In this paper, we present an algorithm for multivariate interpolation of scattered data sets lying in convex domains [Formula: see text], for any [Formula: see text]. To organize the points in a multidimensional space, we build a [Formula: see text]-tree space-partitioning data structure, which is used to efficiently apply a partition of unity interpolant. This global scheme is combined with local radial basis function (RBF) approximants and compactly supported weight functions. A detailed description of the algorithm for convex domains and a complexity analysis of the computational procedures are also considered. Several numerical experiments show the performances of the interpolation algorithm on various sets of Halton data points contained in [Formula: see text], where [Formula: see text] can be any convex domain, like a 2D polygon or a 3D polyhedron. Finally, an application to topographical data contained in a pentagonal domain is presented.
Reference EPFL-ARTICLE-211511 URL: http://bigwww.epfl.ch/publications/unser1403.html Record created on 2015-09-18, modified on 2017-05-10
Cardinal Hermite exponential spline functions are a generalization of the classical cardinal Hermite polynomial splines. In this work we consider the 4-dimensional space $ e _{ 4 } $ = {1, x, $ e ^{ \alpha x } $ , $ e ^{ -\alpha x } $ } with α ∈ $ ℝ ^{ + } $ ∪ i $ ℝ ^{ + } $ , and therefore a generalization of the well-known cubic cardinal Hermite polynomial splines. For this class of Hermite spline functions, here denoted by $ e _{ 4 } $ -Hermite splines, we establish the connection to standard exponential splines, we show stability and approximation power, and we emphasize their capability of reproducing elliptical and circular shapes. Finally, we investigate their multiresolution properties and we propose a non-stationary Hermite interpolatory subdivision scheme for refinement of vector sequences via the repeated application of level-dependent matrix subdivision operators.
