Geometry and Morphology of the Cosmic Web: Analyzing Spatial Patterns in the Universe

Abstract

We review the analysis of the Cosmic Web by means of an extensive toolset based on the use of Delaunay and Voronoi tessellations. The Cosmic Web is the salient and pervasive foamlike pattern in which matter has organized itself on scales of a few up to more than a hundred Megaparsec. First, we describe the Delaunay Tessellation Field Estimator (DTFE). The DTFE formalism is shown to recover the hierarchical nature and the anisotropic morphology of the cosmic matter distribution. The Multiscale Morphology Filter (MMF) uses the DTFE density field to extract the diverse morphological elements - filaments, sheets and clusters - on the basis of a ScaleSpace analysis which searches for these morphologies over a range of scales. Subsequently, we discuss the Watershed Voidfinder (WVF), which invokes the discrete watershed transform to identify voids in the cosmic matter distribution. The WVF is able to determine the location, size and shape of the voids. The watershed transform is also a key element in the SpineWeb analysis of the cosmic matter distribution. It allows the determination of the filamentary spine and connected walls in the cosmic matter density field through the identification of the singularities and corresponding separatrices. Finally, we describe the concept of Alphashapes for assessing the topology of the cosmic matter distribution. Comment: 28 pages, 16 figures, invited review ISVD09 (International Symposium on Voronoi Diagrams and Engineering), Copenhagen, Denmark. IEEE CPS, E3781, ed. F. Anton. For high-res version see http://www.astro.rug.nl/~weygaert/isvd09wey.ieee.pdf

Full text

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arXiv:0912.3448v1 [astro-ph.IM] 17 Dec 2009
Geometry and Morphology of the Cosmic Web:
Analyzing Spatial Patterns in the Universe
(Invited Paper)
Rien van de Weygaert, Bernard J.T. Jones, Erwin Platen
Kapteyn Astronomical Institute
University of Groningen
P.O. Box 800, 9700 AV Groningen, the Netherlands
Email: weygaert@astro.rug.nl
Miguel A. Arag´ on-Calvo
Dept. Physics & Astronomy
The Johns Hopkins University
3701 San Martin Drive, Baltimore, MD 21218, USA
Email: miguel@pha.jhu.edu
Abstract—We review the analysis of the Cosmic Web by
means of an extensive toolset based on the use of Delaunay
and Voronoi tessellations. The Cosmic Web is the salient and
pervasive foamlike pattern in which matter has organized itself
on scales of a few up to more than a hundred Megaparsec.
The weblike spatial arrangement of galaxies and mass into
elongated filaments, sheetlike walls and dense compact clus-
ters, the existence of large near-empty void regions and the
hierarchical nature of this mass distribution are three major
characteristics of the comsic matter distribution.
First, we describe the Delaunay Tessellation Field Esti-
mator.Using the unique adaptive qualities of Voronoi and
Delaunay tessellations, DTFE infers the density field from
the (contiguous) Voronoi tessellation of a sampled galaxy
or simulation particle distribution and uses the Delaunay
tessellation as adaptive grid for defining continuous volume-
filling fields of density and other measured quantities through
linear interpolation. The resulting DTFE formalism is shown to
recover the hierarchical nature and the anisotropic morphology
of the cosmic matter distribution. The Multiscale Morphology
Filter (MMF) uses the DTFE density field to extract the diverse
morphological elements - filaments, sheets and clusters - on
the basis of a ScaleSpace analysis which searches for these
morphologies over a range of scales. Subsequently, we discuss
the Watershed Voidfinder (WVF), which invokes the discrete
watershed transform to identify voids in the cosmic matter
distribution. The WVF is able to determine the location, size
and shape of the voids. The watershed transform is also a
key element in the SpineWeb analysis of the cosmic matter
distribution. Finding its mathematical foundation in Morse
theory, it allows the determination of the filamentary spine
and connected walls in the cosmic matter density field through
the identification of the singularities and corresponding sep-
aratrices. The first results of a direct implementation on the
Delaunay tessellation itself are presented. Finally, we describe
the concept of Alphashapes for assessing the topology of the
cosmic matter distribution.
Keywords-Cosmology: theory - large-scale structure of Uni-
verse - Methods: numerical - Surveys
I. INTRODUCTION: THE COSMIC WEB
The large scale distribution of matter revealed by galaxy
surveys features a complex network of interconnected fil-
amentary galaxy associations. This network, which has
become known as the Cosmic Web [1], contains structures
from a few megaparsecs1up to tens and even hundreds of
Megaparsecs of size. Galaxies and mass exist in a wispy
weblike spatial arrangement consisting of dense compact
clusters, elongated filaments, and sheetlike walls, amidst
large near-empty void regions, with similar patterns existing
at earlier epochs, albeit over smaller scales. The hierarchical
nature of this mass distribution, marked by substructure
over a wide range of scales and densities, has been clearly
demonstrated. Its appearance has been most dramatically
illustrated by the recently produced maps of the nearby
cosmos, the 2dFGRS, the SDSS and the 2MASS redshift
surveys [2], [3], [4]2.
The vast Megaparsec cosmic web is one of the most strik-
ing examples of complex geometric patterns found in nature,
and certainly the largest in terms of sheer size. Computer
simulations suggest that the observed cellular patterns are a
prominent and natural aspect of cosmic structure formation
through gravitational instability [5], the standard paradigm
for the emergence of structure in our Universe.
A. Dynamical Evolution of the Cosmic Web
At least three “universal” characteristics of the resulting
nonlinear cosmic matter distribution can be recognized in
large N-body simulations of cosmic structure formation such
as the Millennium simulation [6]. One prominent aspect
is the hierarchical clustering. The spatial cosmic matter
1The main measure of length in astronomy is the parsec. Technically
a parsec is the distance at which we would see the distance Earth-Sun at
an angle of 1 arcsec. It is equal to 3.262 lightyears = 3.086 × 1013km.
Cosmological distances are substantially larger, so that a Megaparsec (=
106pc) is the regular unit of distance. Usually this goes along with h, the
cosmic expansion rate (Hubble parameter) H in units of 100 km/s/Mpc
(h ≈ 0.71).
2Because of the expansion of the Universe, any observed cosmic object
will have its light shifted redward: its redshift z. According to Hubble’s
law, the redshift z is directly proportional to the distance r of the object, for
z ≪ 1: cz = Hr (with c the velocity of light, and H ≈ 71km/s/Mpc the
Hubble constant). Because it is extremely cumbersome to measure distances
r directly, cosmologists resort to the expansion of the Universe and use z
as a distance measure. Because of the vast distances in the Universe, and
the finite velocity of light, the redshift z of an object may also be seen as
a measure of the time at which it emitted the observed radiation.

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Figure 1.
a ΛCDM scenario. It concerns an N-body simulation in a box of 200h−1Mpc size. The three boxes indicate examples of the main structure components
of the Cosmic Web. Amongst others, the image clarifies the mutual spatial relationship between these elements. Low-density and low contrast walls are
less prominent than the outstanding filamentary channels which define the texture of the Cosmic Web. Near the intersection points of filaments and sheets
we find high-density cluster nodes. The figures demonstrates the significance of the concept “Cosmic Web”. From Arag´ on-Calvo 2007.
The Cosmic Web. The image shows the weblike patterns traced by the Dark Matter distribution, at the present epoch, in a Universe based on
distribution is marked by a large range of scales and den-
sities. It is the product of an evolution in which the first
objects to condense are small, with larger structures forming
through the gradual merging of smaller structures. Another
prominent feature is its anisotropic and weblike spatial
geometry, the consequence of the gravitational tendency
of overdensities to collapse in an anisotropic fashion. It
finds its origin in the intrinsic flattening of the overden-
sities in the primordial density field [7], augmented by the
anisotropy of the gravitational force field induced by the
external matter distribution (i.e. by tidal forces). A third
important characteristic is the dominant presence of Voids,
large roundish underdense, often near-empty, regions. They
form in and around density troughs in the primordial density
field. They have an essential role in the organization of
the cosmic matter distribution. Recently, their emergence
and evolution has been explained within the context of
hierarchical gravitational scenarios [8].
The Cosmic Web theory of Bond et al. [1] succeeded
in synthesizing all relevant aspects into a coherent dy-
namical and evolutionary framework. Instrumental is the
realization that the outline of the cosmic web may already
be recognized in the primordial density field. The statis-
tics of the primordial tidal field explains why the large
scale universe looks predominantly filamentary and why in
overdense regions sheetlike membranes are only marginal
features. Of key importance is the realization that the rare
high peaks which will eventually emerge as clusters are the
dominant agents for generating the large scale tidal force
field: it is the clusters which weave the cosmic tapestry of
filaments [1], [9], [10]. They cement the structural relations
between the components of the Cosmic Web and themselves
form the junctions at which filaments tie up. This relates
the strength and prominence of the filamentary bridges to
the proximity, mass, shape and mutual orientation of the
generating cluster peaks: the strongest bridges are those
between the richest clusters that stand closely together and
point into each other’s direction. Through the direct relation
between the Cosmic Web and the primordial tidal field
one may understand why the large scale universe looks
predominantly filamentary and why in overdense regions
sheetlike membranes are only marginal features [11].
The emerging picture is one of a primordially and hi-
erarchically defined network whose weblike topology is
imprinted over a wide spectrum of scales. Weblike patterns
on ever larger scales get to dominate the density field as
cosmic evolution proceeds, and as small scale structures
merge into larger ones. Within the gradually emptying void
regions, however, the topological outline of the early weblike
patterns remains largely visible.
B. Tessellations and Web Analysis
Despite a large variety of attempts, as yet no generally
accepted descriptive framework has emerged for the objec-
tive and quantitative analysis of the Cosmic Web. Despite
the multitude of elaborate qualitative descriptions it has
remained a major challenge to characterize its structure,
geometry and topology. Many attempts to describe, let alone
identify, the features and components of the Cosmic Web
have been of a rather heuristic nature. The overwhelming
complexity of both the individual structures as well as their
connectivity, the lack of structural symmetries, its intrinsic
multiscale nature and the wide range of densities that one
finds in the cosmic matter distribution has prevented the use
of simple and straightforward toolboxes.
In the observational reality galaxies are the main tracers
of the cosmic web and it is mainly through the measurement
of the redshift distribution of galaxies that we have been able
to map its structure. Likewise, simulations of the evolving
cosmic matter distribution are almost exclusively based upon
N-body particle computer calculations, involving a discrete
representation of the features we seek to study. Both the
galaxy distribution as well as the particles in an N-body
simulation are examples of spatial point processes in that
they are discretely sampled and have an irregular spatial
distribution.
For furthering our understanding of the Cosmic Web,
and to investigate its structure and dynamics, it is of prime
importance to have access to a set of proper and objective
analysis tools. In this contribution we follow the finding that
spatial tessellations – in particular Voronoi and Delaunay
tessellations – generated by the discretely sampled cosmic
density and velocity fields form an ideal basis for an elab-
orate toolset that succesfully deals with several challenging
aspects of the analysis of the Cosmic Web:
• Tessellations may be used to interpolate a sample of
discrete and irregularlydistributed values into a volume-
weighted and volume-covering continuous field.
• In case the point sample is a representative reflection of
an underlying smooth and continuous density/intensity
field, Voronoi tessellations can be used to estimate and
reconstruct the density field.
• Tessellations are highly sensitive to the morphology of
the spatial point distribution. As a result they sensitively
probe the key aspects - hierarchical matter distribution,
anisotropic patterns, and voids – of the Cosmic Web

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Figure 2.
zoom-ins. The frames form a testimony of the strong adaptivity of the Delaunay tessellations to local density and geometry of the spatial point distribution.
From Schaap 2007.
The Delaunay tessellation of a point distribution in and around a filamentary feature. The generated tessellations is shown at three successive
without resorting to user-defined parameters or func-
tions, and without affecting any of the other essential
characteristics.
These issues are all addressed by the Delaunay Tessellation
Field Estimator (DTFE). The DTFE technique, developed
by Schaap & van de Weygaert [12], forms an elaboration of
the velocity interpolation scheme introduced by Bernardeau
& van de Weygaert [13] towards a multidimensional and
fully adaptive scheme for the estimation and interpolation
of density, velocity and other non-uniformly and discretely
sampled quantities to yield a correspondingvolume-covering
continuous field.
The application of DTFE to the observed spatial distri-
bution of galaxies, or that of particles in a computer N-
body simulation, produces a continuous density and velocity
field which retains the crucial aspects of its hierarchical
nature and anisotropic morphology. This may be directly
appreciated from the three consecutive zoom-ins of the
DTFE rendered cosmic web density field in fig. 6, and
its comparison to the more conventional gridbased TSC
interpolation scheme. Because of these characteristics, the
DTFE density field forms the basis for a diverse set of

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tools for the analysis of various aspects of the cosmic
matter distribution. In this contribution we will report on
our work on the Multiscale Morphology Filter for detecting
filaments and sheets, the Watershed Void Finder for detecting
voids, the Cosmic Spine formalism for the analysis of the
filamentary network of the Cosmic Web and, finally, the
related Alphashape analysis of the topological structure of
the cosmic matter distribution.
II. DTFE: FUNDAMENTALS
DTFE obtains optimal local estimates of the spatial density
(see [14], sect. 8.5), while the tetrahedra of its dual Delaunay
tessellation are used as multidimensional intervals for linear
interpolation of the field values sampled or estimated at
the location of the sample points (see [14], ch. 6). With
respect to the aspect of interpolation, the Delaunay Tessel-
lation Field Estimator is the linear version of nn-neighbour
interpolation techniques (see sect. II-B). Its core, however, is
the additional aspect of being able to translate the generating
spatial point distribution into a continuous density field.
The DTFE method has been first defined in the context
of a description and analysis of cosmic flow fields which
are sampled by a set of discretely and sparsely sampled
galaxy peculiar velocities. Bernardeau & van de Weygaert
[13] demonstrated the method’s superior performance with
respect to conventional interpolation procedures. They also
proved that the obtained field estimates involve those of
the proper volume-weighted quantities, instead of the usu-
ally implicit mass-weighted quantities (see sect. II-E). This
corrected a few fundamental biases in estimates of higher
order velocity field moments.
The DTFE technique allows us to follow the same ge-
ometrical and structural adaptive properties of the higher
order nn-neighbour methods while allowing the analysis
of truely large data sets and include the reconstruction of
the most important aspect, that of the cosmological density
field. For three-dimensional samples with large number of
points, akin to those found in large cosmological computer
simulations, the more complex geometric operations in-
volved in the pure nn-neighbour interpolation still represent
a computationally challenging task (see sect. II-B). To deal
with the large point samples consisting of hundreds of
thousands to several millions of points we chose to follow a
related nn-neigbhour based technique that restricts itself to
pure linear interpolation.
A. Voronoi and Delaunay Adaptivity
The primary ingredient of the DTFE procedure, and the
related Natural Neighbour Interpolation methods, is the
Delaunay tessellation of the particle distribution P. The
Delaunay and Voronoi cells adjust themselves to the spatial
characteristics of the point distribution. This concerns its
spatial resolution and its geometry.
DTFE, and its higher-order equivalent Natural Neigbhour
Interpolation [15], [16], [17], [18], [19], exploit three prop-
erties of Voronoi and Delaunay tessellations (see eg. [20],
[21]):
• The tessellations are very sensitive to the local point
density. DTFE uses this to define a local estimate of
the density on the basis of the inverse of the volume of
the tessellation cells.
• The sensitivity of Voronoi and Delaunay tessellations to
the local geometry of the point distribution. This allows
DTFE to accurately trace anisotropic features, such as
seen in the Cosmic Web.
• The adaptive and minimum triangulation properties of
Delaunay tessellations allow them to be used as adap-
tive spatial interpolation intervals for irregular point
distributions. This has also been recognized in a
In sparsely sampled regions the Voronoi and Delaunay
cells are large and the distance between natural neighbours,
points that share a Voronoi simplex, is large. Not only the
size, but also the shape of the Delaunay tetrahedra is fully
determined by the spatial point distribution: if is anisotropic
this will be reflected in the distribution. In fig. 2 we see that
the Delaunay tessellation traces the density and geometry
of the local point distribution at three consecutive spatial
scales to a remarkable degree. The density and geometry
of the local point distribution will therefore determine the
resolution of the spatial interpolation and reconstruction
procedures based on the Voronoi and Delaunay tessellation.
B. Natural Neighbour Interpolation
Natural Neighbour Interpolation formalism is a generic
higher-order multidimensional interpolation, smoothing and
modelling procedure utilizing the concept of natural neigh-
bours to obtain locally optimized measures of system charac-
teristics. Its theoretical basis was developed and introduced
by Sibson [16], while extensive treatments and elaborations
of nn-interpolation may be found in [17], [19]. Natural
neighbour interpolation produces a conservative, artifice-
free, result by finding area-weighted weighted averages, at
each interpolation point, of the functional values associated
with that subset of data which are natural neighbors of
each interpolation point. According to the nn-interpolation
scheme the interpolated value?f(x) at a position x is given
?f(x) =
in which the summation is over the natural neighbours of
the point x, i.e. the sample points j with whom the order-2
Voronoi cells V2(x,xj) are not empty. Sibson interpolation
is based upon the interpolation kernel φ(x,xj) to be equal
to the normalized order-2 Voronoi cell,
by
?
i
φnn,i(x)fi,
(1)
φnn,i(x) =A2(x,xi)
A(x)
,
(2)

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in which A(x) =?
sample P and the volume A2(x,xi) concerns the order-2
Voronoi cell V2(x,xi), the region of space for which the
points x and xi are the closest points. The interpolation
kernels φ are always positive and sum to one. The resulting
function is continuous everywhere within the convex hull of
the data, and has a continuous slope everywhere except at
the data themselves.
jA(x,xj) is the volume of the potential
Voronoi cell of point x if it had been added to the point
C. DTFE Interpolation
The linear interpolation scheme of DTFE exploits the same
spatially adaptive characteristics of the Delaunay tessellation
generated by the point sample P as that of regular natu-
ral neighbour schemes. For DTFE the interpolation kernel
φdt,i(x) is that of regular linear interpolation within the
Delaunay tetrahedron in which x is located (see eq. 5),
?fdt(x) =
?
i
φdt,i(x)fi,
(3)
in which the sum is over the four sample points defining the
Delaunay tetrahedron. Note that for both the nn-interpolation
as well as for the linear DTFE interpolation,the interpolation
kernels φi are unity at sample point location xi and equal
to zero at the location of the other sample points j,
φi(xj) =
?
1
0
if i = j ,
if i ?= j ,
(4)
where xj is the location of sample point j.
In practice, it is convenient to replace eqn. 3 with its
equivalent expression in terms of the (linear) gradient?
?f(x) =?f(xi) +?
The value of?
constituting the vertices of a Delaunay simplex. Given
the location r = (x,y,z) of the four points forming the
Delaunay tetrahedra’s vertices, r0, r1, r2 and r3, and the
value of the sampled field at each of these locations, f0, f1,
f2and f3and defining the quantities
∇f??
m
inside the Delaunay simplex m,
∇f??
m·(x − xi).
(5)
∇f??
mcan be easily and uniquely determined
from the (1 + D) field values fj at the sample points
∆xn
∆yn
∆zn
=
xn− x0;
yn− y0;
zn− z0,
=
=
for n = 1,2,3
(6)
as well as ∆fn ≡ fn− f0(n = 1,2,3) the gradient ∇f
follows from the inversion
∇f
=











∂f
∂x
∂f
∂y
∂f
∂z









= A−1






∆f1
∆f2
∆f3






;
(7)
A
=




∆x1
∆y1
∆z1
∆x2
∆y2
∆z2
∆x3
∆y3
∆z3






Once the value of ∇f has been determined for each De-
launay tetrahedron in the tessellation, it is straightforward
to determine the DTFE field value?f(x) for any location x
Delaunay tetrahedron in which x is located (eqn. 5).
The one remaining complication is to locate the Delaunay
tetrahedron Dmin which a particular point x is located. This
is not as trivial as one might naively think. It not necessarily
concerns a tetrahedron of which the nearest nucleus is a
vertex. Fortunately, a very efficient method, the walking
triangle algorithm [22], [23] has been developed. Details
of the method may be found in [24], [20].
by means of straightforward linear interpolation within the
D. DTFE Density Estimates
The DTFE procedure extends the concept of interpolation of
field values sampled at the point sample P to the estimate
of the density ? ρ(x) from the spatial point distribution itself.
point sample forms a fair and unbiased reflection of the
underlying density field.
It is commonly known that an optimal estimate for the
spatial density at the location of a point xi in a discrete
point sample P is given by the inverse of the volume of
the corresponding Voronoi cell (see [14], for references).
Tessellation-based methods for estimating the density have
been introduced by Brown [25] and Ord [26]. In astron-
omy, Ebeling & Wiedenmann [27] were the first to use
tessellation-based density estimators for the specific purpose
of devising source detection algorithms. This work has
recently been applied to cluster detection algorithms by [28],
[29], [30], [31], [32]. Along the same lines, Ascasibar &
Binney [33] suggested that the use of a multidimensional
binary tree might offer a computationally more efficient
alternative. However, these studies have been restricted to
raw estimates of the local sampling density at the position
of the sampling points and have not yet included the more
elaborate interpolation machinery of the DTFE and Natural
Neighbour Interpolation methods.
The density field reconstruction of the DTFE procedure
consists of two steps, the zeroth-order estimate ? ρ0 of the
This is only feasible if the spatial distribution of the discrete

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Figure 3.
particle distribution in and around a filament. The density values at two sample points within the filamentary structure are indicated (bottom panel). From
Schaap 2007.
Relation between density and volume contiguous Voronoi cells. The figure (top row) zooms in on the Delaunay tessellation defined by the
density values at the location of the points in P and the
subsequent linear interpolation of those zeroth-order density
estimates over the corresponding Delaunay grid throughout
the sample volume. This yields the DTFE density field
estimate ? ρ(x).
our interpolation scheme is that the estimated DTFE density
field ? ρ(x) should guarantee mass conservation: the total
the mass represented by the sample points. Indeed, this is
an absolutely crucial condition for many applications of a
physical nature. Since the mass M is given by the integral
of the density field ρ(x) over space, this translates into the
An essential requirement for cosmological purposes of
mass corresponding to the density field should be equal to
integral requirement
?
M
=
?
?
? ρ(x)dx
mi = M = cst.,
=
N
i=1
(8)
with mi= m is the mass per sample point. It is straightfor-
ward to infer that if the zeroth-order estimate of the density
values would be the inverse of the regular Voronoi volume
the condition of mass conservation would not be met.
Instead, the DTFE procedure employs a slightly modified yet
related zeroth-order density estimate, the normalized inverse
of the volume V (Wi) of the contiguous Voronoi cell Wiof

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Figure 4.
structure. The shear flow along the filaments is meticulously resolved.
The density and velocity field of the LCDM GIF N-body simulation, by means of DTFE zooming in on the flow field around a filamentary
each point i. For D-dimensional space this is
? ρ(xi) = (1 + D)
mi
V (Wi).
(9)
The contiguous Voronoi cell of a point i is the union of
all Delaunay tetrahedra of which point i forms one of the
four vertices. It is straightforward to prove that with the
contiguous Voronoi cell density estimator mass conservation
is guaranteed (see [21], sect. 10).
E. Volume-weighted and Mass-weighted fields
For relating measured quantities to theoretical predictions,
one has to take account of the implicit filtering process
involved in translating measurements (including that in com-
puter simulations) to interpretable measures. When dealing
with a probe of an underlying density field ρ(x) - e.g. the
galaxy distribution in a cosmological context - there are two
possibilities. Theoretically the cleanest measure is that of
the volume-weighted quantity fvol,
?fvol(x) ≡
?
dyf(y)W(x − y)
?
dy W(x − y)
.
(10)
In practice, however, most techniques implicitly (and often
unknowingly) yield the mass-weighted field averages.
?
?
where W(x,y) is the adopted filter function defining the
weight of a mass element in a way that is dependent on its
position y with respect to the position x. It turns out that
the use of the Voronoi and Delaunay tessellation guarantees
the volume-weighted nature of the DTFE fields [13].
An additional crucial ingredient of any reconstruction
procedure are its error characteristics. Within the limited
context of the present contributionit is not feasible to include
a detailed discussion of the noise and error characteristics
?fmass(x) ≡
dyf(y)ρ(y)W(x − y)
dyρ(y)W(x − y)
(11)

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of DTFE reconstructed density and velocity fields. For this
we refer the reader elsewhere [20], [21].
III. THE DTFE PROCEDURE
The complete DTFE reconstruction procedure is shown in
the the schematic diagram of fig. 5. It involves the following
steps:
• Point sample
Defining the spatial distribution of the point sample.
• Delaunay Tessellation
Construction of the Delaunay tessellation from the
point sample.
• Field values point sample
Dependent on whether it concerns the densities at the
sample points or a measured field value there are two
options:
+ General (non-density) field:
(Sampled) value of field at sample point.
+ Density field
• Field Gradient
Calculation of the field gradient estimate?
tetrahedron; D = 2: triangle) by solving the set of
linear equations for the field values at the positions of
the (D + 1) tetrahedron vertices,


• Interpolation.
The final basic step of the DTFE procedure is the field
interpolation. The processing and postprocessing steps
involve numerous interpolation calculations, for each
of the involved locations x. Given a location x, the
Delaunay tetrahedron m in which it is embedded is
determined. On the basis of the field gradient?
?f(x) =?f(xi) +?
also possible.
∇f|m in
each D-dimensional Delaunay simplex m (D = 3:
?
∇f|m
⇐=



f0
f1
f2
f3
r0
r1
r2
r3
(12)
∇f|m
the field value is computed by (linear) interpolation,
∇f??
m·(x − xi).
(13)
In principle, higher-order interpolation procedures are
• Processing.
Though basically of the same character for practical
purposes we make a distinction between straightfor-
ward processing steps concerning the production of
images and simple smoothing filtering operations on
the one hand, and more complex postprocessing on
the other hand. The latter are treated in the next item.
Basic to the processing steps is the determination of
field values following the interpolation procedure(s)
outlined above.
Straightforward “first line” field operations are “Im-
age reconstruction” and, subsequently, “Smooth-
ing/Filtering”.
+ Image reconstruction.
For a set of image points, usually grid points,
determine the image value: formally the average
field value within the corresponding gridcell.
+ Smoothing and Filtering:
• Post-processing.
The real potential of DTFE fields may be found in
sophisticated applications, tuned towards uncovering
characteristics of the reconstructed fields. An impor-
tant aspect of this involves the analysis of structures
in the density field. Some notable examples are:
+ Advanced filtering operations. Potentially inter-
esting applications are those based on the use of
wavelets [34].
+ Cluster, Filament and Wall detection by means
of the Multiscale Morphology Filter [35], [36].
+ Void identification on the basis of the cosmic
watershed algorithm [37], [38].
+ Tracing the filamentary spine of the Cosmic Web,
on the basis of a Morse-theory related analysis,
the Cosmic Spine [39].
+ Halo detection in N-body simulations [32].
+ The computation of 2-D surface densities for the
study of gravitational lensing [40].
In addition, DTFE enables the simultaneous and combined
analysis of density fields and other relevant physical fields.
As it allows the simultaneous determination of density and
velocity fields, it can serve as the basis for studies of the
dynamics of structure formation in the cosmos. Its ability to
detect substructure as well as reproduce the morphology of
cosmic features and objects implies DTFE to be suited for
assessing their dynamics without having to invoke artificial
filters.
IV. DTFE AND THE COSMIC WEB
Figure 4 provides a nice impression of the versatility of
the DTFE formalism. DTFE density and velocity fields may
be depicted at any arbitrary resolution without involving
any extra calculation: zoom-ins represent themselves a real
magnification of the reconstructed fields. This is in stark
contrast to conventional reconstructions in which the reso-
lution is arbitrarily set by the users and whose properties are
dependent on the adopted resolution.
It shows the density and velocity field, at two resolutions,
in and around filaments and other components of the cosmic
web in a GIF ΛCDM simulation. DTFE not only it allows a
study of the patterns in the nonlinear matter distribution but

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Figure 5. Schematic diagram of the DTFE procedure.

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Figure 6.
to reproduce the key characteristics of the Cosmic Web. Top: cosmic web dark matter density field in a 10h−1Mpc wide slice through a ΛCDM computer
simulation of cosmic structure formation. In the subsequent frames zoom-ins focus in on a high-density supercluster region around a massive cluster
(zoom-in boxes indicated).
DTFE cosmic density field illustrating the large dynamic range which is present in the large scale matter distribution, and the ability of DTFE
also a study of the related velocity flows. Because DTFE
manages to follow both the density distribution and the
corresponding velocity distribution into nonlinear features
it opens up the window towards a study of the dynamics
of the formation of the cosmic web and its corresponding
elements [41]. Note that such an analysis of the dynamics
is limited to regions and scales without multistream flows.
A. DTFE characteristics
Within the cosmological context a major – and crucial –
characteristic of a processed DTFE density and velocity field
is that it is capable of delineating the three fundamental
characteristics of the spatial structure of the Megaparsec
cosmic matter distribution:
• It outlines the full hierarchy of substructures present in
the sampling point distribution, relating to the standard
view of structure in the Universe having arisen through

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Figure 7.
contours have been chosen such that 65% of the mass is enclosed. The reconstructions on the basis of the TSC and SPH spline interpolations. The
arrows indicate two structures which are visible in both the galaxy distribution and the DTFE reconstruction, but not in the TSC and SPH spline-based
reconstructions.
.
Three-dimensional visualization of a filamentary particle distribution. The DTFE reconstruction delineates the full weblike network. The density
the gradual hierarchical buildup of matter concentra-
tions (fig. 6).
• DTFE also reproduces any anisotropic patterns in the
density distribution without diluting their intrinsic geo-
metrical properties. This is particularly important when
analyzing the prominent filamentary and planar features
marking the Cosmic Web (fig. 7).
• A third important aspect of DTFE is that it outlines the
presence and shape of voidlike regions. Because of the
interpolationdefinition of the DTFE field reconstruction
voids are rendered as regions of slowly varying and
moderately low density values.
The success of DTFE in following these key characteristics
has been substantiated by quantitative tests. In [42] we
demonstrate DTFE’s ability to infer the fractal dimensions
of a range of fractal point distributions: DTFE follows the
density distribution throughoutthe available spatial and mass
range. Additional tests in [43] have confirmed the ability of

Page 13

DTFE to reconstruct density fields with the correct shape
morphology (inertia tensor), whether it related to distri-
butions dominated by compact high density cluster peaks,
elongated filaments or sheets. Its comparative performance
with respect to other interpolation schemes may be assessed
from fig. 7. Clearly, DTFE remains closer to the actual
structure represented in the particle distribution.
In a thorough analysis of the interpolation and morpho-
logical performance of DTFE, Platen et al. [44] has demon-
strated that DTFE produces better and more reliable results
than Natural Neighbour interpolation and various Kriging
schemes. Even while the latter do take into account long-
range spatial correlations, the self-adaptive local character
of DTFE appears not only to be better suited for uniformly
sampled spatial datasets, but also for dealing with datasets
with a radially varying selection function or datasets beset by
systematic redshift distortions due to the peculiar velocities
of galaxies3.
B. Analysis of the Cosmic Web
Within its cosmological context, DTFE will meet its real po-
tential in more sophisticated applications tuned towards un-
covering morphological characteristics of the reconstructed
spatial patterns. The true potential of DTFE and related
adaptive random tessellation based techniques comes to the
fore when we take DTFE reconstructions as the basis for
a variety of “post-processing” tools. A variety of recent
techniques have recognized the high dynamic range and
adaptivity of tessellations to the spatial and morphological
resolution of the systems they seek to analyze.
We have been working on three different, yet mutually
related, formalisms which use DTFE density fields as the
basis for the identification of various key aspects of the
Cosmic Web. The Multiscale Morphology Filter (MMF),
introduced and defined by Arag´ on-Calvo et al. [36], attempts
to detect weblike anisotropic features over a range of spatial
scales and implicitly incorporates the hierarchical nature of
the Megaparsec matter distribution. While incorporating the
use of a more or less artificial scale filter, the Watershed
Void Finder (WVF, [37]) and the Cosmic Spine technique
[39] turn to the topological structure of the underlying
density field to identify void regions and trace the pervasive
weblike filamentary network. Both techniques are manifest
translations of concepts from Morse Theory, which describes
the topological structure of a density field on the basis of
its singular points - minima, maxima and saddle points.
V. THE MULTISCALE MORPHOLOGY FILTER
The Multiscale Morphology Filter (MMF), introduced by
3Redshift distortions: in addition to their cosmic expansion velocity,
galaxies have velocities with respect to the expanding Universe. The
Doppler redshift corresponding to these velocities add to the cosmic redshift
to yield a redshift z that is not precisely proportional to the distance r of
an object.
Arag´ on-Calvo et al. [36], is used for the identification
and characterization of different morphological elements
of the large scale matter distribution in the Cosmic Web.
An example of the morphological structure of the matter
distribution of a cosmological N-body simulation can be
seen in fig. 8 [36]. The image shows the identified elongated
filaments (dark grey), sheetlike walls (light grey) and clusters
(black dots) in the weblike pattern of the simulation.
The Multiscale Morphology Filter (MMF) method has
been developed on the basis of visualization and feature ex-
traction techniques in computer vision and medical research
[45]. The technology finds its origin in computer vision
research and has been optimized within the context of feature
detections in medical imaging. Frangi et al. [46] and Sato et
al. [47] presented its operation for the specific situation of
detecting the web of blood vessels in a medical image. This
defines a notoriously complex pattern of elongated tenuous
features whose branching make it closely resemble a fractal
network.
The MMF dissects the cosmic web on the basis of the
multiscale analysis of the Hessian of the density field. Its
formulation in terms of its density field scale-space structure
implicitly takes into account the hierarchical nature of the
matter distribution. Scale Space analysis looks for struc-
tures of a mathematically specified type in a hierarchical,
scale independent, manner. It is presumed that the specific
structural characteristic is quantified by some appropriate
parameter (e.g.: density, eccentricity, direction, curvature
components). The data is filtered to produce a hierarchy of
maps having different resolutions, and at each point, the
dominant parameter value is selected from the hierarchy to
construct the scale independent map. We refer to this scale-
filtering processes as a Multiscale morphology filter.
A. Scale Space
Crucial for the ability of the method to identify anisotropic
features such as filaments and walls is the use of a morpho-
logically unbiased and optimized continuous density field
retaining all features visible in a discrete galaxy or particle
distribution. Unless one manages to work directly from
the sampled particle or galaxy distribution, possibly via its
Voronoi or Delaunay tessellation, it is therefore imperative to
translate the discrete particle distribution into its DTFE den-
sity field, The morphological intentions of the MMF method
render DTFE a key element for translating the particle or
galaxy distribution into a representative continuous density
field fDTFE.
Following the determination of the DTFE density field
fDTFE, it is smoothed over a range of scales by means of
a hierarchy of spherically symmetric Gaussian filters WG
having different widths R:
?
fS(? x) =d? yfDTFE(? y)WG(? y,? x)
(14)

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Figure 8.
the largest structures are shown for clarity.
Cosmic web delineated by filaments (dark gray) and walls (light gray). Clusters (dark grey) are located at the intersection of filaments. Only
where WGdenotes a Gaussian filter of width R:
WG(? y,? x) =
1
(2πR2)3/2exp
?
−|? y − ? x|2
2R2
?
.
(15)
A pass of the smoothing filter attenuates structure on scales
smaller than the filter width. The MMF scale-space analysis
involves a discrete number of N + 1 levels, n = 0,...,N.
The base-scale R0is taken to be equal to the pixel scale of
the raw DTFE density map, while the smoothing radii Rnof
the other levels is chosen such that essential (sub)structures
and features in the density field are resolved. We denote
the nthlevel smoothed version of the DTFE reconstructed
field fDTFE by the symbol fn. The Scale Space itself
is constructed by stacking these variously smoothed data
sets, yielding the family Φ of smoothed density maps fn:
Φ =?
easily expanded.
levels nfn. In our implementation [36], we found it
sufficient to use only n = 5 levels, although this may be
B. Density Field Hessian & Eigenvalues
A data point can be viewed at any of the scales where
scaled data has been generated. The crux of the concept is
that the neighbourhood of a given point will look different
at each scale. There are potentially many ways of making a
comparison of the scale dependence of local environment.
The MMF employs the Hessian Matrix ∇ijf(x) of the local
density distribution in each of the smoothed replicas of the
original data,
∂2
∂xi∂xjfS(? x) = fDTFE⊗
?
∂2
∂xi∂xjWG(RS) =
=d? yf(? y)(xi− yi)(xj− yj) − δijR2
S
R4
S
WG(? y,? x)
(16)
where x1,x2,x3= x,y,z and δijis the Kronecker delta. In
other words, the scale space representation of the Hessian
matrix for each level n is evaluated by means of a convolu-
tion with the second derivatives of the Gaussian filter, also

Page 15

Figure 9.
(2007). The filled (grey) circles correspond to clusters with a mass above 1014M⊙. The inserts contain three specific examples of filaments. The gray dots
represent the original (simulation) dark matter particles. The spine of the filaments (black particles) is the result of the filament compression algorithm of
Arag´ on-Calvo (2007).
The filamentary network in a ΛCDM simulation. The filaments were identified by means of the MMF technique of Arag´ on-Calvo et al.
known as the Marr (or, less appropriately, “Mexican Hat”)
Wavelet.
At each point of a dataset in the Scale Space view of the
data we can quantify the local “shape” of the density field
in the neighbourhood of that point by calculating at each
point the eigenvalues λa(? x) of the Hessian Matrix of the
data values,
????
∂2fn(? x)
∂xi∂xj
− λa(? x) δij
????
=0,a = 1,2,3
(17)
where the eigenvalues are arranged so that λ1≤ λ2≤ λ3.
The λi(? x) are coordinate independent descriptors of the
behaviour of the density field in the locality of the point
? x and can be combined to create a variety of morphological