The data energy

As shown in picture 1, filaments of galaxies are locally overdense clouds of points elongated along a principal direction. When the algorithm proposes a segment, it computes its data energy term to decide whether it reasonably suits a filament or not. Ud is computed with respect to the circular neighbourhood depicted in figure 2. Three parameters are then defined as follows :
 

$c = \vert\overline y\vert/r$  as a density term, 
$a = \sum d^{2}(y,s)/d^{2}(x,s)$ as a centering term and 
$n$ as an elongating term

 where $r$ is the number of galaxies is the neighbourhood,$d$  is the length of segment S,  $x,y$ is the Euclidian distance and$$U_D(S)= -\omega_1 g_1(\rho) -\omega_2 g_2(c) -\omega_3 g_3(a)$$  are respectively the longitudinal and latitudinal coordinates from the center of S.

The energy of segment S is then given by:

$\omega_i$

where the $g_i$ are weighting constants and the g functions are quality functions. In our case, these quality functions are simple thresholding functions, whose threshold values have been experimentally determined.

Figure 2: Segment fitting to the local data distribution. Black points represent galaxies.