visualization of scientific data has been developed and possesses extra features to modify the shape of the design curve as and when desired.
37 citations
TL;DR: Embedded Diagonally Implicit Runge-Kutta methods of different orders are used for the treatment of delay differential equations and the Q-stability region of the methods is presented.
Abstract: Embedded Diagonally Implicit Runge-Kutta methods of different orders are used for the treatment of delay differential equations The delay argument is approximated using an appropriate Hermite Interpolation The numerical results based on these methods are compared and the Q-stability region of the methods are presented
35 citations
10 Dec 2002
TL;DR: A new iterative method based on a cubic spline representation of the image based on an objective function taking into account the similarity to the known samples and the regularity of the function is minimized in order to obtain a good approximation.
Abstract: We are concerned with the reconstruction of a regularly-sampled image based on irregularly-spaced samples thereof. We propose a new iterative method based on a cubic spline representation of the image. An objective function taking into account the similarity to the known samples and the regularity of the function is minimized in order to obtain a good approximation. We apply the developed algorithm to motion-compensated image interpolation. Under motion compensation, the resulting sampling grids are irregular and require irregular/regular interpolation. We show experimental results on real-world images and we compare our results with other methods proposed in the literature.
25 citations
TL;DR: In the standard step-by-step cubic spline collocation method for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval, this method is stable in the same region of collocation parameter as in the step- by-step implementation with linear splines.
Abstract: In the standard step-by-step cubic spline collocation method for Volterra integral equations an initial condition is replaced by a not-a-knot boundary condition at the other end of the interval. Such a method is stable in the same region of collocation parameter as in the step-by-step implementation with linear splines. The results about stability and convergence are based on the uniform boundedness of corresponding cubic spline interpolation projections. The numerical tests given at the end completely support the theoretical analysis.
23 citations
TL;DR: 2D subsets of a 3D digital object are transmitted progressively under some ordering scheme, and subsequent reconstructions using the matrix cubic spline algorithm provide an evolving 3D rendering.
Abstract: Mathematical theory of matrix cubic splines is introduced, then adapted for progressive rendering of images. 2D subsets of a 3D digital object are transmitted progressively under some ordering scheme, and subsequent reconstructions using the matrix cubic spline algorithm provide an evolving 3D rendering. The process can be an effective tool for browsing three dimensional objects, and effectiveness is illustrated with a test data set consisting of 93 CT slices of a human head. The procedure has been implemented on a single processor PC system, to provide a platform for full 3D experimentation; performance is discussed. A web address for the complete, documented Mathematica code is given.
22 citations
01 Jan 2002
TL;DR: In this article, the authors used cubic spline interpolation to represent the centerline of a road, for curves in both R and R, and proposed algorithms to create a representation based on arc length and evenly spaced nodes along the center line.
Abstract: We study the use of cubic spline interpolation to represent the centerline of a road, for curves in both R and R . We look at algorithms to create a representation based on arc length and evenly spaced nodes along the centerline. We also consider methods for moving between rectangular coordinates and coordinates based on distance along the centerline and the offset from that centerline (in R) and a related decomposition in R .
18 citations
TL;DR: The geometric characterization introduced by Chung and Yao, which provides simple Lagrange formulae, is here analyzed for interpolation points lying on a line, a conic or a cubic.
17 citations
01 Jan 2002
TL;DR: The natural cubic spline interpolation procedure is introduced in a discursive fashion for sampling of digitized electroencephalographic data and useful applications include compatibility among diverse hardware and software and the customization of data analysis.
Abstract: Resampling of digitized electroencephalographic data allows changing the sampling rate with minimal distortion of the signal. Useful applications of the procedure include compatibility among diverse hardware and software and the customization of data analysis. The natural cubic spline interpolation procedure is introduced in a discursive fashion. A formal presentation is provided in the appendix.
10 Dec 2002
TL;DR: The cubic-spline interpolation is applied to estimate isophotes from sparsely sampled digital images to spread some important light on the nature of interpolation in images and why the well-known zigzag effects are obtained when images are interpolated.
Abstract: We apply the cubic-spline interpolation to estimate isophotes from sparsely sampled digital images. For any non-pixel, we interpolate it by cubic spline, and by solving the yielding cubic function analytically, we find positions of pixels with the same intensity value. Experiment results are given and discussed. This spreads some important light on the nature of interpolation in images and why the well-known zigzag effects are obtained when images are interpolated.
TL;DR: A robust scheme for the numerical solution of the quadratic system with respect to the lengths of the boundary tangent vectors is presented, and the use of the new boundary conditions is illustrated in the context of three examples.
TL;DR: A smooth rational cubic spline interpolation scheme, to preserve the shape of monotonic data, was developed by Sarfraz and motivated by few remarks is devoted towards the compilation of those remarks.
TL;DR: In this article, the generalized Hermite interpolation is derived by us- ing the contour integral and extending the generalizedHermite inter- polation to obtain the sampling expansion involving derivatives for band-limited functions.
Abstract: We derive the generalized Hermite interpolation by us- ing the contour integral and extend the generalized Hermite inter- polation to obtain the sampling expansion involving derivatives for band-limited functions f, that is, f is an entire function satisfying the following growth condition jf(z)jA exp(aejyj) for some A; ae > 0 and any z = x + iy 2C:
01 Jan 2002
TL;DR: Numerical evidences are presented that when the partition is refined, the spline interpolant converges to the function to be approximated and the accuracy of reproduction on a basis of quintic polynomials is tested.
Abstract: We discuss the implementation of a C quintic superspline method for interpolating scattered data in IR based on a modification of Alfeld’s generalization of the Clough-Tocher scheme described by Lai and LeMehaute [4]. The method has been implemented in MATLAB, and we test for the accuracy of reproduction on a basis of quintic polynomials. We present numerical evidences that when the partition is refined, the spline interpolant converges to the function to be approximated. §
TL;DR: This work considers the problem of G^2 two-point Hermite interpolation by a rational cubic and places necessary and sufficient conditions on the weights of the rational cubic curve which ensures that if the data suggest a C -shaped curve, therational cubic interpolates a C-shaped curve without loops, cusps, or inflections.
Abstract: We consider the problem of G^2 two-point Hermite interpolation by a rational cubic. Given two points with tangent vectors and curvatures, the necessary and sufficient conditions are placed on the w...
TL;DR: In this paper, the authors consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic spline only, and the cases of classical cubic interpolatory splines with defect one and Hermite C1 splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.
Abstract: Natural cubic interpolatory splines are known to have a minimal L2-norm of its second derivative on the C2 (or W22) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C1 splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.
07 Nov 2002
TL;DR: It is shown that two-point G/sup 2/ Hermite cubic spline interpolation to a smooth spiral is a spiral and that loops and cusps are impossible within a segment.
Abstract: We show that two-point G/sup 2/ Hermite cubic spline interpolation to a smooth spiral is a spiral. Its unit tangent matches given unit tangents and its signed curvature matches given signed curvatures at end points of the given spiral. Spiral segments are useful in the design of fair curves and have the advantages that there are no unplanned curvature maxima, curvature minima, or inflection points, and that loops and cusps are impossible within a segment.
TL;DR: In this paper, the problem of constructing C2 quartic spline surface interpolation is discussed, and an approach to determining the freedom degrees is given, the continuity equations for constructing C 2 quartic Spline curve are discussed and a new method for constructing the C2 Spline surface is presented.
Abstract: This paper discusses the problem of constructing C2 quartic spline surface interpolation. Decreasing the continuity of the quartic spline to C2 offers additional freedom degrees that can be used to adjust the precision and the shape of the interpolation surface. An approach to determining the freedom degrees is given, the continuity equations for constructing C2 quartic spline curve are discussed, and a new method for constructing C2 quartic spline surface is presented. The advantages of the new method are that the equations that the surface has to satisfy are strictly row diagonally dominant, and the discontinuous points of the surface are at the given data points. The constructed surface has the precision of quartic polynomial. The comparison of the interpolation precision of the new method with cubic and quartic spline methods is included.
01 Jan 2002
TL;DR: In this paper, a non-compactly supported cubic radial basis function implementation of the MLPG method for beam problems is presented, which yields results as accurate or better than those obtained by the conventional MLPG methods for problems with discontinuous and other complex loading conditions.
Abstract: A non-compactly supported cubic radial basis function implementation of the MLPG method for beam problems is presented. The evaluation of the derivatives of the shape functions obtained from the radial basis function interpolation is much simpler than the evaluation of the moving least squares shape function derivatives. The radial basis MLPG yields results as accurate or better than those obtained by the conventional MLPG method for problems with discontinuous and other complex loading conditions.
Journal Article•
TL;DR: In this article, a monotonicity preserving piecewise rational cubic interpolation function is proposed and the interpolation functions is C 1 continuous, where C is the length of the expression.
Abstract: A monotonicity preserving piecewise rational cubic interpolation function is proposed and the interpolation function is C1 continuous. Because the expression has an adjustable parameter, the interpolation curve has more flexibility.
Journal Article•
TL;DR: In this paper, a space cubic curve equation with bi-arc spline was given and the minimum angle of intersection and minimum radius deviation of curvature about consecutive circular arc were solved.
Abstract: The approximation of space cubic curve with bi-arc spline played an important role in mathematics ship-lofting. The rotation of local coordinate system was used. A space cubic curve equation was given tluough two points and two tangents. The minimum angle of intersection and the minimum radius deviation of curvature about consecutive circular arc were solved .An optimal bi\|arc spline was obtained. The error was estimated between space cubic curve and optimal bi\|arc spline using normal error function.
TL;DR: In this article, a new group of smoothness conditions and conformality conditions were employed to determine the dimension of bivariate C 1 cubic spline spaces over a so-called even stratified triangulation.
Abstract: It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.
TL;DR: An error bound for cubic spline approximation of conic section curve is presented and compared to the error bound proposed by Floater (1), which means the overall error bound is sharper than Floater's if the estimating function has the maximum at the mid- point.
Abstract: In this paper we present an error bound for cubic spline approximation of conic section curve. We compare it to the error bound proposed by Floater (1). The error estimating function pro- posed in this paper is sharper than Floater's at the mid-point of parameter, which means the overall error bound is sharper than Floater's if the estimating function has the maximum at the mid- point.
01 Jan 2002
TL;DR: In this paper, it was shown that all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of the half complex planes.
Abstract: It has been observed numerically in [1] that, under certain conditions, all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of the half complex planes. In this paper, we provide a theoretical proof for this spectral result.
TL;DR: In this paper, the applicability of the cubic Hermite function for slope interpolation using the mathematical relations between the trajectories and tangent values at the both ends of the interpolation interval was investigated.
Abstract: The cubic Hermite function was used as an interpolation method to generate the Digital Map 50m Grid (Elevation) by the Geographical Survey Institute of Japan. A Few studies have been reported concerning the reliability of cubic Hermite function for slope interpolation. However, these studies were mostly based on empirical observations without discussions about the cubic Hermite function's mathematical properties. In this paper, we investigate the applicability of the cubic Hermite function for slope interpolation using the mathematical relations between the trajectories of the cubic Hermite function and its tangent values at the both ends of the interpolation interval. The result shows that a proper slope interpolation can be achieved if and if only the trajectories' values are up to three times of the tangent's value.
Journal Article•
TL;DR: Comparisons of the precision of thenew method with other ones showed that when used to construct cubic parametric spline curves, the new method in general gives better approximation than other methods.
Abstract: The construction of parametric spline curves and surfaces plays a very important role in the fields of CAGD, CG, scientific computing and so on. The crux of constructing parametric spline curves is the parameterization of the data points. This paper discusses the continuity equations which the cubic parametric spline curves have to satisfy, and presents a new method to construct GC 2 cubic parametric spline curves. The span of each two adjacent intervals is normalized as one, and then the corresponding three knots can be taken as . Making them be GC 2 continuity at the join point s i sets up the continuity equation of the two curve segments on . The parametric spilne curve is constructed by solving the continuity equations. To set up the continuity equations, an approach for computing knots is presented. The basic idea for computing the knots on the two adjacent intervals is as follows. The five ordered data points in a plane determine a set of cubic polynomial function curves and the six ones determine a cubic polynomial function curve uniquely. The knot s i at the two adjacent intervals is computed by a way that the consecutive six data points are supposed to be taken from a cubic polynomial curve, then a linear relation among s i and other knots is set up, the knot s i is obtained by straightforward constructing the cubic polynomial curve. The constructed parametric spline curve has the precision of cubic polynomial function, i.e., if the given data points are taken from a cubic polynomial function f(t) , then the constructed cubic parametric spline curve reproduces f(t) exactly. The comparisons of the precision of the new method with other ones showed that when used to construct cubic parametric spline curves, the new method in general gives better approximation than other methods.
01 Jan 2002
TL;DR: A table look-up model of thin-film transistors has been de- veloped for circuit simulation and achieves precision for both on-current and off-current simultaneously.
Abstract: A table look-up model of thin-film transistors has been de- veloped for circuit simulation. This model utilizes three schemes. First, the spline interpolation with transformation by achieves precision for both on-current and off-current simultaneously. Second, the square polynomial supplement solves an anomaly near the points where drain voltage equals zero. Third, the linear extrapolation achieves conti- nuities of the current and its derivatives as a function of voltages for areas outside where the spline interpolation is performed and improves conver- gence during circuit simulation.