Investigation of Voronoi diagram based direction choices using uni- and bi-directional trajectory data

In a crowd, individuals make different motion choices such as “moving to destination,” “following another pedestrian,” and “making a detour.” For the sake of convenience, the three direction choices are respectively called destination direction, following direction, and detour direction in this paper. Here, it is found that the featured direction choices could be inspired by the shape characteristics of the Voronoi diagram. To be specific, in the Voronoi cell of a pedestrian, the direction to a Voronoi node is regarded as a potential “detour” direction and the direction perpendicular to a Voronoi link is regarded as a potential “following” direction. A pedestrian generally owns several alternative Voronoi nodes and Voronoi links in a Voronoi cell, and the optimal detour and following direction are determined by considering related factors such as deviation. Plus the destination direction which is directly pointing to the destination, the three basic direction choices are defined in a Voronoi cell. In order to evaluate the Voronoi diagram based basic directions, the empirical trajectory data in both uni- and bi-directional flow experiments are extracted. A time series method considering the step frequency is used to reduce the original trajectories' swaying phenomena which might disturb the recognition of actual forward direction. The deviations between the empirical velocity direction and the basic directions are investigated, and each velocity direction is classified into a basic direction or regarded as an inexplicable direction according to the deviations. The analysis results show that each basic direction could be a potential direction choice for a pedestrian. The combination of the three basic directions could cover most empirical velocity direction choices in both uni- and bi-directional flow experiments.

Figure 1

Illustration of pedestrian motion direction choices. The dashed circles represent the pedestrian and the solid dots represent the target of pedestrian.

Figure 2

Potential following directions and potential detour directions.

Figure 3

Voronoi diagram based direction choices.

Figure 4

(a) Sketch map of the uni-directional flow experiment. (b) Sketch map of the bi-directional flow experiment.

Figure 5

Example of the original trajectories and the smoothed trajectories. The black points represent the original trajectory data and the red points represent the smooth trajectory data. The red and black arrows represent the direction based on the original and smoothed trajectories.

Figure 6

Featured trajectories in uni- and bi-directional flow. The trajectories are colored with blue, green, red, and black to represent destination, following, detour, and inexplicable direction, respectively. (a) Pedestrian trajectories of uni-directional flow experiment. The parameters of the uni-directional flow corridor parameters are $D_{\mathrm{entra}}=0.7\mathrm{m}$ and $D_{\mathrm{corridor}}=D_{\mathrm{exit}}=1.8\mathrm{m}$. (b) Pedestrian trajectories from right to left in bi-directional flow experiment. (c) Pedestrian trajectories from left to right in bi-directional flow experiment. The parameters of the bi-directional flow corridor are $D_{\mathrm{corridor}}=3.6\mathrm{m}$ and $D_{\mathrm{left}}=D_{\mathrm{right}}=0.5\mathrm{m}$.

Figure 7

Distribution of deviation between current velocity direction and basic directions. (a) Uni-directional flow experiment data; (b) bi-directional flow experiment data.

Figure 8

Correlation of deviation between basic directions. Panels (a) and (d) are destination deviation-following deviation relation in uni- and bi-directional flow experiments, respectively. Panels (b) and (e) are following deviation-detour deviation in uni- and bi-directional flow experiments, respectively. Panels (c) and (f) are destination deviation-detour deviation in uni- and bi-directional flow experiments, respectively.

Figure 9

Percentage of basic directions in different densities. (a) Basic direction percentage for uni-directional flow. (b) Basic direction percentage for bi-directional flow. Note that the density of pedestrian is obtained by a Voronoi method which is able to calculate the local density of single pedestrian; the detail definition can be found in Appendix pp1.

Figure 10

Voronoi diagram of pedestrians.

Figure 11

Relationship between ${\alpha }_0$ and percentage of inexplicable direction. Note that ${\alpha }_0$ is measured in degrees in the figure.

Figure 12

Relationship between smoothing steps and average direction deviation. The average step length is marked by the red dashed line.