Showing papers on "Monotone cubic interpolation published in 2001"
TL;DR: In this article, the cubic spline interpolator is proposed to fit cubic polynomials to adjacent pairs of points and choose the values of the two remaining parameters associated with each polynomial such that the polynoms covering adjacent intervals agree with one another in both slope and curvature at their common endpoint.
Abstract: The need to interpolate is widespread, and the approaches to interpolation are just as widely varied. For example, sampling a signal via a sample and-hold circuit at uniform, T-second intervals produces an output signal that is a piecewise-constant (or zero-order) interpolation of the signal samples. Similarly, a digital-to-analog (D/A) converter that incorporates no further post-filtering produces an output signal that is (ideally) piecewise-constant. One very effective, well-behaved, computationally efficient interpolator is the cubic spline. The approach is to fit cubic polynomials to adjacent pairs of points and choose the values of the two remaining parameters associated with each polynomial such that the polynomials covering adjacent intervals agree with one another in both slope and curvature at their common endpoint. The piecewise-cubic interpolating function g(x) that results is twice continuously differentiable. We develop the basic algorithm for cubic-spline interpolation.
98 citations
Proceedings Article•
01 Jan 2001TL;DR: In this paper, the authors give a recipe for deriving local spline approximation methods which reproduce the whole spline space, which are obtained by solving a series of local approximation problems.
Abstract: We give a recipe for deriving local spline approximation methods which reproduce the whole spline space. The methods are obtained by solving a series of local approximation problems. Examples of specific quadratic and cubic approximation methods are given.
60 citations
TL;DR: This scheme uses piecewise rational cubic functions to preserve the shape of positive and/or monotonic data sets and possesses extra features to modify theshape of the design curve as and when desired.
46 citations
TL;DR: An approach to approximation based on interpolating a set of points based on the cubic spline is discussed, which is relatively straightforward to implement and can be made to be computationally efficient.
Abstract: The authors discuss an approach to approximation based on interpolating a set of points. The one interpolator that they concentrated on, the cubic spline, is relatively straightforward to implement and can be made to be computationally efficient. They also point out that another approach to approximation involves fitting a curve to a set of data points without forcing that curve to pass through any of those data points. This approach is especially useful when the data available is recognized to contain an error component.
38 citations
TL;DR: Extensions to interpolation of regularly spaced and scattered bi- and multivariate data by cubic and higher-degree surfaces/hypersurfaces on regular and irregular rectangular/quadrilateral/hexahedral and triangular/tetrahedral grids are outlined.
37 citations
TL;DR: A general parametric local approach for functional C2 Hermite shape preserving interpolation for parametric curves with piecewise cubic components and monotonicity is studied.
Abstract: We propose a general parametric local approach for functional C2 Hermite shape preserving interpolation. The constructed interpolant is a parametric curve which interpolate values, first and second derivatives of a given function and reproduces the behavior of the data. The method is detailed for parametric curves with piecewise cubic components. For the selected space necessary and sufficient conditions are derived to ensure the convexity of the constructed interpolant. Monotonicity is also studied. The approximation order is investigated for both cases. The use of a parametric curves to interpolate data from a function can be considered a disadvantage of the scheme. However, the simple structure of the used curve greatly reduces such a disadvantage.
33 citations
Proceedings Article•
01 Jan 2001TL;DR: In this article, Lagragne interpolation schemes are constructed based on C 1 cubic splines on certain triangulations obtained from checkerboard quadrangulations, and the splines are used to compute the Lagrangians.
Abstract: Lagragne interpolation schemes are constructed based on C1 cubic splines on certain triangulations obtained from checkerboard quadrangulations.
30 citations
TL;DR: It is shown that the problem is with the selection of interpolation parameter values, not with even-degree B-spline curves and surfaces themselves, and a new approach to perform quadratic B- Spline curve interpolation is provided, whose quality is comparable to that of cubic interpolating B- spline curves.
Abstract: Traditional approach in performing even-degree B-spline curve/surface interpolation would generate undesired results. In this paper, we show that the problem is with the selection of interpolation parameter values, not with even-degree B-spline curves and surfaces themselves. We prove this by providing a new approach to perform quadratic B-spline curve interpolation. This approach generates quadratic B-spline curves whose quality is comparable to that of cubic interpolating B-spline curves. This makes quadratic B-spline curves better choices than cubic B-spline curves in some applications in graphics and geometric modeling, since it is cheaper to render/subdivide a quadratic curve and it is easier to find the intersection of two quadratic curves.
22 citations
TL;DR: An algorithm for computing the cubic spline interpolation coefficients without solving the matrix equation involved is presented, which requires only O(n) multiplication or division operations for Computing the inverse of the matrix.
15 citations
01 Jan 2001
14 citations
TL;DR: A reference function f is defined as a cubic smoothing spline which is supposed to have the global form of the data, which is then used to locate the knots of the final approximating spline by using a criterion based on the third derivative of f.
Abstract: This paper presents a data reduction method for functional data. Starting with noisy or not noisy data, we first define a function f, called the “reference function”, as a cubic smoothing spline which is supposed to have the global form of the data. This reference function is then used to locate the knots of the final approximating spline by using a criterion based on the third derivative of f. Then, the least-squares spline approximating all the data is derived with these knots. Numerical results show the effectiveness of the method.
TL;DR: In this article, the convergence of Hermite and Hermite-Fejer interpolation polynomials of higher order at the zeros of Freud orthogonal polynomial on the real line was investigated.
25 Jul 2001
TL;DR: A high degree of smoothness is obtained with automatic estimation of two families of shape preserving parameters for local convexity with necessary and sufficient conditions for a convex data.
Abstract: This paper discusses the distribution of inflection points and singularities on a piecewise parametric rational cubic spline. Necessary and sufficient conditions for a convex data have been presented. A high degree of smoothness is obtained with automatic estimation of two families of shape preserving parameters for local convexity.
TL;DR: In this paper, the convergence rate of the collocation method with cubic splines of minimum deficiency for linear weakly singular integral equations of the second kind was analyzed, and it was shown that the use of a special non-uniform grid in the collocations with minimum deficiency provides the same accuracy order as in a singularity-free problem.
Abstract: It is known (e.g., see [1–10]) that a weak singularity in the kernel of an integral equation of the second kind leads to a singularity in the solution; more precisely, the derivatives of the solution prove to be unbounded near the boundary of the integration domain. This complicates the construction of approximate methods of higher-order accuracy for weakly singular integral equations. It was shown in [3, 4, 8, 10–15] how to condense the grid near points where the solution may prove singular so as to avoid the singularities and construct piecewise polynomial approximations with an arbitrary prescribed accuracy order. A similar approach in the collocation method with cubic splines was considered in [3, 16–18]. The aim of the present paper is to analyze the convergence rate of the collocation method with cubic splines of minimum deficiency for linear weakly singular integral equations of the second kind. Possible singularities of derivatives of the solution of an integral equation are taken into account with the use of a special condensation of the grid, just as in [3]. We generalize and continue the related investigations in [3, 16, 17]. In particular, for integral equations with more general weakly singular kernels, we prove that the use of a special nonuniform grid in the collocation method with cubic splines of minimum deficiency provides the same accuracy order as in a singularity-free problem. We derive new estimates for the error in the approximate solution in the case of a quasiuniform grid as well as in the case of a nonuniform grid depending on the value of the nonuniformity parameter of the grid in question.
Journal Article•
TL;DR: In this paper, the global cubic C2 spline is acguired by using the rule of least weight, and the authors select the method of parameterization, and also consider the storage method for the corresponding graph of the given data.
Abstract: The global cubic C2 spline is acguired by using the rule of least weight. In order to acguire the circle of higher-order approximation, the authors select the method of parameterization, and also consider the storage method for the corresponding graph of the given data.
TL;DR: Test image results indicate that adaptive spline interpolation of a low- resolution image is superior to non-adaptive interpolation if the adjustable parameters are chosen to yield the best match to a known object in a corresponding high-resolution image.
Abstract: Adaptive spline interpolation (which is equivalent to the use of a type of radial basis function neural network) is investigated for digital image interpolation (i.e., for resolution enhancement). Test image results indicate that adaptive spline interpolation of a low-resolution image is superior to non-adaptive interpolation if the adjustable parameters are chosen to yield the best match to a known object in a corresponding high-resolution image.
TL;DR: An easier way to calculate an algorithm for a cubic spline approximation of an offset curve of a given planar cubicspline and a sufficient condition on an offset length for its existence is derived.
Abstract: We derive an easier way to calculate an algorithm for a cubic spline approximation of an offset curve of a given planar cubic spline and a sufficient condition on an offset length for its existence.
TL;DR: It seems to be possible to derive an always working interpolation algorithm by means of the parametric Clough‐Tocher splines recently introduced in ]1[.
Abstract: In ]4[ the Clough-Tocher splines of quadratic precision are shown to allow always nonnegative interpolation of scattered data. The aim of the present note is to point out that Clough-Tocher splines of cubic precision are suitable, too. This result extends to range restricted interpolation if the obstacles are piecewise constant. For piecewise linear and continuous obstacles the Clough-Tocher splines may fail. However, for this type of restrictions it seems to be possible to derive an always working interpolation algorithm by means of the parametric Clough-Tocher splines recently introduced in ]1[.
Journal Article•
TL;DR: In this paper, a modification of the cubic spline is suggested, based on modern adaptive approximation algorithms widely used in constructing TVD schemes, and the case of an arbitrary distribution of interpolation vertexes is considered.
Abstract: A modification of the cubic spline is suggested, based on modern adaptive approximation algorithms widely used in constructing TVD schemes. The case of an arbitrary distribution of interpolation vertexes is considered. The monotonicity of the spline is proved for monotonic input data. For smooth functions without extrema, the modified spline transforms into the original spline as the distance between the interpolation knots decreases. Numerical examples of solutions to test problems are given.
Journal Article•
TL;DR: In this article, the normal interpolation is converted into the rotation transformation of stencils without influencing the effects of vertex interpolation, thus a C 2 continuous surface interpolating given vertices and normals is obtained.
Abstract: Interpolation to vertex positions and normals is one of important contents in parametric surface modeling. This paper presents an approach based on Catmull Clark and Doo Sabin subdivision schemes to generate the control net of bi cubic B spline surface interpolating the given vertices of initial net. The notion of stencils is introduced such that the normal interpolation is converted into the rotation transformation of stencils without influencing the effects of vertex interpolation, thus a C 2 continuous surface interpolating given vertices and normals is obtained. Compared to traditional methods through stitching patches piece by piece, our method is more compact and has smoothness of higher degree. In addition, the method can be easily extended to high degree B spline surfaces with arbitrary topology nets and it is also fairly useful for practical applications.
TL;DR: This paper proposes a novel image interpolation algorithm based on the quadratic B-spline basis function that preserves the original edges while not destroying the smoothness in flat area using the adaptive interpolation method according to the directional edge pattern of input image.
Abstract: In this paper we propose a novel image interpolation algorithm based on the quadratic B-spline basis function. Our interpolation algorithm preserves the original edges while not destroying the smoothness in flat area using the adaptive interpolation method according to the directional edge pattern of input image, significantly improving the overall performance of the interpolation. Our experimental result shows that it can produce higher quality and resolution than the currently existing image interpolation methods.
Journal Article•
TL;DR: A method can be presented which can do the conversion above the Function of 3order sampling interpolation and a very smoothy curve can be obtained.
Abstract: the data that it was tested are commonly a series of discrete sampling data in CAT . How do this data can be converted into a smoothy curve seemly with no disorder and also make it print out? This paper presents a method which can do the conversion above the Function of 3order sampling interpolation. At the same time , This paper also presents the limitation of the method and its improvent. As a result, a very smoothy curve can be obtained.
TL;DR: A reconstruction algorithm by the cubic spline function interpolation, which is used to recover a signal from its dyadic wavelet transform extrema (instead of modulus maxima), is proposed and Numerous experiment results demonstrate its satisfying properties.
Abstract: A reconstruction algorithm by the cubic spline function interpolation, which is used to recover a signal from its dyadic wavelet transform extrema (instead of modulus maxima), is proposed. Numerous experiment results demonstrate its satisfying properties. Compared with previous algorithms it is simpler, more straightforward and faster.
Journal Article•
TL;DR: In this article, a 3D quartic Bezier curve for given GC 2 Hermite interpolation conditions is constructed, where the solution exists locally and possesses one degree of freedom.
Abstract: A 3-dimensional quartic Bezier curve for given GC 2 Hermite interpolation conditions is constructed. Study shows that the solution exists locally and possesses one degree of freedom. Moreover, the approximation order of the interpolation is proven to be 6.
Journal Article•
TL;DR: In this article, the effect of errors of both the boundary conditions and interpolation dates on interpolation functions of cubic spline was discussed and the error estimation formulas were given. And numerical stability of the interpolation function of cubic Spline was proved.
TL;DR: In this article, the authors construct a smooth surface through n points in the Euclidean space with prescribed tangent planes in these points, which is also optimal in a certain sense.
Abstract: We construct a smooth surface through n points in the Euclidean space with prescribed tangent planes in these points, which is also optimal in a certain sense.
TL;DR: Following a precise definition of shape-preserving interpolating functions to data, a cubic spline S2 ϵ C2 is constructed in a new and elementary manner, letting two additional knots per interval.