Pedestrian Flow Analysis in High-Density Crowds: Continuity Equation with Voronoi-Based Fields

Since the beginning of the century, capturing trajectories of pedestrian streams precisely from video recordings has been possible. To enable measurements at high density, the heads of the pedestrians are marked and tracked, thus providing a complete representation of the phase space. However, classical definitions of flow, density, and velocity of pedestrian streams are based on different segments in phase space. In addition, traditional methods fail with high densities of people, as heads move even when a crowd is blocked and standing still. In this article, Voronoi decomposition is used to construct density and velocity fields from pedestrian trajectories to solve this problem. Combined with the continuity equation, a flow equation on the basis of trajectories is derived satisfying the conservation of particle numbers exactly. The proposed method allows definitions of all quantities in the same segment of phase space even on scales smaller than the dimensions of a pedestrian. It is shown that these new definitions of flow, density, velocity are consistent with classical measurements and make it possible to determine standstill in pedestrian flows even when individual body parts are moving. These properties allow to scrutinize inconsistencies in the state of the art of pedestrian fundamental diagrams.