The spatial model

We model each pathlet within a group by using a linear statistical model, assuming a Gaussian distribution. Each pathlet is described by a vector, which is a simple concatenation of the pathlet points. A pathlet group is a set of vectors, on which we can apply a principal component analysis. Each pathlet ${\bf s}$ is approximated by:

$${\bf s}=\overline{\bf s}+{\bf Q_s} {\bf b_s}$$ (29)

where $\overline{\bf s}$ is a vector representing the mean pathlet of the group, $\bf Q_s$ is a matrix computed by the principal component analysis describing how the data varies, and $\bf b_s$ is the vector of parameters for that particular pathlet. The probability density function of the set of parameters $\bf b_s$ is modelled by a Gaussian, by computing the mean and variance of $\bf b_s$ for all pathlet $\bf s$ in the group. Work by Makris and Ellis [64] has shown that for pedestrian trajectories, the distribution of paths around a mean path is approximatively Gaussian.

In order to be able to perform the principal component analysis on a group of pathlets, it is required that all the pathlets are encoded with the same number of points. Therefore, we interpolate all the pathlets by cubic splines and homogeneously resample them to a given number of points [65]. Resampling to 50 points has been found to give good results in our experiments. Figure 5.9 shows the result of resampling a hand drawn trajectory using cubic splines.

Figure 5.9: Resampling using cubic splines. The trajectory on figure 5.9(b) represents the trajectory on figure 5.9(a) resampled to $50$ points using cubic splines. The data on figure 5.9(a) were created by manual drawing.

[Before resampling] [After resampling]

Unfortunately the resampling process discards timing information. In the original pathlets, we know that the time spent between two successive points is constant. In the resampled pathlets, we do not know the time between two successive points. However, we need this information if we want to synthesise new pathlet from the pathlet model. The next section describes how we can extend the current model with timing information.