Powell-Sabin splines in range restricted interpolation of scattered data
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 \(\Delta\).
scientific article; zbMATH DE number 683321
scientific article