Powell-Sabin splines in range restricted interpolation of scattered data
type: Item
instance of: scholarly article
title: Powell-Sabin splines in range restricted interpolation of scattered data
zbMATH Open document ID: 0811.65004
DOI: 10.1007/BF02252986
author: Jochen W. Schmidt
author: Bernd Mulansky
published in: Computing
publication date: 1995-05-02T00:00:00Z
review text: Let be given a set of data sites \(\{P_ i= (X_ i, Y_ i)\in \mathbb{R}^ 2; i= 0,\dots, n\}\) and corresponding data values \(\{Z_ i\}\). Let \(\Delta\) be an admissible triangulation of the convex hull of the data sites. The triangle with vertices \(P_ i\), \(P_ j\) and \(P_ k\) is denoted by \(\Delta_{ijk}\). The authors suppose that lower and upper bounds \(L_{ijk}\) and \(U_{ijk}\) for all triangles \(\Delta_{ijk}\in \Delta\) are known, which fulfill the conditions \(L_{ijk}\leq Z_ p\leq U_{ijk}\) for \(p= i,j,k\), \(\Delta_{ijk}\in \Delta\). The aim of this paper is the construction of an interpolating \(C^ 1\) spline \(s\), satisfying the interpolation conditions \(s(X_ i, Y_ i)= Z_ i\) for \(i= 0,\dots, n\) and the range restrictions \(L_{ijk}\leq s(X,Y)\leq U_{ijk}\) for \((X,Y)\in \Delta_{ijk}\), \(\Delta_{ijk}\in \Delta\). It is shown that range restricted interpolation is always successful using quadratic \(C^ 1\) splines on Powell-Sabin refinements of \(\Delta\).
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 683321
zbMATH Keywords: Powell-Sabin splines
zbMATH Keywords: interpolation of scattered data
zbMATH Keywords: interpolating splines
zbMATH Keywords: quadratic splines
zbMATH Keywords: range restrictions
reviewed by: Nicolae Ţăndăreanu
MaRDI profile type: Publication
cites work: Finite element approximation of the volume-matching problem
cites work: Post-Processing Piecewise Cubics for Monotonicity
cites work: Q3199955
cites work: Q5758624
cites work: An Analysis of Two Algorithms for Shape-Preserving Cubic Spline Interpolation
cites work: Determination of shape preserving spline interpolants with minimal curvature via dual programs
cites work: Nonnegativity-, Monotonicity-, or Convexity-Preserving Cubic and Quintic Hermite Interpolation
cites work: Best interpolation in a strip
cites work: On the characterization of non-negative volume-matching surface splines
cites work: A local algorithm for constructing non-negative cubic splines
cites work: A Smoothest Curve Approximation
cites work: Positive Hermite Interpolation by Quadratic Splines
cites work: Q3487672
cites work: Interpolation and approximation from convex sets. II: Infinite-dimensional interpolation
cites work: Q4325824
cites work: Nonnegativity of bivariate quadratic functions on a triangle
cites work: The derivation of cubic splines with obstacles by methods of optimization and optimal control
cites work: Piecewise Quadratic Approximations on Triangles
cites work: Positive interpolation with rational splines
cites work: \(C^ 1\) splines for interpolation of rapidly varying data
cites work: Dual algorithms for solving convex partially separable optimization problems
cites work: Positive, monotone, and \(S\)-convex \(C^ 1\)-interpolation on rectangular grids
cites work: Positive interpolation with rational quadratic splines
cites work: Positivity of cubic polynomials on intervals and positive spline interpolation
cites work: Spline Interpolation under Two-Sided Restrictions on the Derivatives
cites work: On Shape Preserving Quadratic Spline Interpolation
cites work: Q3726690
cites work: Q3772657
cites work: Non-negative exponential splines
cites work: Q4325813
full work available at URL: https://doi.org/10.1007/bf02252986
OpenAlex ID: W89737844
Recommended article: A \(C^{1}\)-rational spline in range restricted interpolation of scattered data
Recommended article: Scattered data interpolation subject to piecewise quadratic range restrictions
Recommended article: Convexity preserving scattered data interpolation using Powell-Sabin elements
Recommended article: Lobachevsky spline functions and interpolation to scattered data
Recommended article: Scattered data interpolation using minimum energy Powell-Sabin elements and data dependent triangulations
Recommended article: Interpolation and scattered data fitting on manifolds using projected Powell-Sabin splines
Recommended article: Q3974340
Recommended article: Range restricted interpolation using Gregory's rational cubic splines
Recommended article: Q3346737
Recommended article: Scattered data interpolation and approximation with error bounds
description: scientific article; zbMATH DE number 683321