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:

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

Similarly, the Voronoi velocity is defined as:

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