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.
In this talk we show a construction for characterising developable surfaces in the form of Bézier triangular patches. It is shown that constructions used for rectangular patches are not useful, since they provide degenerate triangular patches. Explicit constructions of non-degenerate developable triangular patches are provided.
One of the main reasons why polynomial splines play an important role in computer--aided design as well as in diverse areas of approximation theory and numerical analysis is the fact that they can be represented as linear combination of B-splines. There are nice and stable algorithms for evaluation of such splines and their derivatives and integrals. The well known tools of knot insertion and degree raising can be enhanced by introducing still more additional parameters, and relaxing the continuity conditions at the knots by prescribing jumps in their derivatives. The purpose of this paper is to derive recurrence formulae for some related B-splines, and to exploit the underlying connection with the theory of Chebyshev splines. The cubic version of the jump spline is then recognized as Foley"s $ u-$spline, often used in minimizing functionals like $V(f):,=sum_{i=1}^n (w_i int_{t_i}^{t_{i+1}}[D^2f(t)]^2dt+ u_iint_{t_i}^{t_{i+1}}[Df(t)]^2 dt)$, $ u_i geq 0$, $w_i > 0$. The parametric version is often used as a polynomial alternative to the exponential spline in tension in computer--aided geometric design. It is shown how the associated B-splines can be calculated by a knot--insertion algorithm, and this in turn motivates a definition of certain generalized discrete splines.
CAD systems are usually based on a tensor product representation of free form surfaces. Trimmed patches provide a reasonable solution for the representation of general topologies, provided that the gap between equivalent trimming curves in the euclidean space is small enough. Several commercial CAD systems, however, represent certain non-rectangular surface regions through degenerate rectangular patches. Degenerate patches produce rendering artifacts and can lead to malfunctions in the subsequent geometric operations. In this paper, two algorithms for converting degenerate tensor-product patches into triangular and trimmed rectangular patches are presented. The algorithms are based on specific degree reduction algorithms for Bezier curves. In both algorithms, the final surface approximates the initial one in a quadratic sense while inheriting its boundary curves. In the second one, e-G^1 continuity is achieved. Approximation errors are analyzed and some examples are presented and discussed. Approximation errors can be arbitrarily decreased through the degree elevation of the degenerate patches.
