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: Illustration of the Voronoi diagrams

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