Delaunay tessellation field estimator  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 point distribution. 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 the first step of the DTFE, the Delaunay tessellation of the point distribution is constructed. This is a volume-covering division of space into triangles (tetrahedra in three dimensions), whose vertices are formed by the point distribution (see figure, upper right-hand frame). The Delaunay tessellation is defined such that 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 of the DTFE. In the figure it is clearly visible that the tessellation automatically adapts to both the local density and geometry of the point distribution: where the density is high, the triangles are small and vice versa. The size of the triangles is therefore a measure of the local density of the point distribution. This property of the Delaunay tessellation is exploited in step 2 of the DTFE, in which the local density is estimated at the locations of the sampling points. For this purpose the density is defined at the location of each sampling point as the inverse of the area of its surrounding Delaunay triangles (times a normalization constant, see figure, 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 varies linearly (see figure, lower left-hand frame).