MULTIVARIATE INTERPOLATION USING POLYHARMONIC SPLINES

Data measuring and further processing is the fundamental activity in all branches of science and technology. Data interpolation has been an important part of computational mathematics for a long time. In the paper, we are concerned with the interpolation by polyharmonic splines in an arbitrary dimension. We show the connection of this interpolation with the interpolation by radial basis functions and the smooth interpolation by generating functions, which provide means for minimizing the L2 norm of chosen derivatives of the interpolant. This can be useful in 2D and 3D, e.g., in the construction of geographic information systems or computer aided geometric design. We prove the properties of the piecewise polyharmonic spline interpolant and present a simple 1D example to illustratethem.

Shape Designing of Engineering Images Using Rational Spline Interpolation

In modern days, engineers encounter a remarkable range of different engineering problems like study of structure, structure properties, and designing of different engineering images, for example, automotive images, aerospace industrial images, architectural designs, shipbuilding, and so forth. This paper purposes an interactive curve scheme for designing engineering images. The purposed scheme furnishes object designing not just in the area of engineering, but it is equally useful for other areas including image processing (IP), Computer Graphics (CG), Computer-Aided Engineering (CAE), Computer-Aided Manufacturing (CAM), and Computer-Aided Design (CAD). As a method, a piecewise rational cubic spline interpolant, with four shape parameters, has been purposed. The method provides effective results together with the effects of derivatives and shape parameters on the shape of the curves in a local and global manner. The spline method, due to its most generalized description, recovers various existing rational spline methods and serves as an alternative to various other methods includingv-splines, gamma splines, weighted splines, and beta splines.

Positivity Preserving Interpolation Using Rational Bicubic Spline

This paper discusses the positivity preserving interpolation for positive surfaces data by extending theC1rational cubic spline interpolant of Karim and Kong to the bivariate cases. The partially blended rational bicubic spline has 12 parameters in the descriptions where 8 of them are free parameters. The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail. Based on Root Mean Square Error (RMSE), our partially blended rational bicubic spline is on a par with the established methods.

QUASI-INTERPOLATION BY SPLINES ON THE UNIFORM KNOT SETS

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.

Hermite spline interpolents ― New methods for constructing and compressing Hermite interpolants

International audience In this paper, we present a quite simple recursive method for the construction of classical tensor product Hermite spline interpolant of a function defined on a rectangular domain. We show that this function can be written under a recursive form and a sum of particular splines that have interesting properties. As application of this method, we give an algorithm which allows to compress Hermite data. In order to illustrate our results, some numerical examples are presented. Dans ce travail, nous présentons une méthode simple permettant de construire le produit tensoriel des interpolants splines d'Hermite d'une fonction définie sur un domaine rectangulaire. Nous montrons que cette fonction peut être décrite de manière récursive sous la forme d'une somme de fonctions splines qui vérifiant des propriétés intéressantes. Comme application de cette décomposition, nous décrivons un algorithme qui permet de compresser des données d'Hermite. Pour illustrer nos résultats théoriques, nous donnons quelques exemples numériques.