Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

Abstract

Abstract. In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

Optimal higher order Delaunay triangulations of polygons

“…As the parameter k is increased, more points inside the circumcircles imply a reduction in the shape quality of the triangles, but also an increase in the number of triangulations that are considered, and hence greater flexibility to optimize some other criterion, while limiting the badness of the shape of the triangles. The concept of higher order Delaunay triangulation has been successfully applied to several areas, including terrain modeling [7], minimum interference networks [1] and multivariate splines [20].…”

“…Note that in this paper we use the standard definition of order of a triangle, as in [11,20], which does not take the boundary edges of the polygon into account. This implies that for a given polygon and value k, our algorithm may find that no order-k triangulation of that polygon exists.…”

Adaptive parametrization of multivariate B-splines for image registration

“…Recently, Wang et al [15] introduced DMS-splines in computer vision for nonrigid registration with rigid parts that defined by manual landmarks. The most recent multivariate B-splines were introduced by Neamtu [7], and they rely heavily on the new concept of Delaunay configurations [8].…”

Optimal Higher Order Delaunay Triangulations of Polygons

“…As the parameter k is increased, more points inside the circumcircles imply a reduction in the shape quality of the triangles, but also an increase in the number of triangulations that are considered, and hence greater flexibility to optimize some other criterion, while limiting the badness of the shape of the triangles. The concept of higher order Delaunay triangulation has been successfully applied to several areas, including terrain modeling [7], minimum interference networks [1] and multivariate splines [20].…”

“…Note that in this paper we use the standard definition of order of a triangle, as in [11,20], which does not take the boundary edges of the polygon into account. This implies that for a given polygon and value k, our algorithm may find that no order-k triangulation of that polygon exists.…”