Showing papers on "Monotone cubic interpolation published in 2004"

Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations

Abstract: We describe local Lagrange interpolation methods based on C1 cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.

Interactive Shape Control with Rational Cubic Splines

Abstract: A rational cubic spline, with shape parameters, has been discussed with the view to 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. The rational cubic spline attains parametric C2 smoothness whereas the stitching of the conic segments preserves visually reasonable smoothness (C1) at the neighboring knots. A very simple distance-based approximated derivative scheme is also presented to calculate control points. 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. We discuss difficult cases of elliptic arcs in space and introduce intermediate point interpolation scheme which can force the curve to pass through given point between any segment.

On cubic operators defined on finite-dimensional simplexes

Abstract: We introduce the concept of cubic operator. For one class of cubic operators defined on finite-dimensional simplexes, a complete description of the behavior of their trajectories is given. The convergence of Cesaro means is established.

A fast computation of 2-D cubic-spline interpolation

Abstract: Based on a cross-zonal filter in the two-dimensional (2-D) cubic-spline interpolation (CSI) and a symmetric extension method, an efficient algorithm is proposed for image coding. Experimental results show that the proposed method is superior in performance and yields a better quality of reconstructed image than other interpolation methods.

A new edge preserving pixel-by-pixel (PBP) cubic image interpolation approach

Abstract: A new pixel-by-pixel (PBP) cubic image interpolation algorithm that preserves image edges is suggested in this paper. The PBP approach is based on optimizing the standard cubic image interpolation formula at each estimated pixel. Thus, the mean square error (MSE) in the entire image is minimized. A study of the effect of optimizing the cubic image interpolation formula with respect to the separated or combined parameters of the formula is presented The optimum values of the parameters are estimated iteratively at each pixel. The performance of the suggested approach is tested in the presence of different noise levels and is compared with the traditional warped distance adaptive image interpolation technique. The obtained results proves the superiority of the suggested PBP cubic image interpolation algorithm as compared to the traditional algorithm from both of the MSE and edge preservation points of view.

Range Restricted Interpolation Using Cubic Bézier Triangles

Abstract: A range restricted C 1 interpolation local scheme to scattered data is derived. Each macro triangle of the triangulated domain is split into three mini triangles and the interpolating surface on each mini triangle is a cubic Bezier triangle. Sufficient conditions derived for the non-negativity of these cubic Bezier triangles are expressed as lower bounds to the Bezier ordinates. The non-negativity preserving interpolation scheme extends to the construction of a range restricted interpolating surface with lower or upper constraints which are polynomial surfaces of degree up to three. The scheme is illustrated with graphical examples.

An Efficient Algorithm for Generating Univariate Cubic L 1 Splines

Abstract: An active set based algorithm for calculating the coefficients of univariate cubic L1 splines is developed. It decomposes the original problem in a geometric-programming setting into independent optimization problems of smaller sizes. This algorithm requires only simple algebraic operations to obtain an exact optimal solution in a finite number of iterations. In stability and computational efficiency, the algorithm outperforms a currently widely used discretization-based primal affine algorithm.

Curve and surface interpolation and approximation: knowledge unit and software tool

Abstract: This paper describes a knowledge unit and the use of a software tool, DesignMentor, for teaching a very challenging topic in computer graphics and visualization, namely: curve and surface interpolation and approximation. Topics include global and local interpolation, global approximation, and curve network interpolation. For the past six years, a junior-level course has successfully used this approach.

Study of empirical mode decomposition based on high-order spline interpolation

Abstract: The time-frequency analysis method based on empirical mode decomposition (EMD) was introduced. The series data were separated into intrinsic mode functions (IMFs) with different time scale using EMD. The spectrum of Hilbert transformation was yielded by applying Hilbert transformation to every IMF. Based on three-order spline interpolation, high-order spline interpolation was used to improve the precision of the algorithm. The simulation result shows that the precision of the time-frequency analysis can be improved effectively using the proposed new algorithm.

Thin-Plate Spline Interpolation

Abstract: The purpose of this chapter is to present an introduction to thin-plate spline interpolation and indicate how it can be a useful tool in medical imaging applications. After a brief review of the strengths and weaknesses of polynomial and Fourier interpolation, the ideas fundamental to the success of cubic spline interpolation are discussed. These ideas include convergence rates of both the interpolants and the derivatives as well as the fact that the clamped cubic spline is the solution of a minimization problem, where the optimal solution is the one exhibiting the fewest “wiggles.” This measure is important because it helps ensure that if slowly oscillating data is interpolated by a spline technique, then the interpolation will also be reasonable. The classical examples of Runge are presented, which dramatically demonstrate the dangers of polynomial interpolation for even the least oscillatory data. While the Fourier interpolants are more stable than the polynomial ones, they have the problem that while the interpolants converge, their derivatives do not. Thus, the spline approach has definite advantages

Image resizing with raised cosine pulses

Abstract: In this work, an interpolation filter using truncated raised cosine pulses for image resizing, is investigated. The rectangular-truncated raised cosine pulse with rolloff factor /spl beta//spl ges/0.5 and truncation length T/spl ges/4T/sub s/ is shown to be a good choice for interpolation. The experimental result reveals that with a comparable computational complexity, the truncated raised cosine interpolator is superior to the widely used interpolator with Keys cubic function. The result also reveals that the performance of the truncated raised cosine interpolator is only slightly worse than that of the least-square cubic B-spline interpolator, but with much lower computational or hardware complexity.

Efficient Implementation of Higher Order Image Interpolation.

Abstract: This work presents a new method of fast cubic and higher order image interpolation. The evaluation of the piecewise n-th order polynomial kernels is accelerated by transforming the polynomials into the interval [0,1], which has the advantage that some terms of the polynomials disappear, and that several coecients could be precalculated, which is proven in the paper. The results are exactly the same as using standard n-th order interpolation, but the computational complexity is reduced. Calculating the interpolation weights for the cubic convolution only needs about 60% of the time compared to the classical method optimized by the Horner’s rule. This allows a new ecient implementation for image interpolation.

Variational Interpolation of Subsets

Abstract: We consider the problem of the variational interpolation of subsets of Euclidean spaces by curves such that the L2 norm of the second derivative is minimized. It is well-known that the resulting curves are cubic spline curves. We study geometric boundary conditions arising for various types of subsets such as subspaces, polyhedra, and submanifolds, and we indicate how solutions can be computed in the case of convex polyhedra.

Parametric smoothing of spline interpolation

Abstract: Cubic spline interpolation is commonly applied in signal reconstruction problems. However, overshooting between samples is normally observed, and typically the reconstructed signal does not preserve the statistical properties of the original data or other desired properties such as monotonicity or convexity. These undesirable effects are minimized in the case of piecewise linear (PWL) interpolation, of course with a discontinuous derivative. In this paper we use a parameterized family of splines, named /spl alpha/splines, that allows a smooth transition from PWL (/spl alpha/ = 0) to cubic spline interpolation (/spl alpha/ = 1). Closed-form expressions that relate /spl alpha/ to the smoothness and variance of the interpolation are derived. Moreover, a fast interpolation technique based on digital filtering can be applied.

RC_SPLINE: Stata module to generate restricted cubic splines

Abstract: rc_spline creates variables that can be used for regression models in which the linear predictor f(xvar) is assumed to equal a restricted cubic spline function of an independent variable xvar. In these regressions, the user explicitly or implicitly specifies k knots located at xvar = t1, t2, ..., tk. f(xvar) is defined to be a continuous smooth function that is linear before t1, is a piecewise cubic polynomial between adjacent knots, and is linear after tk. See Harrell (2001) for additional details.

Numerical error patterns for a scheme with Hermite interpolation for 1 + 1 linear wave equations

Abstract: Numerical error patterns were presented when the fourth-order scheme based on Hermite interpolation was used to solve the 1 + 1 linear wave equation. Since most non-linear equations for real systems can be converted into linear forms by using proper transformations, this study certainly pertains its practical significance. The analytical solution was obtained under inhomogeneous initial and boundary conditions. It was found that not only the Hurst index of an error train at a given position but also its spatial distribution is dependent on the ratio of temporal to spatial intervals. The solution process with the fourth-order scheme based on Hermite interpolation diverges as the ratio is greater than unity. The results show that regular error pattern and smaller maxima of absolute values of numerical errors can be obtained when the ratio is set as unity; while chaotic phenomena for the numerical error propagation process can appear when the ratio is less than unity. It was found that it is better to choose the ratio as unity for the numerical solution of 1 + 1 linear wave equation with the scheme; while other selections for the ratio in the scheme can bring about chaotic patterns for the numerical errors. Copyright © 2004 John Wiley & Sons, Ltd.

A formal linearization for time-variant nonlinear systems by the cubic Hermite interpolation and its applications

Abstract: A computational method of the formal linearization for time-variant nonlinear systems by using the cubic Hermite interpolation is proposed. We introduce a linearizing function that consists of the state variables, their squares, and the cubes. The nonlinear terms are approximated by the cubic Hermite interpolation and thus a formal linear time-variant system with respect to the linearizing function is acquired. The execution of this method is easily carried out by simple matrix multiplications. A nonlinear observer and a nonlinear filter are synthesized as applications of this method and are verified through numerical examples.

Hybrid linear compensation and improved Hermite interpolation for optical measurement of 3D airfoil surface

Abstract: Non-contact optical measurement systems such as laser scanner have been widely used to measure 3D profiles. In the case of measuring a polished turbine airfoil, however, specular reflection on the surface causes "non-measurable" gaps in each optical scan. Conventional techniques like linear interpolation and spline interpolation become ineffective in reconstructing the surface from such "corrupted" measurement data sets. This work develops a hybrid technique that combines piecewise cubic Hermite interpolation with linear compensation. The technique preserves the shape well, and generates smooth curvature in the specular reflection gap. In comparison with spline interpolation, it achieves better and more accurate profile fitting.

Hermite cubic collocation method for optimal control of two-integrator systems

Abstract: The validity of the Hermite cubic collocation algorithm was verified using the time-optimal control problem of the classical two-integrator dynamics system. The Hermite cubic collocation method and the sequential quadric programming method were used for the time-optimal control algorithm. The simulation results agree well with the analytical results showing the effectiveness of the Hermite cubic collocation method for two-point boundary value problems in dynamic systems. The method can be applied for astrodynamics and optimal control.

Improvements of volume computation from nonparallel cross-sections

Abstract: We describe an algorithm for volume evaluation of a 3D object from area measurements done in non-parallel cross-sections. The algorithm is based on Watanabe formula for volume computation and uses an interpolation by cubic splines. The same splines are applied also for computation of object area and centroid in every cross-section. It allowed us to derive an explicit formula for volume computation and to extend the algorithm described in GM Treece, et al. (1999).

Unbalanced hermite interpolation with tschirnhausen cubics

Abstract: A method for constructing a cubic Pythagorean hodograph (PH) curve (called a Tschirnhausen cubic curve as well) satisfying unbalanced Hermite interpolation conditions is presented. The resultant curve interpolates two given end points, and has a given vector as the tangent vector at the starting point. The generation method is based on complex number calculation. Resultant curves are represented in a Bezier form. Our result shows that there are two Tschirnhausen cubic curves fulfilling the unbalanced Hermite interpolation conditions. An explicit formula for calculating the absolute rotation number is provided to select the better curve from the two Tschirnhausen cubic curves. Examples are given as well to illustrate the method proposed in this paper.

On the parameter dependence of the real cubic surfaces in Arnold's form

Abstract: We consider the topological classification of cubic surfaces which are obtained as intersection of the sphere\(\mathbb{S}^3 \) with the algebraic variety defined by the zeroes of a homogeneous cubic polynomial in Arnold’s normal form. This classification is based on the parameters appearing in this normal form, obtaining a correspondence between the parameters of the surface and its topological type. General classifications of cubic surfaces are made in the projective space ℙ3(ℝ), but our method, based on a very simple combinatorial procedure is easier to implement in\(\mathbb{S}^3 \). We split the cubic surfaces parameter space into ten equivalence classes.