Showing papers on "Monotone cubic interpolation published in 2003"
TL;DR: A link between classical osculatory interpolation and modern convolution-based interpolation is established and it is shown that two well-known cubic convolution schemes are formally equivalent to two osculation interpolation schemes proposed in the actuarial literature about a century ago.
Abstract: We establish a link between classical osculatory interpolation and modern convolution-based interpolation and use it to show that two well-known cubic convolution schemes are formally equivalent to two osculatory interpolation schemes proposed in the actuarial literature about a century ago. We also discuss computational differences and give examples of other cubic interpolation schemes not previously studied in signal and image processing.
113 citations
01 Jan 2003
TL;DR: In this paper, the authors present a parameterization and an interpolation method for quintic splines, which result in a smooth and consistent feedrate profile for C3 splines.
Abstract: This paper presents a parameterization and an interpolation method for quintic splines, which result in a smooth and consistent feedrate profile. The discrepancy between the spline parameter and the actual arc length leads to undesirable feed fluctuations and discontinuity, which elicit themselves as high frequency acceleration and jerk harmonics, causing unwanted structural vibrations and excessive tracking error. Two different approaches are presented that alleviate this problem: The first approach is based on modifying the spline toolpath so that it is optimally parameterized with respect to its arc length. The second approach is based on scheduling the spline parameter to accurately yield the desired arc displacement (i.e. feedrate), either by approximation of the relationship between the arc length and the spline parameter with a feed correction polynomial, or by solving the spline parameter iteratively in real-time at each interpolation step. The two approaches are compared to nearly arc length parameterized C3 quintic spline interpolation in terms of feedrate consistency and experimental tracking accuracy.Copyright © 2003 by ASME
51 citations
TL;DR: In this paper, the authors used parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems.
Abstract: We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.
40 citations
TL;DR: The new rational spline scheme, like the old one in Sarfraz, 2000, has a unique representation and the degree of smoothness attained is C 2 and the method of computation is robust.
34 citations
TL;DR: In this article, a fractal technique generalizing cubic spline functions is proposed, where the fixed point of a map between spaces of functions is defined as the corresponding fractal function and its derivatives.
Abstract: Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).
33 citations
TL;DR: This paper characterize MPH curves in R2,1 by the roots of the hodographs of their complexified spine curves, and presents two schemes for this interpolation problem: one is a subdivision scheme using direct C1 interpolation and the other is a two step scheme using a new concept, C1/2 interpolation.
28 citations
TL;DR: The usual third- order spline is modified into a fourth-order spline, called mc-spline, which provides a smooth and faithful continuous interpolation of the original data that is well suited for its graphical representations or for the forcing of numerical models.
25 citations
TL;DR: An improved method for selecting the knots is described that results in a visually satisfactory curve, and the ideas are extended to show how a curve can be drawn to satisfy arbitrary lower and upper bounds—thus allowing a curve to be drawn between two other curves.
25 citations
TL;DR: In this paper, the authors presented a new iterative method, derived from Hermite interpolation, with order of convergence p = 1+, which requires, at each step, only two function evaluations, which is better than those of classical methods, such as the secant method or Newton's method, and those of the recent methods introduced by Costabile et al.
Abstract: We present a new iterative method, derived from Hermite interpolation, with order of convergence p = 1+ , which requires, at each step, only two function evaluations. The efficiency index of the method is better than those of classical methods, such as the secant method or Newton's method, and those of the recent methods introduced by Costabile et al. [1,2] as well. This method has the best efficiency index in a family of methods derived from Hermite interpolation.
24 citations
TL;DR: In this paper, a path-planning interpolation methodology is presented with which the user may analytically specify the desired path to be followed by any planar industrial robot, and the trajectory-planner can be implemented as part of kinematic and kinetic simulation software, and also has the potential application for controlling machine tools in cutting along free-form curves.
Abstract: A new path-planning interpolation methodology is presented with which the user may analytically specify the desired path to be followed by any planar industrial robot. The user prescribes a set of nodal points along a general curve to be followed by the chosen working point on the end-effector of the mechanism. Given these specified points along the path and additional prescribed kinematical requirements, Overlapping Cubic Arcs are fitted in the Cartesian domain and a cubic Spline interpolation curve is fitted in the time-domain. Further user-specified information is used to determine how the end-effector orientation angle should vary along the specified curve. The proposed trajectory-planning methodology is embodied in a computer-algorithm (OCAS), which outputs continuous graphs for positions, velocities and accelerations in the time-domain. If a varying end-effector orientation angle is specified, the OCAS-algorithm also generates continuous orientation angle, orientation angular velocity and orientation angular acceleration curves in the time-domain. The trajectory-planning capabilities of the OCAS-algorithm are tested for cases where the prescribed nodal points lie along curves defined by analytically known non-linear functions, as well as for nodal points specified along a non-analytical (free-form) test-curve. The proposed trajectory-planner may be implemented as part of kinematic and kinetic simulation software, and it also has the potential application for controlling machine tools in cutting along free-form curves. Copyright © 2003 John Wiley & Sons, Ltd.
17 citations
DOI•
01 Jan 2003
TL;DR: The Z-splines as discussed by the authors are moment conserving cardinal splines of compact support constructed using Hermite-Birkhoff curves that reproduce explicit finite difference operators computed by Taylor series expansions.
Abstract: The Z-splines are moment conserving cardinal splines of compact support. They are constructed using Hermite-Birkhoff curves that reproduce explicit finite difference operators computed by Taylor series expansions. These curves are unique. The Z-splines are explicit piecewise polynomial interpolation kernels of cumulative regularity and accuracy. They are succesive spline approximations to the perfect reconstruction filter sinc(x). It is found that their interpolation properties: quality, regularity, approximation order and discrete moment conservation, are related to a single basic concept: the exact representation of polynomials by a long enough Taylor series expansion.
TL;DR: This higher order predictor is described based upon the clamped cubic spline interpolation function using previously computed points on the curve to compute the coefficients via divided differences.
01 Jan 2003
TL;DR: Differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support are designed, given in closed form, and provide stable, orthogonal decompositions of L2 (R2) .
Abstract: Dedicated to Professor M. J. D. Powell on the occasion
of his sixty-fifth birthday and his retirement.
In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L
2
(R
2
) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.
01 May 2003
TL;DR: In this paper, the authors prove convergence rates for spherical spline Hermite interpolation on the sphere Sd−1 via an error estimate given in a technical report by Luo and Levesley.
Abstract: In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere Sd−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.
18 May 2003
TL;DR: A curve design method has been proposed which, in addition to enjoying the good features of cubic splines, possesses interested shape design features too.
Abstract: A curve design method has been proposed which, in addition to enjoying the good features of cubic splines, possesses interested shape design features too. Two families of shape parameters have been introduced in such a way that one family of parameters is associated with intervals and the other with points. These parameters provide a variety of shape controls like point and interval tension. This is an interpolatory curve scheme, which utilizes a piece-wise rational cubic function in its description. The proposed method enjoy ideal geometric properties and geometric continuity of order two is also achieved.
09 Mar 2003
TL;DR: This paper presents a discretization technique for particle dynamics equation based on the B-spline interpolation of the solution based in the general framework recently proposed by the authors.
Abstract: This paper presents a discretization technique for particle dynamics equation based on the B-spline interpolation of the solution. The method is developed in the general framework recently proposed by the authors. Numerical tests include the coagulation-growth of the exponential distribution and of a cosine hill in logarithmic coordinates.
08 Aug 2003
TL;DR: The improved method of two-dimensional cubic convolution has three parameters that can be tuned to yield maximal fidelity for specific scene ensembles characterized by autocorrelation or power-spectrum.
Abstract: This paper presents results of image interpolation with an improved method for two-dimensional cubic convolution. Convolution with a piecewise cubic is one of the most popular methods for image reconstruction, but the traditional approach uses a separable two-dimensional convolution kernel that is based on a one-dimensional derivation. The traditional, separable method is sub-optimal for the usual case of non-separable images. The improved method in this paper implements the most general non-separable, two-dimensional, piecewise-cubic interpolator with constraints for symmetry, continuity, and smoothness. The improved method of two-dimensional cubic convolution has three parameters that can be tuned to yield maximal fidelity for specific scene ensembles characterized by autocorrelation or power-spectrum. This paper illustrates examples for several scene models (a circular disk of parametric size, a square pulse with parametric rotation, and a Markov random field with parametric spatial detail) and actual images -- presenting the optimal parameters and the resulting fidelity for each model. In these examples, improved two-dimensional cubic convolution is superior to several other popular small-kernel interpolation methods.
TL;DR: In this article, the authors proposed a method of constructing the equi-valued surface from quantized data based on the face-centered cubic lattice, which provides a polyhedron approximation of the Equi-Value surface by linear interpolation.
Abstract: In this paper, the authors propose a method of constructing the equi-valued surface from quantized data based on the face-centered cubic lattice. Hitherto, the quantization of the three-dimensional scalar field has been generally based on the simple cubic lattice, and the technique called the marching cubes method is widely used in constructing equi-valued surfaces. It is known theoretically, on the other hand, that lattice quantization of the three-dimensional scalar field is best based on the face-centered cubic lattice. The proposed method provides a polyhedron approximation of the equi-valued surface by linear interpolation from the quantized data of the three-dimensional scalar field based on the face-centered cubic lattice. An experiment on equi-valued surface construction was performed for the scalar field generated algebraically by using a meta-ball. The equi-valued surface obtained by the proposed method is compared to the equi-valued surfaces obtained by the marching cubes method and other related cube decomposition methods. The effectiveness and range of application of the proposed method are demonstrated. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(12): 1–13, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10095
Journal Article•
TL;DR: The research shows that two kinds of cubic spline interpolation functions can be solved in this algorithm and the detailed derivation is derived in detail without error and convergence.
Abstract: The aim is to solve two kinds of cubic spline interpolation functions in a gener al algorithm and show the detailed derivation. By combining the spline equation s based on the initial conditions for two kinds of splines, the calculation form ula for cubic spline interpolation functions is derived in detail without changi ng the error and convergence. The research shows that two kinds of cubic spline interpolation functions can be solved in this algorithm.
Journal Article•
TL;DR: In this article, a double cubic spline function is analyzed by means of taylor expansion, and the remainder term of the function can be estimated by using the taylor expansions.
Abstract: There are wide applications of the spline function in the engineering. The estimation of its remainder is a fundamental problem of approximating theory of spline function. If function f(x,y) has the nth derivatives(n is enough), Practical method of double cubic spline funciton is given in this research. If a function can be analyzed by means of taylor expansion, remainder term R(x,y) can be estimated.
Journal Article•
TL;DR: This paper investigates the monotonicity-preserving interpolation of a kind of plane parameter curve with a shape control parameter, which can be obtained succinctly and is hopeful to be widely applied to engineering and practice.
Abstract: In the geometric shape design, shape preserving interpolation of curve/surface is an important and difficult subject in which both monotonicity-preserving and convexity-preserving interpolation are two basic contents. In this paper, the monotonicity-preserving interpolation of a kind of plane parameter curve with a shape control parameter is investigated. The basic idea is as follows: first, a kind of plane α-B-spline interpolation curve with a shape control parameter a is constructed; then, by converting the first derivatives of the curve into Bernstein polynomial, the positive conditions of Bernstein polynomial can be used to get the necessary and sufficie nt conditions for the monotonicity of α-B-spline interpolation curves, i.e., the range of the parameter a. Therefore, monotone-preserving interpolating curves can be obtained succinctly. Numerical examples illustrate the correctness and the validity of theoretical reasoning. In virtue of its convenience and efficiency, this method is hopeful to be widely applied to engineering and practice.
Journal Article•
TL;DR: Principals of optimal piecewise cubic spline interpolation and applications of interpolant fitting to the thermocouple characteristic curves are introduced and the fitting accuracy is increased.
Abstract: Principals of optimal piecewise cubic spline interpolation and applications of interpolant fitting to the thermocouple characteristic curves are introduced.By analyzing nonlinearity of the cubic spline, the optimal interpolant knots are calculated,and the thermocouple characteristic curve is fitted by the nonuniform least square interpolation.By continual adjusting the interpolant knots, the fitting accuracy is increased.With the fitting result, only thrice multiplications and additions are needed to realize the optimal cubic spline interpolant by the ordinary microprocessor.
Journal Article•
TL;DR: A novel approach of image resampling in motion compensated prediction used cubic convolution interpolation instead of bilinear interpolation is proposed and conception of Adaptive Motion Accuracy is given.
Abstract: Based on the research of H.26L coding structure, a novel approach of image resampling in motion compensated prediction used cubic convolution interpolation instead of bilinear interpolation is proposed and conception of Adaptive Motion Accuracy is given. Compared with the bilinear interpolation in H.263, cubic convolution interpolation has its advantages. Then we realized the H.26L software codec based on cubic convolution interpolation. Implementation results show that cubic convolution interpolation is effective.
Journal Article•
TL;DR: This paper presents a method which is based on 1D interpolation to reconstruct the 3D smooth surface of head based on the several part-line-head of profiles, and has several advantages, such as simple arithmetic, and little time spending for reconstruction.
Abstract: This paper presents a method which is based on 1D interpolation to reconstruct the 3D smooth surface of head. Based on the several part-line-head of profiles, a 3D line-head is reconstructed, and then three steps are needed, including distilling the control points, interpolating based on 1D-cubic-spline, reconstructing the 3D-surface. After these steps, the 3D digital surface of head with the arbitrary resolution is formed, and the surface can be described by triangle-grids, rectangle-grids or other forms. Compared with traditional 3D surface reconstruction methods, such as two-dimension interpolation, the algorithm has several advantages, such as simple arithmetic, and little time spending for reconstruction. The algorithm has been implemented on a PIII 800 PC coming out with good results
Journal Article•
TL;DR: This paper program three procedures in Matlab language, then they can find cubic spline interpolation functions automatically under three boundary conditions, and this method is very troublesome in most textbooks of numerical analysis.
Abstract: In the Spline Toolbox of Matlab, it does not give the expressions of cubic spline interpolation functions , but we need to know the expressions in teaching process and some practical problems. In most textbooks of numerical analysis, the expressions be got by solving some equations, this method is very troublesome. In this paper, we program three procedures in Matlab language, then we can find cubic spline interpolation functions automatically under three boundary conditions.