Bernd Mulansky

Powell-Sabin splines in range restricted interpolation of scattered data

The construction of range restricted bivariateC1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a Powell-Sabin refinement of a triangulation of the data sites subject to piecewise constant lower and upper bounds on the values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a solvable system of linear inequalities for the gradients as parameters, which is separated with respect to the data sites. Since there exists an infinite number of spline interpolants meeting the constraints, the selection of a visually pleasant solution is based on the minimum norm modification of a suitable initial interpolant or on the minimization of the thin plate functional. While the first proposal reduces to the solution of independent local quadratic programs, the second proposal results in a global quadratic optimization problem.ZusammenfassungBehandelt wird die Interpolation unregelmäßig verteilter Daten durch quadratische Powell-Sabin-Splines, wobei stückweise konstante obere und untere Schranken für die Funktionswerte der Interpolierenden vorgegeben sind. Die hergeleiteten hinreichenden Bedingungen für die Einhaltung der zweiseitigen Schranken ergeben ein lösbares lineares Ungleichungssytem für die Gradienten als Parameter, welches bezüglich der Datenpunkte separiert ist. Die Auswahl von visuell gefälligen Interpolierenden erfolgt durch die Minimierung geeigneter Funktionale.

Scattered data interpolation subject to piecewise quadratic range restrictions

Abstract The constriction of range restricted univariate and bivariate C1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a refined univariate grid (respectively on a Powell-Sabin refinement of a triangulation of the data sites) subject to piecewise quadratic lower and upper obstacles to the values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a system of linear inequalities for the slopes (respectively gradients) as parameters, which is separated with respect to the data sites. This system is shown to be always solvable for important special forms of the obstacles. If at all, in general there exist an infinite number of spline interpolants meeting the constraints. The selection of a visually pleasant one is based on the minimization of a suitable choice functional.

Constructive methods in convex C 2 interpolation using quartic splines

Using quartic splines on refined grids, we present a method for convexity preservingC2 interpolation which is successful for all strictly convex data sets. In the first stage, one suitable additional knot in each subinterval of the original data grid is fixed dependent on the given data values. In the second stage, a visually pleasant interpolant is selected by minimizing an appropriate choice functional.

Tensor Product Spline Interpolation subject to Piecewise Bilinear Lower and Upper Bounds

This paper is concerned with range restricted interpolation of gridded data by biquadratic and biquartic tensor product splines on refined rectangular grids. In particular, the given lower and upper bounds are assumed to be continuous and piecewise bilinear with respect to the original grid. Sufficient conditions for the fulfillment of the range restrictions are derived utilizing the tensor product structure as well as corresponding results for univariate quadratic and quartic splines with additional knots. The solvability of this system of sufficient conditions, hence the existence of interpolants meeting the constraints, can always be achieved for strictly compatible data by constructing the refined grid appropriately. The selection of a visually improved range restricted interpolant is based on a fit-and-modify approach or on the minimization of a bivariate Holladay functional.

Convex interval interpolation using a three-term staircase algorithm

Abstract. Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation, the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants. In particular, we refer to methods based on polynomial splines defined on grids with additional knots.

Tensor Products of Convex Cones

Motivated by problems of shape preserving tensor product interpolation, tensor products of convex cones in finite dimensional linear spaces are studied in this paper. We recall the notions of projective and injective tensor product cones and derive some of their properties. It is shown that the cones usually considered in shape preserving tensor product interpolation can be represented as intersections of injective tensor product cones. Consequently, sufficient conditions for the fulfillment of the shape constraints are easily derived from corresponding conditions in the univariate case.

Composition based staircase algorithm and constrained interpolation with boundary conditions

Summary. Weakly coupled systems of inequalities arise frequently in the consideration of so-called direct methods for shape preserving interpolation. In this paper, a composition based staircase algorithm for bidiagonal systems subject to boundary conditions is developed. Using the compositions of the corresponding relations instead of their projections, we are able to derive a necessary and sufficient solvability criterion. Further, all solutions of the system can be constructed in a backward pass. To illustrate the general approach, we consider in detail the problem of convex interpolation by cubic

Nonnegative Volume Matching by Cubic C 1 Splines on Clough--Tocher Splits

C^1

Interpolation and approximation from convex sets. II infinite-dimensional interpolation

splines. For this problem, an algorithm of the complexity O(n) in the number n of data points is obtained.

Interpolation and Approximation from Convex Sets

We describe a numerical method to construct volume matching C1 splines satisfying the frequently occurring requirement of nonnegativity. The approximated histogram is assumed to be defined on a partition of a polygonal domain into polygonal subdomains. Our idea consists of applying cubic C1 splines on Clough--Tocher refinements of a triangulation that is compatible with the given partition. The proposed algorithm always works successfully for nonnegative histograms.