The Delaunay tessellation field estimator (DTFE), (or Delone tessellation field estimator (DTFE)) is a mathematical tool for reconstructing a volume-covering and continuous density or intensity field from a discrete point set. The DTFE has various astrophysical applications, such as the analysis of numerical simulations of cosmic structure formation, the mapping of the large-scale structure of the universe and improving computer simulation programs of cosmic structure formation. It has been developed by Willem Schaap and Rien van de Weijgaert. The main advantage of the DTFE is that it automatically adapts to (strong) variations in density and geometry. It is therefore very well suited for studies of the large scale galaxy distribution.
The DTFE consists of three main steps:
The starting point is a given discrete pointdistribution. In the upper left-hand frame of the figure, a point distribution is plotted in which at the center of the frame an object is located whose density diminishes radially outwards. In thefirst step of the DTFE, the Delaunay tessellation of the pointdistribution is constructed. This is a volume-covering divisionof space into triangles (tetrahedra in three dimensions), whosevertices are formed by the point distribution (see figure, upper right-hand frame). The Delaunay tessellation is defined suchthat inside the interior of the circumcircle of each Delaunay triangle no other points from the defining point distribution are present.
The Delaunay tessellation forms the heart ofthe DTFE. In the figure it is clearly visible that the tessellationautomatically adapts to both the local density and geometry of thepoint distribution: where the density is high, the triangles are smalland vice versa. The size of the triangles is therefore a measure ofthe local density of the point distribution.This property of the Delaunay tessellation is exploited in step 2 ofthe DTFE, in which the local density is estimated at the locations ofthe sampling points. For this purpose the density is defined at thelocation of each sampling point as the inverse of the area of itssurrounding Delaunay triangles (times a normalization constant, seefigure, lower right-hand frame).
In step 3 these density estimates are interpolated to any other point,by assuming that inside each Delaunay triangle the density field varieslinearly (see figure, lower left-hand frame).
One of the main applications of the DTFE is the rendering of our cosmic neighborhood. Below the DTFE reconstruction of the 2dF Galaxy Redshift Survey is shown, revealingan impressive view on the cosmic structures in the nearby universe. Several superclusters stand out, such as the Sloan Great Wall, one of the largest structures in the universe.
Most algorithms for simulating cosmic structure formation are particle hydrodynamics codes. At the core of these codes is thesmoothed particle hydrodynamics (SPH) density estimation procedure.Replacing it by the DTFE density estimate will yield a major improvement for simulations incorporating feedback processes, which play a major role in galaxy and star formation.
The DTFE has been designed for reconstructing density or intensityfields from a discrete set of irregularly distributed points samplingthis field. However, it can also be used toreconstruct other continuous fields which have been sampled at thelocations of these points, for example the cosmic velocity field. The use of the DTFE for this purpose hasthe same advantages as it has for reconstructing density fields. Thefields are reconstructed locally without the application of an artificial or user-dependent smoothing procedure, resulting in an optimal resolution and the suppression of shot noise effects. Theestimated quantities are volume-covering and allow for a directcomparison with theoretical predictions.
The DTFE has been specifically designed for describing thecomplex properties of the cosmic web. It can therefore be used to study the evolution of voids and superclusters in the large scale matter galaxy distribution.