Showing papers on "Monotone cubic interpolation published in 1999"

Monotonic cubic spline interpolation

Abstract: This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C/sup 2/ continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic cubic spline interpolation results are presented.

Cubic spline algorithms for orientation interpolation

Abstract: This article presents a class of spline algorithms for generating orientation trajectories that approximately minimize angular acceleration. Each algorithm constructs a twice-differentiable curve on the rotation group SO(3) that interpolates a given ordered set of rotation matrices at specified knot times. Rotation matrices are parametrized, respectively, by the unit quaternion, canonical co-ordinate, and Cayley–Rodrigues representations. All the algorithms share the common feature of (i) being invariant with respect to choice of fixed and moving frames (bi-invariant), and (ii) being cubic in the parametrized co-ordinates. We assess the performance of these algorithms by comparing the resulting trajectories with the minimum angular acceleration curve. Copyright © 1999 John Wiley & Sons, Ltd.

Constrained interpolation using rational Cubic Spline with linear denominators

Abstract: In this paper, a rational cubic interpolant spline with linear denominator has been constructed, and it is used to constrain interpolation curves to be bounded in the given region Necessary and sufficient conditions for the interpolant to satisfy the constraint have been developed The existence conditions are computationally efficient and easy to apply Finally, the approximation properties have been studied

The approximation properties of some rational cubic splines

Abstract: This paper deals with the approximation properties of some typical rational cubic splines, including the case with cubic, quadratic or linear denominator. From the point of view of the magnitude of the optimal error constant, it is found that the rational cubic spline with linear denominator gives the best approximation to the function being interpolated.

Hermite interpolation with radial basis functions on spheres

Abstract: We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented.

An investigation of Eulerian-Lagrangian methods for solving heterogeneous advection-dominated transport problems

Abstract: Numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity. Eulerian-Lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected. This paper compares the performance of four Eulerian-Lagrangian solvers that rely on linear, quadratic, cubic spline, and taut spline interpolators. In each case a tensor product decomposition is used to reduce the general n -dimensional interpolation problem to a sequence of n one-dimensional problems. Comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory. The cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives. Simulation experiments in two- and three-dimensional heterogeneous media indicate that the taut spline interpolator, which is applied here for the first time to a solute transport problem, is able to yield accurate essentially nonoscillatory solutions for high grid Peclet numbers. The cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods.

Piecewise polynomial kernels for image interpolation: a generalization of cubic convolution

Abstract: A well-known approach to image interpolation is cubic convolution, in which the ideal sine function is modelled by a finite extent kernel, which consists of piecewise third order polynomials In this paper we show that the concept of cubic convolution can be generalized We derive kernels of up to ninth order and compare them both mutually and to cardinal splines of corresponding orders From spectral analyses we conclude that the improvements of the higher order schemes over cubic convolution are only marginal We also conclude that in all cases, cardinal splines are superior

Spectral Analysis of Hermite Cubic Spline Collocation Systems

Abstract: In this paper, we present an eigenvalue analysis of the first-order and second-order Hermite cubic spline collocation differentiation matrices with arbitrary collocation points. Some important features are explored and compared with some other discrete methods, such as finite difference methods.

A Parametric Cubic Element with Tension Properties

Abstract: In this paper we present the construction and study the properties of a new parametric cubic element having tension properties. The element is based on the Clough--Tocher split of a given triangle. Due to its tension properties the element can be used to obtain a local interpolation scheme for scattered data.

Compensation of unidirectional geometric distortion in EPI using spline warping

Abstract: Due to magnetic field inhomogeneities, EPI images are geometrically distorted, predominantly along the phase-encoding direction. Currently, the distortion is either ignored or compensated manually using a warping function defined through a set of landmarks. We propose an automatic method to unwarp the geometric distortion of EPI images by registering them with corresponding undistorted anatomical MRI images. We show that cubic splines are optimal interpolating functions for both landmark interpolation and approximation. We will consequently use the same warping space in our algorithm which replaces landmarks by an image difference criterion. B-splines are used as generating functions, which leads to a fast and accurate computation. Multiresolution gives robustness and additional speedup. The algorithm performance was evaluated using both real and synthetic data and was found superior to the manual method.

Conic representation of a rational cubic spline

Abstract: A rational cubic spline with a family of shape parameters is discussed from the viewpoint of its application in computer graphics. It incorporates both conic sections and parametric cubic curves as special cases. The parameters (weights), in the description of the spline curve can be used to modify the shape of the curve, locally and globally, at the knot intervals. The rational cubic spline attains parametric C/sup 2/ smoothness, whereas the stitching of the conic segments preserves visually reasonable smoothness at the neighboring knots. The curve scheme is interpolatory and can plot parabolic, hyperbolic, elliptic, and circular splines independently, as well as bits and pieces of a rational cubic spline.

Gregory’s Rational Cubic Splines in Interpolation Subject to Derivative Obstacles

Abstract: This paper is concerned with the interpolation of gridded data subject to lower and upper bounds on the first order derivatives. We apply Gregory’s rational cubic C 1 splines and corresponding tensor products. The occurring rationality parameters can always be determined such that the considered interpolation problems turn out to be solvable.

A digital convergence system using cubic splines

Abstract: In a convergence control system with Lagrange interpolation of high degree, the unexpected movement problem of other sampling points in the case of a change of a seed point was found This paper proposes a new digital convergence system based on cubic spline interpolation to increase the quality of interpolation as well as to solve the problem The system was modeled in VHDL, implemented with a chip, and applied for projection TVs

A smooth rational spline for visualizing monotone data

Abstract: A C/sup 2/ curve interpolation scheme for monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The monotone rational cubic spline scheme has a unique representation.

A digital convergence system using cubic splines

Abstract: In the convergence control system with the Lagrange interpolation of high degree, the unexpected movement problem of other sampling points in case of a change of a seed point was found. This paper proposes a new digital convergence system based on cubic spline interpolation to increase the quality of interpolation as well as to solve the problem. The system was modeled in VHDL, implemented with a chip, and applied for projection TVs.

Piecewise Cubic Interpolation for Reverse Engineering

Abstract: This paper presents a new fitting method with cubic Bezier curves with a monotone curvature. The slopes at the endpoints are imposed, which allows to generate G 1 -connected curves. The proposed method is based on a parametrization free interpolation scheme, whose study leads to the determination of areas for the intermediate interpolating points, so that the generated curve is convex, close to the data points, with a monotone curvature. Examples of application are presented and discussed.

Cubic constraint approximation based adaptive algorithms for envelope-constrained filtering

Abstract: The design of envelope-constrained filters is formulated as a constrained optimization problem. In this paper the constraint approximation is realized through a cubic natural spline, which results in an unconstrained optimization problem for envelope-constrained filter design. The solution of this unconstrained problem is suitable for real-time update. It is shown that, compared with the previously used quadratic natural spline, the cubic natural spline leads to the establishment of the adaptive algorithms with much more desirable performance. In particular, the step size of the cubic constraint approximation based adaptive algorithms can be chosen in a more flexible manner so as to achieve faster convergence. Numerical examples illustrate the main results.