Showing papers on "Monotone cubic interpolation published in 2007"
TL;DR: This paper presents a fast algorithm for cubic spline interpolation, which is based on the precondition of equidistant interpolation nodes, faster and easier to implement than the often applied B-Spline approach.
Abstract: Computing numerical solutions of household's optimization, one often faces the problem of interpolating functions. As linear interpolation is not very good in fitting functions, various alternatives like polynomial interpolation, Chebyshev polynomials or splines were introduced. Cubic splines are much more flexible than polynomials, since the former are only twice continuously differentiable on the interpolation interval. In this paper, we present a fast algorithm for cubic spline interpolation, which is based on the precondition of equidistant interpolation nodes. Our approach is faster and easier to implement than the often applied B-Spline approach. Furthermore, we will show how to loosen the precondition of equidistant points with strictly monotone, continuous one-to-one mappings. Finally, we present a straightforward generalization to multidimensional cubic spline interpolation.
96 citations
TL;DR: A cubic spline polynomial is presented for the solution of non-linear singular two-point boundary value problems using the quesilinearization technique to reduce the non- linear problem to a sequence of linear problems.
61 citations
TL;DR: Two super cubic convergence methods to solve systems of nonlinear equations are presented and a theorem is proved which shows the cubic convergence for the Adomian decomposition method.
59 citations
TL;DR: The rational cubic function is extended to rational bicubic partially blended function (Coons Patches) and simple constraints are derived on free parameters in the description of rational bimodal partially blended patches to visualize the monotone data in the view ofmonotone surfaces.
35 citations
TL;DR: In this paper, the authors developed a dynamic multi-point rational interpolation method based on Krylov subspace techniques to generate reduced order models for passive components and interconnects that are accurate across a wide-range of frequencies.
Abstract: The efficient modeling of integrated passive components and interconnects is vital for the realization of high performance mixed-signal systems. In this paper, we develop a dynamic multi-point rational interpolation method based on Krylov subspace techniques to generate reduced order models for passive components and interconnects that are accurate across a wide-range of frequencies. We dynamically select interpolation points by applying a cubic spline-based algorithm to detect complex regions in the system's frequency response. The results indicate that our method provides greater accuracy than techniques that apply uniform interpolation points.
34 citations
Proceedings Article•
01 Jan 2007
TL;DR: Tests on real images demonstrate that the thin-plate spline method can estimate the color of the incident illumination quite accurately, and the proposed training set pruning significantly decreases the computation.
31 citations
Book•
17 Oct 2007
TL;DR: Computer-Aided Reverse Engineering Using Evolutionary Heuristics on NURBS and Multiresolution Framework for B-Splines for Rational Cubic Splines is presented.
Abstract: Weighted Nu Splines.- Rational Cubic Spline with Shape Control.- Rational Sigma (? )Splines.- Linear, Conic and Rational Cubic Splines.- Shape-Preserving Rational Interpolation for Planar Curves.- Visualization of Shaped Data by a Rational Cubic Spline.- Visualization of Shaped Data by Cubic Spline Interpolation.- ApproximationwithB-SplinesCurves.- Spirals.- Corner Detection for Curve Segmentation.- Linear Capture of Digital Curves.- Digital Outline Capture with Cubic Curves.- Computer-Aided Reverse Engineering Using Evolutionary Heuristics on NURBS.- Multiresolution Framework for B-Splines.
29 citations
TL;DR: The purpose of this paper is to provide sufficient geometric conditions that imply the existence of a cubic parametric polynomial curve which interpolates six points in the plane.
21 citations
TL;DR: This article introduces two new algorithms that, at first, look quite different from what has been done so far, but it will turn out that they actually do fit into the solution scheme.
Abstract: In the previous four columns, the properties of the homogeneous cubic polynomial were studied. In this article, the author introduces two new algorithms that, at first, look quite different from what we've done so far. It will turn out, though, that they actually do fit into our solution scheme. In showing this, he has taken good ideas from a variety of authors and translated them into a common notation while also converting them to deal with homogeneous polynomials
21 citations
TL;DR: Cubic spline collocation method with the far-side boundary condition has been proposed as a numerical method for the convection-dominant convectiondiffusion problem as discussed by the authors, and it has been shown that the proposed method can give highly accurate result for very large Peclet number problems by effectively suppressing the undesired ripple that is commonly observed in ordinary orthogonal collocations method.
Abstract: Cubic spline collocation method with the far-side boundary condition has been proposed as a numerical method for the convection-dominant convection-diffusion problem. It has been shown that the proposed method can give highly accurate result for very large Peclet number problems by effectively suppressing the undesired ripple that is commonly observed in ordinary orthogonal collocation method.
20 citations
26 Nov 2007
TL;DR: An efficient local cubic spline interpolation curve modeling algorithm is proposed and an approach to apply the algorithm to model arbitrary shape built from free form curves is provided.
Abstract: We present an approach to model handwriting like curves with the cubic spline interpolation function. Different from NURBS such as Bezier and B-spline curve modeling, the huge complexity of the traditional spline interpolation have been obstructed and limited the application of spline curve modeling. We propose an efficient local cubic spline interpolation curve modeling algorithm and provide an approach to apply the algorithm to model arbitrary shape built from free form curves.
TL;DR: It is proved that the univariate interpolating cubic L1 spline to the Heaviside function at three sites to the left of the jump and three sites with zero slopes at the end point entirely agrees with the He Daviside function, and shows that there is no oscillation near the discontinuous point.
Abstract: We prove that the univariate interpolating cubic L1 spline to the Heaviside function at three sites to the left of the jump and three sites to the right of the jump entirely agrees with the Heaviside function except in the middle interval where it is the interpolating cubic with zero slopes at the end point. This shows that there is no oscillation near the discontinuous point i.e. no Gibbs’ phenomenon.
TL;DR: The necessary and sufficient conditions for linearizability of an eight-parameter family of two-dimensional system of differential equations in the form of linear canonical saddle perturbed by polynomials with four quadratic and four cubic terms are obtained.
TL;DR: In this paper, a cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points, is introduced.
Abstract: This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.
TL;DR: A trivariate Lagrange interpolation method based on C 1 cubic splines is described, which is local and stable, provides optimal order approximation, and has linear complexity.
Abstract: A trivariate Lagrange interpolation method based on C 1 cubic splines is described. The splines are defined over a special refinement of the Freudenthal partition of a cube partition. The interpolating splines are uniquely determined by data values, but no derivatives are needed. The interpolation method is local and stable, provides optimal order approximation, and has linear complexity.
TL;DR: In this article, the Hermite interpolation and its derivatives of any order are provided, assuming that the interpolated function f has continuous derivatives of sufficiently high order on some sufficiently small neighborhood of a given point x and any group of nodes also given on the neighborhood.
Abstract: Explicit representations for the Hermite interpolation and their derivatives of any order are provided. Furthermore, suppose that the interpolated function f has continuous derivatives of sufficiently high order on some sufficiently small neighborhood of a given point x and any group of nodes are also given on the neighborhood. If the derivatives of any order of the Hermite interpolation polynomial of f at the point x are applied to approximating the corresponding derivatives of the function f(x), the asymptotic representations for the remainder are presented.
TL;DR: In this article, it was shown that the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4, and the converse is also true.
Abstract: Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4.
TL;DR: In this article, a wide class of sufficient conditions that admit a G1 cubic spline interpolant is determined, and the existence requirements are based upon geometric properties of data entirely, and can be easily verified in advance.
Abstract: In this paper, geometric interpolation by G1 cubic spline is studied. A wide class of sufficient conditions that admit a G1 cubic spline interpolant is determined. In particular, convex data as well as data with inflection points are included. The existence requirements are based upon geometric properties of data entirely, and can be easily verified in advance. The algorithm that carries out the verification is added.
TL;DR: Harten’s framework for multiresolution analysis to the vector case for cell-averaged data is extended, focusing on Hermite interpolatory techniques.
Abstract: Harten’s interpolatory multiresolution representation of data has been extended in the case of point-value discretization to include Hermite interpolation by Warming and Beam in [17]. In this work we extend Harten’s framework for multiresolution analysis to the vector case for cell-averaged data, focusing on Hermite interpolatory techniques.
TL;DR: In this paper, a semi-analytic technique using two-point Hermite interpolation is compared with conventional methods via a series of examples and is shown to be generally superior, particularly for problems involving nonlinear equations and/or boundary conditions.
Abstract: This paper is concerned with constructing polynomial solutions to ordinary boundary value problems. A semi-analytic technique using two-point Hermite interpolation is compared with conventional methods via a series of examples and is shown to be generally superior, particularly for problems involving nonlinear equations and/or boundary conditions.
TL;DR: In this article, the authors show how to represent the Hermite Martensen spline recursively and explicitly in terms of the B-spline by using the famous Marsden identity.
TL;DR: In the case of uniform grids, the error of the spline interpolant of a function defined on R has been well estimated as mentioned in this paper, and the difference between the interpolant and the quasi-interpolants has been estimated.
Abstract: In the case of uniform grids, the error of the spline interpolant of a function defined on R has been well estimated On the basis of the spline interpolation formula for functions defined on R we derive quasi‐interpolation formulae for functions defined on R or in a vicinity of a bounded interval, say [0,1], and we estimate the difference between the interpolant and the quasi‐interpolants
TL;DR: In this article, it was shown that the space of plane curves of degree at most d having singularities of multiplicity ≤ m in general position has the expected dimension if the number of low order singularities (of multiplicity k ≤ 12) is greater then some r(m, k).
Abstract: The main goal of the paper is to find an effective estimation for the minimal number of points in 𝕂2 in general position for which the basis for Hermite interpolation consists of the first U003F1 terms (with respect to total degree ordering). As a result we prove that the space of plane curves of degree at most d having singularities of multiplicity ≤ m in general position has the expected dimension if the number of low order singularities (of multiplicity k ≤ 12) is greater then some r(m, k). Additionally, the upper bounds for r(m, k) are given.
TL;DR: This paper provides upper bound estimates for the minimum number of hidden units that can solve the Hermite interpolation problem exactly using single-layer feedforward networks.
TL;DR: Two new functional Monte Carlo algorithms are constructed for the numerical solution of three-dimensional Dirichlet boundary value problems for the linear and nonlinear Helmholtz equations using first Monte Carlo methods followed by an appropriate interpolation scheme.
Abstract: Here we construct two new functional Monte Carlo algorithms for the numerical solution of three-dimensional Dirichlet boundary value problems for the linear and nonlinear Helmholtz equations. These algorithms are based on estimating the solution and, if necessary, its partial derivatives at grid nodes using first Monte Carlo methods followed by an appropriate interpolation scheme. This allows us to obtain an approximation of the solution in the entire domain, which is not commonly done with Monte Carlo. The Monte Carlo methods used in this paper include the random walk on spheres method and the walk in balls process (with possible branching in the nonlinear case) and the stochastic application of Green's formula. For global approximation, cubic spline interpolation is used. One of the proposed approximation algorithms is based on Hermite cubic spline interpolation and utilizes estimates of the solution and its first partial derivatives. The other algorithm is based on Lagrange tricubic spline interpolation on a uniform grid and needs only estimates of the solution. An important problem is to find the optimal values of the interpolation algorithm parameters, such as the number of grid nodes and the sample volume. For this we use a stochastic optimization approach; i.e., for both of the proposed approximation algorithms we construct upper bounds of the approximation errors and minimize computational cost functions constrained by a fixed error criterion with a stochastic technique. To study the effectiveness of these proposed methods, we make a comparison between three functional algorithms, which are based on the use of the Hermite cubic splines, on the Lagrange tricubic splines, and on more common multilinear interpolation.
Journal Article•
TL;DR: Panoramic annular lens (PAL) project the view of the entire 360° around the optical axis onto an annular plane as mentioned in this paper, which plays important roles in the applications of robot vision, surveillance and virtual reality.
Abstract: Panoramic annular lens(PAL) project the view of the entire 360° around the optical axis onto an annular plane.Due to the super wide field of view,panoramic imaging system plays important roles in the applications of robot vision,surveillance and virtual reality.An annular image from PAL needs to be unwrapped to conventional rectangular image without distortion.The problem of the decreased resolution from outer circles to inner ones needs to be resolved during unwrapping procedure.The PAL image is restored according to the imaging feature of panoramic annular lens.Referenced to the highest resolution,the image adopts cubic spline interpolation function with the optimal parameter.Compared with nearest and bilinear interpolations,cubic spline interpolation with optimal parameter can restore the detail of the image better,and decrease the computational cost.
15 Apr 2007
TL;DR: This work gives a closed form solution to the H∞ optimization in the case of the cubic spline and shows that the optimal FIR (finite impulse response) filter can be designed by an LMI (linear matrix inequality), which can also be effectively solved numerically.
Abstract: Spline interpolation systems generally contain non-causal filters, and hence such systems are difficult to use for real-time processing. Our objective is to design a causal system which approximates spline interpolation. This is formulated as a problem of designing a stable inverse of a system with unstable zeros. For this purpose, we adopt H00 optimization. We give a closed form solution to the H∞ optimization in the case of the cubic spline. For higher order splines, the optimal filter can be effectively solved by a numerical computation. We also show that the optimal FIR (finite impulse response) filter can be designed by an LMI (linear matrix inequality), which can also be effectively solved numerically. A design example is presented to illustrate the result.
01 Jan 2007
TL;DR: In this paper, the boundary conditions and the class of spline functions were changed from first derivative to fourth derivative to minimize the error bounds for spline interpolation by spline function.
Abstract: We changed the boundary conditions and the class of spline functions from first derivative to fourth derivative. It was observed that and we show that the change of the boundary conditions and the class of spline functions affect in minimizing the error bounds for lacunary interpolation by spline function. Key words: Spline function INTRODUCTION During the late middle of twentieth century theory of splines has received considerable .According to Schoenberg [9] ,the interest in spline functions is due to the fact that ,spline functions are a good tool for the numerical approximation of functions on the one hand and that they suggest new, challenging and rewarding problems on the other. Piecewise linear functions, as well as step functions ,have long been an important theoretical and practical tools for approximation of functions. A notable exception was the work done by actuarial mathematicians on the so called 'osulatory interpolation 'that began soon after Hermite's work on interpolation. For more information about a spline function ,one is referred to Alberg, Nilson and Walsh
TL;DR: This work presents a new and simple formulation of a two-dimensional grid generation method that uses a special cubic spline algorithm developed to generate the analytical representation of arbitrary boundaries for two- dimensional domains.