All Figures

$\begin{figure} \par\mbox{ \includegraphics[width=7.3cm,clip]{7880fig1a.eps} \includegraphics[width=7.3cm,clip]{7880fig1b.eps} } \end{figure}$ Figure 1: DTFE image of a slice through the N-body simulation used in this work. Left: DTFE density field in a central slice. Right: the corresponding particle distribution in a slice of width $5~h^{-1}~\hbox{Mpc}$ .
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{7880fig2.eps} \end{figure}$ Figure 2: Scale-space: a particle distribution ( left) is translated by DTFE into a density field ( centre), followed by the determination of the field, by means of filtering, at a range of scales ( righthand).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=18cm,clip]{7880fig3.eps} \end{figure}$ Figure 3: Maps of the eigenvalues of the Hessian matrix at 3 different scales (levels). From top to bottom: the 3 eigenvalues $\lambda _1$ , $\lambda _2$ and $\lambda _3$ ( $\lambda _1 > \lambda _2 > \lambda _3$ ). From left to right: 3 different scales R₁R₃ and R₅, ( R₁>R₂>R₅). Positive values are represented as gray shades in logarithmic scale while negative values are indicated by contour lines also in logarithmic scale.
Open with DEXTER

In the text

$\begin{figure} \par\mbox{\includegraphics[width=17cm]{7880fig4.eps} } \end{figure}$ Figure 4: Morphology Mask $\mathcal{E}$ : on the basis of the 3 eigenvalues $\lambda _1$ , $\lambda _2$ and $\lambda _3$ at each location we determine whether the morphological criterion - here whether it corresponds to a filament (Table 1) - is valid. If so ${\mathcal E}=1$ , otherwise it is ${\mathcal E}=0$ . Top row: maps of the three eigenvalues; bottom row: the Morphology Mask $\mathcal{E}$ .
Open with DEXTER

In the text

$\begin{figure*} \par\includegraphics[width=17cm,clip]{7880fig5.eps} \end{figure*}$ Figure 5: Morphology Filter $\mathcal{T}$ . The Morphology Response function $\mathcal{M}$ ( top centre) is the soft thresholded version of the Shape Significance map $\mathcal{S}$ ( left frame), determined from the values of the eigenvalues $\lambda _1$ , $\lambda _2$ and $\lambda _3$ . The Morphology Intensity function $\mathcal{I}$ ( bottom centre) is also computed from the $\lambda$ 's using Eq. (11). Finally, the Morphology Filter $\mathcal{T}$ ( right frame) is obtained by combining $\mathcal{M}$ with $\mathcal{I}$ .
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=16.7cm,clip]{7880fig6.eps} \end{figure}$ Figure 6: The Feature Map $\mathcal{F}$ ( righthand frame) is computed for each scale and is equal to the Morphology Filter $\mathcal{T}$ at the locations where the Morphology Mask $\mathcal{E}$ is unity (and nonzero).
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=16.5cm,clip]{7880fig7.eps} \end{figure}$ Figure 7: The Scale Space Map Stack $\Psi$ : the formalism selects at any location the scale with the optimal signal of the feature map. Depicted are the Feature maps $\mathcal{F}$ for three different scales ( top row), and the resulting Map Stack $\Psi$ ( bottom row), determined over the full range of scales.
Open with DEXTER

In the text

$\begin{figure} \par\subfigure[Threshold determination for blobs.] { \mbox{\hski... ...0truecm\includegraphics[width=0.71\linewidth]{7880fig8c.eps} } } \end{figure}$ Figure 8: Thresholds for feature isolation based on the feature erosion criterion. The selected value is shown as a dotted vertical line. The object count to the right of the line declines due to erosion.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=16.7cm,clip]{7880fig9.eps} \end{figure}$ Figure 9: Recovered particles in Blobs, Filaments and Walls from a voronoi particle distribution. Particles inside blobs are detected ( left), at 90/15 percent real/false detections. From the new blob-free distribution we detect particles in filaments ( center) at 90/10 percent real/false detections. Finally the blob-filament-free distribution is used to find the particles inside walls ( right) at 80/10 percent real/false detections.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=17.5cm,clip]{7880fig10.eps} \end{figure}$ Figure 10: Reals versus false detections for different voronoi models (see Table 2) (A: solid, B: dotted, C: dashed, D: dotted-dashed) for blobs ( left), filaments ( center) and walls ( right). We applied the MMF (black) and simple density thresholding (grey) in order to compare both methods.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=16.5cm,clip]{7880fig11.eps} \end{figure}$ Figure 11: MMF applied to N-body simulation. The top row shows a subsample a) consisting of 10 $\%$ of the total number of particles together with, in panels b) and c), the structures resulting from simple density thresholding using two different thresholds. Panels b) and c) both contain both spherical and elongated structures: there is a large amount of cross contamination between morphologies. Simple density thresholding is not an effective morphological discriminator. The second row shows the results of applying the MMF procedure showing clearly segregated a) blobs, b) filaments and c) walls (for clarity we display only the largest structures. The third row shows the particles associated with the MMF defined structures.
Open with DEXTER

In the text

$\begin{figure} \par\includegraphics[width=15.5cm,clip]{7880fig12.eps} \end{figure}$ Figure 12: Comparing blobs found from HOP and from MMF. A): Particles. B) Isosurfaces of the blob identiffied with MMF. C) Particles inside the blobs (black) and background particles in grey. D) The position of the HOP Haloes (circles) and the particles inside the MMF blobs (dark grey). Light grey particles are just the rest.
Open with DEXTER

In the text

$\begin{figure} \par\mbox{\includegraphics[width=17cm]{7880fig13.eps} } \end{figure}$ Figure 13: Particles defining filamentary structures in a slice of an N-body model. The grayscale images show the MMF detection of filamentary features on various filtering scales. Top lefthand: the filament volume occupancy (number of sample grid cells with a filament signal) as a function of smoothing scale.
Open with DEXTER

In the text

$\begin{figure} \par\subfigure[Volume occupied by each of the structural features... ...] { \includegraphics[width=5.0cm,angle=0.0]{7880fig14b.eps} } \par \end{figure}$ Figure 14: Occupancy of Cosmic Web features, by volume ( top) and mass ( bottom) for a $\Lambda$ CDM N-body simulation (see text).
Open with DEXTER

In the text

$\begin{figure} \par\mbox{\includegraphics[width=18cm]{7880fig15.eps} } \end{figure}$ Figure A.1: Three different patterns of Voronoi element galaxy distributions, shown in a 3D cubic setting. The depicted spatial distributions correspond to a wall-dominated Voronoi Universe ( top), a filamentary Voronoi Universe ( centre) and a cluster-dominated Voronoi Universe ( bottom).

In the text

$\begin{figure} \par\vskip 0.5truecm \mbox{\hskip 0.0truecm\includegraphics[height=7.0cm]{7880fig16.eps} } \vskip 0.2cm \end{figure}$ Figure B.1: Schematic illustration of the galaxy projections in the Voronoi clustering model. See text.

In the text

$\begin{figure} \par\mbox{ \includegraphics[width=7.3cm,clip]{7880fig1a.eps} \includegraphics[width=7.3cm,clip]{7880fig1b.eps} } \end{figure}$	Figure 1: DTFE image of a slice through the N-body simulation used in this work. Left: DTFE density field in a central slice. Right: the corresponding particle distribution in a slice of width $5~h^{-1}~\hbox{Mpc}$ .
Open with DEXTER

$\begin{figure} \par\includegraphics[width=18cm,clip]{7880fig2.eps} \end{figure}$	Figure 2: Scale-space: a particle distribution ( left) is translated by DTFE into a density field ( centre), followed by the determination of the field, by means of filtering, at a range of scales ( righthand).
Open with DEXTER

$\begin{figure} \par\includegraphics[width=16.7cm,clip]{7880fig6.eps} \end{figure}$	Figure 6: The Feature Map $\mathcal{F}$ ( righthand frame) is computed for each scale and is equal to the Morphology Filter $\mathcal{T}$ at the locations where the Morphology Mask $\mathcal{E}$ is unity (and nonzero).
Open with DEXTER

$\begin{figure} \par\includegraphics[width=16.5cm,clip]{7880fig7.eps} \end{figure}$	Figure 7: The Scale Space Map Stack $\Psi$ : the formalism selects at any location the scale with the optimal signal of the feature map. Depicted are the Feature maps $\mathcal{F}$ for three different scales ( top row), and the resulting Map Stack $\Psi$ ( bottom row), determined over the full range of scales.
Open with DEXTER

$\begin{figure} \par\subfigure[Threshold determination for blobs.] { \mbox{\hski... ...0truecm\includegraphics[width=0.71\linewidth]{7880fig8c.eps} } } \end{figure}$	Figure 8: Thresholds for feature isolation based on the feature erosion criterion. The selected value is shown as a dotted vertical line. The object count to the right of the line declines due to erosion.
Open with DEXTER

$\begin{figure} \par\includegraphics[width=17.5cm,clip]{7880fig10.eps} \end{figure}$	Figure 10: Reals versus false detections for different voronoi models (see Table 2) (A: solid, B: dotted, C: dashed, D: dotted-dashed) for blobs ( left), filaments ( center) and walls ( right). We applied the MMF (black) and simple density thresholding (grey) in order to compare both methods.
Open with DEXTER

$\begin{figure} \par\includegraphics[width=15.5cm,clip]{7880fig12.eps} \end{figure}$	Figure 12: Comparing blobs found from HOP and from MMF. A): Particles. B) Isosurfaces of the blob identiffied with MMF. C) Particles inside the blobs (black) and background particles in grey. D) The position of the HOP Haloes (circles) and the particles inside the MMF blobs (dark grey). Light grey particles are just the rest.
Open with DEXTER

$\begin{figure} \par\mbox{\includegraphics[width=17cm]{7880fig13.eps} } \end{figure}$	Figure 13: Particles defining filamentary structures in a slice of an N-body model. The grayscale images show the MMF detection of filamentary features on various filtering scales. Top lefthand: the filament volume occupancy (number of sample grid cells with a filament signal) as a function of smoothing scale.
Open with DEXTER

$\begin{figure} \par\subfigure[Volume occupied by each of the structural features... ...] { \includegraphics[width=5.0cm,angle=0.0]{7880fig14b.eps} } \par \end{figure}$	Figure 14: Occupancy of Cosmic Web features, by volume ( top) and mass ( bottom) for a $\Lambda$ CDM N-body simulation (see text).
Open with DEXTER