Method J

This method calculates the density based on Voronoi diagrams, which are a special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space. At any time the positions of the pedestrians can be represented as a set of points, from which the Voronoi diagram can be generated.

The density distribution of the space $\rho_{xy}$ can be defined as

$\rho_{xy} = 1/A_i \qquad \mbox{if} \quad (x,y) \in A_i,$

The Voronoi density for the measurement area is defined as:

$\langle \rho \rangle_v=\frac{\iint{\rho_{xy}dxdy}}{\Delta x \cdot \Delta y}.$

For a given trajectory $\vec{x_i}(t)$ , the velocity $v_i(t)$ is calculated by use of the displacement of pedestrian $i$ in a small time interval $\Delta t^\prime$ around $t$ :

$v_i(t)=\frac{\vec{x_i}(t+\Delta t^\prime/2)-\vec{x_i}(t-\Delta t^\prime/2))}{\Delta t^\prime}.$

The spatial mean velocity is the average of the instantaneous velocities $v_i(t)$ for all pedestrians $N$ who have an intersection with the measurement area at the time $t$ :

$\langle v \rangle_{\Delta x}=\frac{1}{N_{(x,y) \in A_i}}\sum_{i=1}^{N_{(x,y) \in A_i}}{v_i(t)}.$