Showing papers on "Monotone cubic interpolation published in 2005"
Book•
08 Sep 2005
TL;DR: This paper presents a probabilistic procedure called "Spline Interpolation–Bezier Approximation–Subdivision Methods", which automates the very labor-intensive and therefore time-heavy and expensive process of solving the inequality of the following types of inequality:.
Abstract: Basic Theory.- Linear Interpolation.- Polynomial Interpolation.- Hermite Interpolation.- Spline Interpolation.- Bezier Approximation.- B-Spline Approximation.- Subdivision Methods.- Sweep Surfaces.
317 citations
TL;DR: This paper has presented a method based on cubic splines for solving a class of singular two-point boundary value problems and the tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm.
90 citations
TL;DR: It is shown that, depending upon the orientation of the end tangents t0,t1 relative to the end point displacement vector Δp=p1−p0, the problem of G1 Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions.
Abstract: It is shown that, depending upon the orientation of the end tangents $\t_0,
\t_1$ relative to the end point displacement vector $\Delta\p=\p_1-\p_0$, the
problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero,
one, or two distinct solutions. For cases where two interpolants exist, the
bending energy may be used to select among them. In cases where no solution
exists, we determine the minimal adjustment of one end tangent that permits a
spatial PH cubic Hermite interpolant. The problem of assigning tangents to a
sequence of points $\p_0,\ldots,\p_n$ in $\mathbb{R}^3$, compatible with a
$G^1$ piecewise--PH--cubic spline interpolating those points, is also briefly
addressed. The performance of these methods, in terms of overall smoothness
and shape--preservation properties of the resulting curves, is illustrated by
a selection of computed examples.
52 citations
TL;DR: An efficient scheme is presented which constructs a curve interpolating a set of given data points and allows subsequent interactive alteration of the shape of the curve by changing the shape control and shape preserving parameters associated with each curve segment.
46 citations
TL;DR: Four different Eulerian grid-based Vlasov solvers are discussed, namely a second order method and a fourth order method (symplectic integrator) using cubic splines for interpolation, the CIP (cubic interpolated propagation) method, and an Euler–Lagrange method applying two-dimensional cubic interpolants.
40 citations
TL;DR: A method for interpolating three-dimensional kinematic data, minimizing error while maintaining ease of calculation is described, which uses cubic quaternion and hermite interpolation to fill gaps between kinematics data points.
Abstract: Kinematic interpolation is an important tool in biomechanics. The purpose of this work is to describe a method for interpolating three-dimensional kinematic data, minimizing error while maintaining ease of calculation. This method uses cubic quaternion and hermite interpolation to fill gaps between kinematic data points. Data sets with a small number of samples were extracted from a larger data set and used to validate the technique. Two additional types of interpolation were applied and then compared to the cubic quaternion interpolation. Displacement errors below 2% using the cubic quaternion method were achieved using 4% of the total samples, representing a decrease in error over the other algorithms.
30 citations
TL;DR: In this paper, a simplified dual-to-primal transformation for a geometric programming model for cubic L1 splines is developed, which allows one to establish in a transparent manner relationships between the shape-preserving properties of a cubic L2 spline and the solution of the dual geometric-programming problem.
25 citations
TL;DR: A weighted rational cubic spline interpolation has been constructed using two kinds of rational cubicspline with quadratic denominator and the degree of smoothness of this spline is C^2 in the interpolating interval when the parameters satisfy a continuous system.
22 citations
TL;DR: A local convexity preserving interpolation scheme using parametricC2 cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives is given.
Abstract: We give a local convexity preserving interpolation scheme using parametricC2 cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.
14 citations
TL;DR: A difference scheme based on cubic spline in tension for second-order singularly perturbed boundary-value problem of the form λ1 and λ2 is considered and the method is tested and the results are found to be in agreement with the theory.
Abstract: We consider a difference scheme based on cubic spline in tension for second-order singularly perturbed boundary-value problem of the form The method is shown to have second- and fourth-order convergence depending on the choice of parameters λ1 and λ2 involved in the method. The method is tested on an example and the results found to be in agreement with the theory.
14 citations
TL;DR: A cubic Hermite approximation for two-dimensional boundary integral analysis is presented, which differs from previous Hermite interpolation algorithms in that the gradient equations are sparse, significantly reducing the computational cost.
TL;DR: A sextic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of the spline, which lead to uniform convergence of O(h 7) throughout the interval of interpolation.
Abstract: In this paper a sextic spline is defined for interpolation at equally spaced knots along with the end conditions required to complete the definition of the spline. These conditions are in terms of given functional values at the knots and lead to uniform convergence of O(h 7) throughout the interval of interpolation. The main objective of defining the end conditions for the sextic spline is to use the sextic spline not only for interpolation purposes, but also for the solution of the fifth-order boundary value problem, with the change consistent with the boundary value problem. †Dedicated to the memory of Dr. M. Rafique.
TL;DR: This work considers the interpolation of fuzzy data by a differentiable fuzzy-valued function by setting some conditions on the interpolant and its first derivative and gives a numerical method for calculating this function.
Abstract: We consider the interpolation of fuzzy data by a differentiable fuzzy-valued function. We do it by setting some conditions on the interpolant and its first derivative. We give a numerical method fo...
Journal Article•
TL;DR: The problem of envelope algorithm of HHT is introduced and the shortages of two shorts of classical envelope algorithms are analyzed and an important conclusion, called section slide theorem is proved with eyeable geometry meaning and a new envelope algorithm-subsection power function method is presented.
Abstract: Hilbert-Huang transform (HHT) is a new method for signal analysis developed by NordenEHuang et al in 1998 The key problem is the envelope (-fitting) algorithm, however, the existing algorithms are still not very perfect This paper first introduces the problem of envelope algorithm of HHT and analyzes the shortages of two shorts of classical envelope algorithms, ie cubic spline interpolation algorithm and Akima interpolation algorithm According to the principle of the parabola parameter spline interpolation algorithm, an important conclusion, called section slide theorem is proved with eyeable geometry meaning and a new envelope algorithm-subsection power function method is presented The new envelope algorithm is worthy of being referred and applied since some examples have shown that the new envelope algorithm sometimes is softer than cubic spline interpolation algorithm and slider than Akima interpolation algorithm, and it brings less persuade frequency in HHT analysis
05 Dec 2005
TL;DR: If the main purpose for high resolution satellite resampling is to obtain an optimal smooth final image, intuitive and experimental justifications are provided for preferring splines interpolation to nearest-neighbor, linear and cubic interpolation.
Abstract: In this paper some insights into the behavior of interpolation functions for resampling high resolution satellite images are presented. Using spatial and frequency domain characteristics, splines interpolation performance is compared to nearest-neighbor, linear and cubic interpolation. It is shown that splines interpolation injects spatial information into the final resample image better than the other three methods. Splines interpolation is also shown to be faster than cubic interpolation when the former is implemented with the LU decomposition algorithm for its tridiagonal system of linear equations. Therefore, if the main purpose for high resolution satellite resampling is to obtain an optimal smooth final image, intuitive and experimental justifications are provided for preferring splines interpolation to nearest-neighbor, linear and cubic interpolation.
Posted Content•
TL;DR: In this paper, the Hermite interpolation problem via splines of odd-degree splines has been considered and it is shown that the interpolation error is bounded in the supremum norm independently of the locations of the knots.
Abstract: We present a Hermite interpolation problem via splines of odd-degree which, to the best knowledge of the authors, has not been considered in the literature on interpolation via odd-degree splines. In this new interpolation problem, we conjecture that the interpolation error is bounded in the supremum norm independently of the locations of the knots. Given an integer k ≥ 3, our spline interpolant is of degree 2k − 1 and with 2k − 4 (interior) knots. Simulations were performed to check the validity of the conjecture. We present strong numerical evidence in support of the conjecture for k = 3, � � � ,10 when the interpolated function belongs to C (2k) [0,1], the class of 2ktimes continuously differentiable functions on [0,1]. In this case, the worst interpolation error is proved to be attained by the perfect spline of degree 2k with the same knots as the spline interpolant. This interpolation problem arises naturally in nonparametric estimation of a multiply monotone density via Least Squares and Maximum Likehood methods.
TL;DR: In this paper, the minimization principle for bivariate cubic L"1 splines was reformulated as a generalized geometric programming problem and a geometric dual with a linear objective function and convex cubic constraints was derived.
Abstract: Bivariate cubic L"1 splines provide C^1-smooth, shape-preserving interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude. The minimization principle for bivariate cubic L"1 splines results in a nondifferentiable convex optimization problem. This problem is reformulated as a generalized geometric programming problem. A geometric dual with a linear objective function and convex cubic constraints is derived. A linear system for dual-to-primal conversion is established. The results of computational experiments are presented.
TL;DR: In this article, the algebraic structure of bivariate C^1 cubic spline spaces over nonuniform type-2 triangulation and its subspaces with boundary conditions is discussed.
Abstract: In this paper, we discuss the algebraic structure of bivariate C^1 cubic spline spaces over nonuniform type-2 triangulation and its subspaces with boundary conditions. The dimensions of these spaces are determined and their local support bases are constructed.
TL;DR: In this paper, a numerical solution of the RLW equation using a cubic spline collocation method is presented using two test problems to show the robustness of the proposed procedure.
Abstract: A numerical solution of the RLW equation is presented using a cubic spline collocation method. Basic cubic spline relations are outlined and incorporated into the numerical solution procedure. Two test problems are studied to show the robustness of the proposed procedure.
TL;DR: The minimal dimension of a subspace of C^1(R^2) needed to interpolate an arbitrary function and some of its prescribed partial derivatives at two arbitrary points is investigated.
TL;DR: It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order.
Abstract: Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this ‘semi-cardinal’ box-spline subspace satisfy certain boundary conditions extending the ‘not-a-knot’ end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4.
Journal Article•
TL;DR: In this paper, a class of polynomial blending functions of degree four is presented, which is an extension of the cubic non-uniform B-spline curve based on the blending functions.
Abstract: A class of polynomial blending functions of degree four is presented,which is an extension of the cubic non-uniform B-spline curve Based on the blending functions,a method of generating piecewise polynomial curves with several shape parameters is obtained By changing the value of the shape parameters,the approaching degree of the curves to their control polygon can be adjusted For the given value of the shape parameters, the generated curves can be G~2 continuous and have the same properties as the cubic non-uniform B-spline curves
Journal Article•
TL;DR: In this paper, the existence, uniqueness and convergence properties of discrete quartic spline interpolation over non-uniform mesh have been studied which match the given functional values at mesh points, interior points and second difference at boundary points.
Abstract: In the present paper, the existence, uniqueness and convergence properties of discrete quartic spline interpolation over non-uniform mesh have been studied which match the given functional values at mesh points, interior points and second difference at boundary points.
TL;DR: In this paper, the end conditions for cubic spline interpolation with equidistant knots are defined so as to make the (slightly modified) B-spline coefficients minimal, which produces good approximation results as compared e.g. with the not-a-knot spline.
Abstract: The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.
TL;DR: In this article, a weighted rational cubic spline interpolation with quadratic numerator was constructed using two kinds of rational cubic interpolations with different types of splines, and the sufficient conditions that constrain the interpolant curves to be in a given region were derived.
Abstract: A weighted rational cubic spline interpolation has been constructed using two kinds of rational cubic spline with quadratic denominator. The sufficient conditions that constrain the interpolant curves to be in the given region are derived, also the error estimate formulas of this interpolation are obtained. Copyright © 2005 John Wiley & Sons, Ltd.
01 Jan 2005
TL;DR: In this article, the mathematical algorithm of B-spline interpolation is described and its use for interpolation of soil water characteristic data is demonstrated and implemented in JAVA 2.
Abstract: Summary Soil water characteristic data are of fundamental importance for all calculations of soil water movement. Values between measurements can be obtained by curve approximation or interpolation techniques. In this paper, the mathematical algorithm of B-spline interpolation is described and its use for interpolation of soil water characteristic data is demonstrated. A computer program for interpolation with B-splines and natural cubic splines was implemented in JAVA 2. In cases were VAN GENUCHTEN equations cannot be fitted to the data with sufficient accuracy, splines provide a helpful alternative. In comparison to natural cubic splines, uniform B-splines of degree 2 and 3 tend to exhibit less oscillations between data points. However, they do not guarantee to follow the monotony of the data in all cases. The local modification property of B-splines can be used to adjust segments of the curve while the rest of the curve stays unchanged.
28 Aug 2005
TL;DR: Overall performance of three pass algorithms is better than two pass algorithms as they have no signal contraction and interpolation can be obtained by convolution.
Abstract: This paper analyses two-dimensional image rotation algorithms on the basis of execution time and mean square error produced. Effect of forward and reverse mapping on algorithm complexity is discussed. Single and multipass algorithms have been analyzed. Various interpolation techniques including nearest neighbor, linear, cubic, sinc and cubic spline interpolation are applied to the problem of image rotation. Sinc and cubic spline produce good quality images but are computationally intensive and are not suitable for real time rotation applications. Overall performance of three pass algorithms is better than two pass algorithms as they have no signal contraction and interpolation can be obtained by convolution. The real time rotation algorithms were implemented on MDSP architecture for X-ray images.
23 Aug 2005
TL;DR: An iterative hybrid interpolation method is proposed in this study, which is an integration of the bilinear and the bi-cubic interpolation methods and implemented by an iterative scheme and the effectiveness of the proposed method is verified based on the experimental study.
Abstract: An iterative hybrid interpolation method is proposed in this study, which is an integration of the bilinear and the bi-cubic interpolation methods and implemented by an iterative scheme. First, the implement procedure of the iterative hybrid interpolation method is described. This covers (a) a low resolution image is interpolated by using the bilinear and the bi-cubic interpolators respectively; (b) a hybrid interpolated result is computed according to the weighted sum of both bilinear interpolation result and bi-cubic interpolation result and (c) the final interpolation result is obtained by repeating the similar steps for the successive two hybrid interpolation results by a recursive manner. Second, a further discussion on the method – the relation between hybrid parameter and details of an image is provided from the theoretical point of view, at the same time, an approach used for the determining of the parameter is proposed based on the analysis of error parameter curve. Third, the effectiveness of the proposed method is verified based on the experimental study.