Showing papers on "Monotone cubic interpolation published in 1993"
TL;DR: A class of recursive filtering algorithms for the efficient implementation of B-spline interpolation and approximation techniques is described and it is shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next.
Abstract: For pt.I see ibid., vol.41, no.2, p.821-33 (1993). A class of recursive filtering algorithms for the efficient implementation of B-spline interpolation and approximation techniques is described. In terms of simplicity of realization and reduction of computational complexity, these algorithms compare favorably with conventional matrix approaches. A filtering interpretation (low-pass filter followed by an exact polynomial spline interpolator) of smoothing spline and least-squares approximation methods is proposed. These techniques are applied to the design of digital filters for cubic spline signal processing. An efficient implementation of a smoothing spline edge detector is proposed. It is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next. In terms of common measures of fidelity, this data structure appears to be superior to the Gaussian/Laplacian pyramid. >
632 citations
TL;DR: In this article, the authors relax the monotonicity constraint in a geometric framework in which the median function plays a crucial role, and present algorithms for piecewise cubic interpolants, which preserve monoticity as well as uniform third and fourth-order accuracy.
Abstract: Monotone piecewise cubic interpolants are simple and effective. They are generally third-order accurate, except near strict local extrema where accuracy degenerates to second-order due to the monotonicity constraint. Algorithms for piecewise cubic interpolants, which preserve monotonicity as well as uniform third and fourth-order accuracy are presented. The gain of accuracy is obtained by relaxing the monotonicity constraint in a geometric framework in which the median function plays a crucial role.
124 citations
TL;DR: A simple algorithm for generating a C 1 piecewise cubic Hermite interpolant that preserves positivity is given, which is local in nature, and unlike other algorithms does not require modification of the slope data.
109 citations
TL;DR: In this paper, a globally C 2 interpolatory cubic spline containing free parameters is derived and its properties established, and sufficient conditions are given for choosing the parameters to control monotonicity; convexity is discussed.
Abstract: A globally C 2 interpolatory cubic spline containing free parameters is derived and its properties established. Sufficient conditions are given for choosing the parameters to control monotonicity; convexity is discussed. An algorithm is developed and tested on several examples
37 citations
35 citations
TL;DR: The results confirm the validity of constant velocity motion as a first-order model for frequency domain analysis of motion.
Abstract: Translations with piecewise cubic trajectories are studied in the frequency domain. This class of motion has as an important subcase: cubic spline trajectories. Translations with trajectories depending on time with general polynomial law are preliminarily considered, and a general theorem concerning this type of motion is introduced. The application of this theorem to the case of cubic time dependence and the consideration of finite-duration effects lead to the solution of the piecewise cubic trajectory case. The results, which are remarkably different from those concerning constant velocity translations, clearly indicate the importance of the role of velocity and time duration. In this respect, they confirm the validity of constant velocity motion as a first-order model for frequency domain analysis of motion. >
24 citations
TL;DR: An algorithm for creating tangent continuous splines from segments of algebraic cubic curves, which naturally contains conic splines as a subfamily is presented.
Abstract: We present an algorithm for creating tangent continuous splines from segments of algebraic cubic curves. The curves used are cubic ovals, and thus are guaranteed convex. Each segment is given by an equation which has five coefficients, thus four degrees of freedom available for shape control. We describe shape handles that work via the coet%cients ta control the curve. Each segment can be chosen to interpolate one more point and slope and has two additional fullness parameters to control the shape. This family of curves naturally contains conic splines as a subfamily.
19 citations
TL;DR: In this paper, a cubic spline method for linear second-order boundary value problems is presented, which is a Petrov-Galerkin method using a cubic trial space, a piecewise-linear test space, and a simple quadrature rule for the integration.
Abstract: A cubic spline method for linear second-order two-point boundary-value problems is analysed. The method is a Petrov-Galerkin method using a cubic spline trial space, a piecewise-linear test space, and a simple quadrature rule for the integration, and may be considered a discrete version of the H 1 -Galerkin method. The method is fully discrete, allows an arbitrary mesh, yields a linear system with bandwidth five, and under suitable conditions is shown to have an o(h 4−i ) rate of convergence in the W p i norm for i = 0, 1, 2, 1 ≤ p ≤ ∞. The H 1 -Galerkin method and orthogonal spline collocation with Hermite cubics are also discussed
17 citations
TL;DR: C cubic nonsplitting macro patches that do not split the triangular facets of the polyhedron are presented, which lead to an efficient and powerful spline surface scheme using implicit cubics.
Abstract: Macro patches are important for generating quadric or cubic implicit spline surfaces from the input of a polyhedron. All existing macro patches split the triangular facets of the polyhedron; this paper presents cubic nonsplitting macro patches (NMP) that do not split these facets. The NMP's are based on a necessary and sufficient condition for nonsplitting constructions of implicit cubic spline surfaces. This condition can be satisfied for most practical applications, so the NMP's lead to an efficient and powerful spline surface scheme using implicit cubics. The free parameters in an NMP are set using a new technique for excluding topological anomalies such as extraneous sheets, splits, unwanted holes, self-intersections, and unwanted handles. Each cubic patch obtained by this technique best approximates, in a least-squares sense, a quadric patch from a single algebraic component of a monotone polynomial derived from the input data.
10 citations
TL;DR: In this article, necessary and sufficient conditions under which a quadratic spline preserves the positivity of a set of function values in the Hermite interpolation were derived, and it was shown that positive interpolation is always possible over nonnegative function values with suitable parameters.
Abstract: Necessary and sufficient conditions are derived under which a quadratic spline preserves the positivity of a set of function values in the Hermite interpolation. As a corollary it is seen that positive interpolation is always possible over a set of nonnegative function values when a quadratic spline with suitable parameters is used.
10 citations
TL;DR: In this article, a third-order variable-mesh method based on cubic spline approximation for nonlinear singularly perturbed boundary-value problems of the form ey″ = f(x,y), y(a) = α, y(b) = β are presented.
TL;DR: In this paper, the authors obtained new results on Hermite interpolation based on Jacobi and generalized Jacobi zeros in C1 space and proved error estimates in uniform and weighted Lp norms.
Abstract: The authors obtain new results on Hermite interpolation based on Jacobi and generalized Jacobi zeros in C1 space and prove error estimates in uniform and weighted Lp norms. The paper gives also the state of art on the topic.
24 Feb 1993
TL;DR: Some well known theorems of algebraic geometry are used in reducing polynomial Hermite interpolation and approximation in any dimension to the solution of linear systems to provide mathematical models for scattered data sampled in three or higher dimensions.
Abstract: In this paper we use some well known theorems of algebraic geometry in reducing polynomial Hermite interpolation and approximation in any dimension to the solution of linear systems. We present a mix of symbolic and numerical algorithms for low degree curve ts through points in the plane, surface ts through points and curves in space, and in general, hypersuface ts through points, curves, surfaces, and sub-varieties in n dimensional space. These interpolatory and (or) approximatory ts may also be made to match derivative information along all the sub varieties. Such multi-dimensional hypersurface interpolation and approximation provides mathematical models for scattered data sampled in three or higher dimensions and can be used to compute volumes, gradients, or more uniform samples for easy and realistic visualization.
01 Jan 1993
TL;DR: In this article, the authors discuss the basic ideas behind piecewise cubic interpolation and a large variety of interpolation methods exist that are designed to cope with special problems, such as surface generation.
Abstract: This chapter discusses the basic ideas behind piecewise cubic interpolation. A large variety of interpolation methods exists that are designed to cope with special problems. The most popular class of methods is that of piecewise polynomial schemes. All these methods construct curves that consist of polynomial pieces of the same degree and that are of a prescribed overall smoothness. The given data are usually points and parameter values; sometimes, tangent information is added as well. In a surface generation environment, one is often given a set of points p i ∈ 3 and a surface normal vector n i at each data point. Thus, one only knows the tangent plane of the desired surface at each data point, not the actual endpoint derivatives of the patch boundary curves.
TL;DR: The Hermite interpolation problem in the plane considered here is to join two points and to match given unit tangent vectors and signed curvatures at the two points with various G 2 curves consisting of a pair of spirals.
01 Jan 1993-Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering
TL;DR: In this article, the authors compare cubic spline interpolation with experiment, using the magnetization tables as a source of carefully measured experimental data, and show that in all examined cases, cubic splines interpolation introduces errors large enough to invalidate a design, and propose a simple solution to the problem, thus combining the best of all worlds: the speed and convergence properties of Newton-Raphson, the accuracy of a good interpolation scheme, and the convenient mathematical properties of cubic spines.
Abstract: The accurate interpolation of magnetization tables is of paramount importance in the design of high‐precision magnets used for particle accelerators or for magnetic resonance imaging of the human body. Cubic spline interpolation is normally used in combination with the fast converging Newton‐Raphson scheme in the two‐dimensional finite element modelling of such magnets. We compare cubic spline interpolation with experiment, using the magnetization tables as a source of carefully measured experimental data. We show that, in all examined cases, cubic spline interpolation introduces errors large enough to invalidate a design. We also propose a simple solution to the problem, thus combining the best of all worlds: the speed and convergence properties of Newton‐Raphson, the accuracy of a good interpolation scheme, and the convenient mathematical properties of cubic splines. We examine both two‐dimensional and three‐dimensional cases.
TL;DR: An efficient algorithm for the computation of the Hermite spline interpolant is obtained, which is mainly based on the fast Fourier transform, and a simple representation of the fundamental splines can be given.
Abstract: Generalized Hermite spline interpolation with periodic splines of defect 2 on an equidistant lattice is considered. Then the classic periodic Hermite spline interpolation with shifted interpolation nodes is obtained as a special case.
TL;DR: The order of convergence of the interpolatory splines with these end conditions is O(h 4 ) for the cubic spline, and O( h 6 ) for a quintic spline.
TL;DR: In this article, it was shown that the cardinal interpolation operator is bounded and invertible on l p, 1 ≤ ρ ≤ ∞, and the finite linear combination of translates of a conditionally positive definite function.
Abstract: In this paper, we discuss “cardinal” as well as “infinite scattered data ” interpolation using finite linear combination of translates of a conditionally positive definite function. We prove that for a subclass of such functions, the cardinal interpolation operators are bounded and invertible on l p, 1 ≤ ρ ≤ ∞. In some special cases, including $ldquo;Hardy's multiquadrics” and “The Thin Plate Spline,” we show that the scattered data interpolation operators are bounded and invertible on l 2.
01 Jan 1993
TL;DR: In this article, a complete characterization of the Hermite interpolation problem by spline functions with multiple knots is given, and the B-spline representation of s leads to the study of the corresponding collocation matrix.
Abstract: Even an elementary study of the interpolation problem
$$s\left( {{t_i}} \right) = {f_i},\quad i = 0,...,n + 1 $$
in the simplest linear case, i.e., when s ∈ S 1(x 1,... ,x n ), shows that the solvability of the corresponding system depends entirely on the mutual location of the interpolation nodes \(t = \left\{ {{t_i}} \right\}_0^{n + 1}\) and the spline knots \(x = \left\{ {{x_i}} \right\}_i^n\) For example, in the case t i = x i , i = 1,... ,n, the problem has a unique solution: the piecewise linear function with vertices at (t i, f i ), i = 0,..., n + 1 On the other hand, in the case where three or more interpolation nodes are situated between two consecutive x i ’s, the problem becomes unresolvable. We shall give here a complete characterization of the Hermite interpolation problem by spline functions with multiple knots. The B-spline representation of s leads us to the study of the corresponding collocation matrix {B i (t j)}.
01 Jan 1993
TL;DR: In this article, rational Hermite interpolation is used in two different ways in order to derive and analyze quadrature rules, one approach yields quadratures of Gaussian-type and the other one generalizes Engels' dual Quadratures.
Abstract: Rational Hermite interpolation is used in two different ways in order to derive and analyze quadrature rules. One approach yields quadratures of Gaussian-type whereas the other one generalizes Engels’ dual quadratures exhibiting the close connection between rational Hermite interpolation and quadrature in general.
TL;DR: The interpolation of a discrete set of data on a uniform mesh, representing the derivative values of a smooth function f, between consecutive nodes, is considered using quadratic C 1 -splines.
01 Jan 1993
TL;DR: This chapter presents two entirely independent derivations of cubic interpolatory splines: the B-spline form and the Hermite form, which are able to model complex shapes easily and fulfill the task of interpolation.
Abstract: In this chapter, we discuss what is probably the most popular curve scheme: C2 cubic interpolatory splines. We have seen how polynomial Lagrange interpolation fails to produce acceptable results. On the other hand, we saw that cubic B-spline curves are a powerful modeling tool; they are able to model complex shapes easily. This “modeling” is carried out as an approximation process, manipulating the control polygon until a desired shape is achieved. We will see how cubic splines can also be used to fulfill the task of interpolation, the task of finding a spline curve passing through a given set of points. Cubic spline interpolation was introduced into the CAGD literature by J. Ferguson [183] in 1964, while the mathematical theory was studied in approximation theory (see de Boor [112] or Holladay [264]). For an outline of the history of splines, see Schumaker [423].
Because of the subject's importance, we present two entirely independent derivations of cubic interpolatory splines: the B-spline form and the Hermite form.
01 Jun 1993
TL;DR: The technique devised decomposes a standard cubic interpolation into a set of two more cheap bilinear interpolations using the two windows surrounding any pixel in the interpolated image to compute a local contrast by measuring the decrease or increase of the luminance in a given neighbourhood.
Abstract: This paper describes the development of a transputer based system for interpolating 2D images using a modified cubic interpolation technique. The technique devised decomposes a standard cubic interpolation into a set of two more cheap bilinear interpolations using the two windows surrounding any pixel in the interpolated image. The two bilinear interpolation results are then used to compute a local contrast by measuring the decrease or increase of the luminance in a given neighbourhood. A local contrast enhancement is then used to compute and enhance the value of the interpolated pixel. Implementation on a network of transputers using 'image parallelism' has been carried out and a significant speedup factor obtained. >