Showing papers on "Monotone cubic interpolation published in 1989"
TL;DR: In this article, the authors describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates, which can help prevent the problems that result from inappropriate linearity assumptions.
Abstract: We describe the use of cubic splines in regression models to represent the relationship between the response variable and a vector of covariates. This simple method can help prevent the problems that result from inappropriate linearity assumptions. We compare restricted cubic spline regression to non-parametric procedures for characterizing the relationship between age and survival in the Stanford Heart Transplant data. We also provide an illustrative example in cancer therapeutics.
2,186 citations
TL;DR: In this article, the shape-preserving interpolation method is applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry, and the derivative estimates are modified to ensure that the interpolant is monotonic.
Abstract: The more attractive one dimensional, shape-preserving interpolation schemes as determined from a companion study are applied to two-dimensional semi-Lagrangian advection in plane and spherical geometry. Hermite cubic and a rational cubic are considered for the interpolation form. Both require estimates of derivatives at data points. A cubic derivative form and the derivative estimates of Hyman and Akima are considered. The derivative estimates are also modified to ensure that the interpolant is monotonic. The modification depends on the interpolation form. Three methods are used to apply the interpolators to two-dimensional semi-Lagrangian advection. The first consists of fractional time steps or time splitting. The method has noticeable displacement errors and larger diffusion than the other methods. The second consists of two-dimensional interpolants with formal definitions of a two-dimensional monotonic surface and application of a two-dimensional monotonicity constraint. This approach is exam...
335 citations
Patent•
29 Sep 1989TL;DR: In this paper, a spline interpolation method of subjecting given points to interpolation by using a cubic spline curve is provided, where a first-derivative vector is derived from a preset number of points including a starting point (P1), and a cubic equation between the starting point and a next point is derived based on the coordinate values of the preset points including the starting position (P 1) and the extreme point condition of the starting location (P 2 ).
Abstract: A spline interpolation method of subjecting given points to interpolation by using a cubic spline curve is provided. A first-derivative vector is derived from a preset number of points including a starting point (P1), and a cubic equation between the starting point and a next point is derived based on the coordinate values of the preset points including the starting point (P1) and the extreme point condition of the starting point (P1), to derive a spline curve between the starting point (P1) and a point (P2) next to the starting point (P1). Next, the first-derivative vector at P2 and a new next point are used instead of the starting point (P1), to derive a cubic curve between P2 and P3. In this way, a cubic equation between points is sequentially derived to obtain a cubic spline curve, and as a result, a spline curve posing no practical problems can be obtained without previously receiving all of the sequential points, while sequentially receiving the sequential points in a forward direction.
36 citations
TL;DR: Approximate Gaussian filtering of equidistant data can be obtained by regularizing the data with Tikhonov's second order stabilizing functionals with good performance compared with that of single- and multi-stage FIR Gaussian filter approximations.
34 citations
24 citations
TL;DR: In this article, the authors discute deux algorithmes de construction de l'interpolant par spline cubique sous la contrainte de positivite ou de monotonie.
Abstract: On discute deux algorithmes de construction de l'interpolant par spline cubique sous la contrainte de positivite ou de monotonie. On analyse en detail la convergence qu'on illustre par des tests numeriques
23 citations
01 Nov 1989
TL;DR: An analysis of a recently-proposed two-parameter piecewise-cubic convolution algorithm for image reconstruction indicates that the additional parameter does not improve the reconstruction fidelity - the optimal two- parameter convolution kernel is identical to the optimal kernel for the traditional one-parameters algorithm.
Abstract: This paper presents an analysis of a recently-proposed two-parameter piecewise-cubic convolution algorithm for image reconstruction. The traditional cubic convolution algorithm is a one-parameter, interpolating function. With the second parameter, the algorithm can also be approximating. The analysis leads to a Taylor series expansion for the average square error due to sampling and reconstruction as a function of the two parameters. This analysis indicates that the additional parameter does not improve the reconstruction fidelity - the optimal two-parameter convolution kernel is identical to the optimal kernel for the traditional one-parameter algorithm. Two methods for constructing the optimal cubic kernel are also reviewed.
20 citations
TL;DR: Etude des meilleures bornes d'erreur d'interpolation par des fonctions splines du quatrieme degre. as discussed by the authors.
20 citations
15 citations
Patent•
29 Sep 1989TL;DR: In this article, a spline interpolation system which interpolates a given point by a cubic spline curve is described. But this is not a practical problem, since the line of all the points at a time is unknown.
Abstract: This invention relates to a spline interpolation system which interpolates a given point by a cubic spline curve. A linear differential vector is determined from a predetermined number of points including a start point (P1) and a cubic equation between the start point (P1) and a next point (P2) is determined from the coordinates value of predetermined points including the start point (P1) and the end condition of the start point (P1). Then, a spline curve between the start point (P1) and the next point (P2) continuing from the former is determined. Next, a spline curve between points P2 and P3 is determined by adding the linear differential vector at a new point in place of the start point (P1) and (P2). In this manner, the cubic equations between points are obtained sequentially and a cubic spline curve is obtained. A spline curve free from a practical problem can be determined by sequentially reading points starting with the first without reading in advance the line of all the points at a time.
14 citations
TL;DR: In this paper, a class of end conditions is derived for cubic spline functions by use of integration, and these end conditions are given in terms of function values at the knots (data), and give rise to O(h^4) spline approximation.
TL;DR: In this paper, shape-preserving approximation and interpolation of functions by box spline surfaces on three and four directional meshes are considered. And a characterization of the grid spacing is given which guarantees the preservation of these properties for functions in certain Lipschitz classes.
TL;DR: In this article, the spline-one-spline interpolation is applied to numerical integration of a sufficiently smooth function f on [0, 1] and the ratio of the asymptotic error estimates is calculated.
01 Jan 1989
TL;DR: In this paper, the vector spherical splines are introduced in analogy to the well-known scalar theory, and the main tool is the theory of vector spherical harmonics, which is used to define vector spherical harmonic splines.
Abstract: Vector spherical splines are introduced in analogy to the well-known scalar theory The main tool is the theory of vector spherical harmonics
TL;DR: Formulas are given for representing a C1 piecewise cubic curve as a B-spline curve for various classes of parametric cubic splines.
TL;DR: In this article, the theory of Hermite-spline interpolation on the equidistant latticeZ is written in purly real terms and this for an arbitrary polynomial degree, Hermitian order and node-shift parameter.
Abstract: The theory of Hermite-spline interpolation on the equidistant latticeZ is written in purly real terms and this for an arbitrary polynomial degree, Hermitian order and node-shift parameter. An explicit representation formula for the Hermitian fundamental splines (Lagrangians) is presented and the convergence of the corresponding Lagrange-series is discussed.
TL;DR: In this article, a cubic spline that fits the first derivatives of a function at mesh points and the function value and its second derivative at the beginning of the interval is presented.
Abstract: This paper presents a formulation of a cubic spline that fits the first derivatives of a function at mesh points and the function value and its second derivative at the beginning of the interval. Error bounds for the function and its first three derivatives are derived over the whole interval. This formulation can be applied, in particular, to quadratures.
23 May 1989
TL;DR: The proposed interpolation formula differs from standard techniques such as cubic spline convolution in that the image samples are modified by a discrete convolution operator prior to the reconstruction summation.
Abstract: A novel method for the interpolation of sampled images is presented. It makes use of a recently discovered formula for the least-squares projection of an arbitrary function onto a repetitive basis. The proposed interpolation formula differs from standard techniques such as cubic spline convolution in that the image samples are modified by a discrete convolution operator prior to the reconstruction summation. The visual performance of the method is shown to be superior to that of cubic spline convolution, which is the best current algorithm. The main attraction of the method is that the algorithm is automatically tailored to the spatial resolution of the image sensor. The exact computational cost of the method, in terms of reconstruction sum size, depends on the sensor PSF (point spread function) but is likely to be only slightly greater than that for spline convolution. All the results given hold good in a general N-dimensional space. >
TL;DR: In this paper, two special Hermite-Birkhoff interpolation problems in the space of spline functions are treated and some recursion relations for the calculation of the solutions of these problems are developed for continuous approximation of the solution of a nonlinear two-point boundary value problem.
18 Oct 1989
TL;DR: This paper generalizes a number of recent results by constructing C 1 piecewise cubic polynomial interpolants to non-monotone data, which have a minimum number of changes in sign in the first derivative and approximate an underlying function and its first derivative with optimal order.
Abstract: A number of recent papers have addressed the problem of constructing monotone piecewise cubic interpolants to monotone data. These have focused not only on the monotonicity of the interpolant, but also on properties such as “visual pleasure”, and optimal order error bounds. We review some of these results, and generalize them by constructing C 1 piecewise cubic polynomial interpolants to non-monotone data.These interpolants have a minimum number of changes in sign in the first derivative and approximate an underlying function and its first derivative with optimal order.