Pedestrian flow analysis in high-density crowds: continuity equation with Voronoi-based fields

Abstract Since the beginning of the century, it has been possible to precisely capture trajectories of pedestrian streams from video recordings. To enable measurements at high density, the heads of pedestrians are marked and tracked, thereby providing a complete representation of the phase space. However, the 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 study, Voronoi decomposition was used to construct density and velocity fields from pedestrian trajectories to solve this problem. Combined with the continuity equation, a flow equation based on trajectories was derived, exactly satisfying the conservation of particle numbers. The proposed method allows all quantities to be defined within the same segment of phase space, even at scales smaller than the dimensions of a pedestrian. It is shown that these new definitions of flow, density, and velocity are consistent with classical measurements and enable the determination of standstill in pedestrian flows, even when individual body parts are moving. These properties allow inconsistencies in the state of the art of pedestrian fundamental diagrams to be scrutinised.