Modeling crowd dynamics through coarse-grained data analysis

Figure 1.  Experiments and data acquisition. (a) A typical experiment where bi-directional circulation is analyzed. (b) Participants equipped with reflexive markers were tracked by means of an optoelectronic motion capture system (VICON MX-40, Oxford Metrics, UK). (c) The area-weighting assignment procedure: The particle $i$ (red dot) is located in the cell ${\mathcal C}^n(i) = [\theta_k, \theta_{k+1}]$ and is assigned to the two nodes $\theta_k$ and $\theta_{k+1}$ in proportion to the area enclosing the opposite node. For instance, on the picture, the assignment to the node $\theta_k$ is in proportion to the ratio of the shaded area to the area of ${\mathcal C}^n(i)$

Figure 2.  Characteristic speeds in the BM model. (a) Perspective view of the BFD $f(\rho_+, \rho_-)$ as a function of $\rho_+$ and $\rho_-$. (b) Sketchy fundamental diagram (in red) of a one-way flow $f(\rho)$ as a function of the single density $\rho$. The function $f(\rho)$ has the same monotony as cuts of the BFD $f(\rho_+, \rho_-)$ along lines $\rho_- = \mbox{Constant}$. The average velocity $u(\rho) = f(\rho)/\rho$ and the cluster velocity $\lambda(\rho) = f'(\rho)$ are respectively the slopes of the secant (in red) and tangent (in green) lines to this curve

Figure 3.  Bi-directional Fundamental Diagram (BFD). (a) Estimated BFD expressing the flux $f$ in one direction as a function of the density of pedestrians moving in the same direction $\rho_+$ and in the opposite direction $\rho_-$. (b) Parametrized BFD used in the model. The values of $f$, in $(m.s)^{-1}$, are color coded according to the color bar

Figure 4.  Model results and comparisons with experimental data. (a) Setting for the model: the density at the entry of the corridor ($x = 0$) is taken from the experiment and the model is used to predict the occupancy inside the corridor. (b-c) Clockwise and anti-clockwise (resp. b and c) pedestrian densities as functions of position (horizontal axis) and time (vertical axis running downwards), for one of the replications with $60$ pedestrians corresponding to balanced fluxes ($50 \%$ of pedestrians walking in each direction). Left picture of each panel: experiment; right picture: BM model, run with initial and boundary data and BFD estimated from the experimental data. The density is color-coded according to the lateral scale (in m$^{-2}$)

Figure 5.  Model results and comparisons with experiment data with the $75 \%\! -\! 25\%$ flux balance: $\%75$ are moving to the left ($\rho_-$) and $\%25$ are moving to the right ($\rho_+$). See Fig. 4b, c for balanced flux

Figure 6.  Pedestrian and cluster velocities. Comparison between the pedestrian velocity $u$ (red curve) and cluster velocity $\lambda$ (green curve) as functions of the local density of co-moving pedestrians $\rho$ from the experiments (circles) and from the BM model (solid line) with balanced flux ($50\%\! -\! 50\%$) on the left figure and with $75\%\! -\! 25\%$ flux balance on the right figure. Since $\lambda <u$, information is propagating upstream as predicted by the BM model

Figure 7.  Efficiency segregation strategy. Estimation of the relative gain using the segregation strategy (in %) as a function of the densities $\rho_+, \, \rho_-$ (in m$^{-2}$) of the two types of pedestrians (level curve representation). The strategy is efficient when both densities $\rho_+$ and $\rho_-$ are roughly balanced and large

Figure 8.  Cluster and cluster velocity. (a) Illustration of a cluster defined from a density distribution $\rho(x, t^n)$ for a given threshold $h$. The edges of the cluster (i.e. $X^+$ and $X^-$) described the level curves of $\rho(x, t)$. (b) Graphical representation of the cluster velocity using the level curves of $\rho$. The picture depicts the edges (in yellow) of the cluster (in red). The cluster edge velocity i.e. the slope of the cluster edge in the position-time plane is given by $f'(h)$ and is illustrated by the green segment

Figure 9.  Level curves of the density. Level curves $X(t)$ for various levels $h$ for the replication displayed in Fig. 4 and clockwise pedestrian density $\rho_+$. Horizontal axis is space and vertical axis is time, running downwards. The color code corresponds to the level height $h$, from blue (lower levels) to red (higher levels). (a): with no cutoff and no filtering. (b): after cutoff (only level curves with life-time greater than $3$ seconds are kept) but before filtering. (c): after cutoff and filtering