## 2. The Delaunay tessellation field estimatorGiven a set of field values sampled at a discrete number of locations along one dimension we are familiar with various prescriptions for reconstructing the field over the full spatial domain. The most straightforward way involves the partition of space into bins centered on the sampling points. The field is then assumed to have the - constant - value equal to the one at the sampling point. Evidently, this yields a field with unphysical discontinuities at the boundaries of the bins. A first-order improvement concerns the linear interpolation between the sampling points, leading to a fully continuous field. In more than one dimension, the equivalent spatial intervals of the 1-D bins are well-known in stochastic geometry. A point process defines a Voronoi tessellation by dividing space into a unique and volume-covering network of mutually disjunct convex polyhedral cells, each of which comprises that part of multidimensional space closer to the defining point than to any of the other (see van de Weygaert 1991and references therein). These Voronoi cells (see Fig. 1) are the multidimensional generalization of the 1-D bins in which the zeroth-order method approximates the field value to be constant. The natural extension to a multidimensional linear interpolation interval then immediately implies the corresponding Delaunay tessellation (Delone 1934). This tessellation (Fig. 1) consists of a volume-covering tiling of space into tetrahedra (in 3-D, triangles in 2-D, etc.) whose vertices are formed by four specific points in the dataset. The four points are uniquely selected such that their circumscribing sphere does not contain any of the other datapoints. The Voronoi and Delaunay tessellation are intimately related, being each others dual in that the centre of each Delaunay tetrahedron's circumsphere is a vertex of the Voronoi cells of each of the four defining points, and conversely each Voronoi cell nucleus a Delaunay vertex (see Fig. 1). The "minimum triangulation" property of the Delaunay tessellation has in fact been well-known and abundantly applied in, amongst others, surface rendering applications such as geographical mapping and various computer imaging algorithms.
Consider a set of in which is the estimated constant
field gradient within the tetrahedron. Given the
field values
, the value of the The one factor complicating a trivial and direct implementation of
above procedure in the case of density (intensity) field estimates is
the fact that the number density of data points itself is the measure
of the underlying density field value. Unlike the case of velocity
fields, we therefore cannot start with directly available field
estimates at each datapoint. Instead, we need to define appropriate
estimates from the point set itself. Most suggestive would be to base
the estimate of the density field at the location
of each point on the inverse of the
volume of its Voronoi cell,
. Note that in this we take every
datapoint to represent an equal amount of mass Having computed these density estimates, we subsequently proceed to determine the complete volume-covering density field reconstruction through the linear interpolation procedure outlined in Eq. (1). © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |