Different steps of the algorithms. Top left: initial distribution of galaxies in a redshift slice. Top centre: a density distribution is estimated from the galaxies. Top right: a grid of points {y_j} is selected as the initial positions of the iterative steps; they correspond to the centres of HEALPix pixels. Bottom left: the positions of the points after one step of the iteration. Bottom centre: the positions of the points after they converge, with the initial galaxy distribution overlapped. Bottom right: same as bottom centre, but the uncertainty of each point in the filaments is calculated with Algorithm 2; a redder colour represents a lower certainty. All images are gnomic projections centred at RA = 180°, Dec = 40° with a side size of 60°. The data used in this example corresponds to BOSS data at z = 0.570 (see Sect. 3).