Method D

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 Voronoi cell area, $A_i$ , for each person $i$ can be obtained.

Method_D

Then, the density and velocity distribution of the space $\rho_{xy}$ and $v_{xy}$ can be defined as

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

where $v_i(t)$ is the instantaneous velocity of each person.

The Voronoi density for the measurement area is defined as:

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

Similarly, the Voronoi velocity is defined as:

$\langle v \rangle_v=\frac{\iint{v_{xy}dxdy}}{b_\text{cor}\cdot\Delta x}.$