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.

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:

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

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 $$t$$:

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

For calculating the mean velocity in the measurement area two approaches can be applied.

1. The Voronoi velocity is defined as:
$\langle v \rangle_v=\frac{\iint{v_{xy}dxdy}}{b_\text{cor}\cdot\Delta x}.$
1. The Arithmetic 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 $$t$$:
$\langle v \rangle_{\Delta x}=\frac{1}{N_{(x,y) \in A_i}}\sum_{i=1}^{N_{(x,y) \in A_i}}{v_i(t)}.$
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