Continuity equation and fundamental diagram of pedestrians

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 and local measurements of flow, density, and velocity of pedestrian streams using trajectories are based on different segments in phase space (Lagrangian representation). The flow is defined as an average value over time, while the density is defined as the average value of an area. This leads to inconsistencies in central relations, such as the flow equation or the fundamental diagram. These have a particular effect in inhomogeneous states, such as the stop-and-go waves, where, in addition, the pedestrians do not change their position in the stop phase, but the head of the body moves. In order to obtain a local and spatio-temporally consistent measurement of the quantities flow, density, and velocity while ensuring particle number conservation fields (Euler representation) and the continuity equation could be used. To map trajectories of pedestrians heads parameter free and unambiguously to fields, this article introduces a method based on the Voronoi decomposition. These new definitions of flow, density, speed, and the particle number conserving flow equation are consistent with classical measurements. They are able to scrutinise inconsistencies in the state of the art of pedestrian fundamental diagrams.