From Wikipedia, the free encyclopedia Delaunay tessellation field estimator
Delaunay tessellation field estimator
The reconstruction of a density field from a discrete set of
points sampling this field.
The Delaunay tessellation field estimator (DTFE) is
a mathematical tool for reconstructing a volume-cover-
ing and continuous density or intensity field from a dis-
crete point set. The DTFE has various astrophysical ap- Overview of the DTFE procedure.
plications, such as the analysis of numerical simulations
of cosmic structure formation, the mapping of the large
scale structure in the universe and improving computer
Step 2
simulation programs of cosmic structure formation. It The Delaunay tessellation forms the heart of the DTFE.
has been developed by Willem Schaap and Rien van de In the figure it is clearly visible that the tessellation au-
Weijgaert. The main advantage of the DTFE is that it au- tomatically adapts to both the local density and geome-
tomatically adapts to (strong) variations in density and try of the point distribution: where the density is high,
geometry. It is therefore very well suited for studies of the triangles are small and vice versa. The size of the tri-
the large scale galaxy distribution. angles is therefore a measure of the local density of the
point distribution. This property of the Delaunay tessel-
lation is exploited in step 2 of the DTFE, in which the lo-
Method cal density is estimated at the locations of the sampling
The DTFE consists of three main steps: points. For this purpose the density is defined at the lo-
cation of each sampling point as the inverse of the area
Step 1 of its surrounding Delaunay triangles (times a normaliza-
tion constant, see figure, lower right-hand frame).
The starting point is a given discrete point distribution.
In the upper left-hand frame of the figure a point distri-
bution is plotted in which at the center of the frame an
Step 3
object is located whose density diminishes radially out- In step 3 these density estimates are interpolated to any
wards. In the first step of the DTFE the Delaunay tessel- other point, by assuming that inside each Delaunay trian-
lation of the point distribution is constructed. This is a gle the density field varies linearly (see figure, lower left-
volume-covering division of space into triangles (tetra- hand frame).
hedra in three dimensions), whose vertices are formed
by the point distribution (see figure, upper right-hand
frame). The Delaunay tessellation is defined such that in-
Applications
side the interior of the circumcircle of each Delaunay tri-
angle no other points from the defining point distribu-
An atlas of the nearby universe
tion are present. One of the main applications of the DTFE is the rendering
of our cosmic neighborhood. Below the DTFE reconstruc-
tion of the 2dF Galaxy Redshift Survey is shown, reveal-
ing an impressive view on the cosmic structures in the
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From Wikipedia, the free encyclopedia Delaunay tessellation field estimator
nearby universe. Several superclusters stand out, such as
the Sloan Great Wall, the largest structure in the universe
known to date.
• The 2dF Galaxy Redshift Survey
• DTFE reconstruction of the inner parts of the 2dF DTFE velocity field reconstructions of superclusters and voids
Galaxy Redshift Survey in the large scale galaxy distribution.
Numerical simulations of structure for- Evolution and dynamics of the cosmic
mation web
Most algorithms for simulating cosmic structure forma- The DTFE has been specifically designed for describing
tion are particle hydrodynamics codes. At the core of the complex properties of the cosmic web. It can there-
these codes is the smoothed particle hydrodynamics fore be used to study the evolution of voids and super-
(SPH) density estimation procedure. Replacing it by the clusters in the large scale matter galaxy distribution.
DTFE density estimate will yield a major improvement • Evolution of a void
for simulations incorporating feedback processes, which • Evolution of a supercluster
play a major role in galaxy and star formation.
Cosmic velocity field External links
The DTFE has been designed for reconstructing density • DTFE: the Delaunay Tessellation Field Estimator,
or intensity fields from a discrete set of irregularly dis- Willem Schaap, 2007, PhD Thesis, Rijksuniversiteit
tributed points sampling this field. However, it can also Groningen, The Netherlands
be used to reconstruct other continuous fields which • The Sloan Great Wall: Largest Known Structure? on
have been sampled at the locations of these points, for APOD
example the cosmic velocity field. The use of the DTFE • Probing cosmic velocity flows in the local universe,
for this purpose has the same advantages as it has for Emilio Romano-Diaz, 2004, PhD Thesis,
reconstructing density fields. The fields are reconstruct- Rijksuniversiteit Groningen, The Netherlands
ed locally without the application of an artificial or user- • The cosmic web: geometric analysis, Rien van de
dependent smoothing procedure, resulting in an optimal Weygaert and Willem Schaap, 2004
resolution and the suppression of shot noise effects. The
estimated quantities are volume-covering and allow for a
direct comparison with theoretical predictions.
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From Wikipedia, the free encyclopedia Delaunay tessellation field estimator
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Categories:
• Large-scale structure of the cosmos
• Computational science
• Astronomical surveys
• Geometric algorithms
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