Method of comparison

In order to compare the two models, we need some measure of behavioural similarity between two video sequences. Our approach is to compare the distribution of the generated points in the parameter space with the distribution of points extracted from the original video sequence.

We construct two dimensional histograms to approximate the distribution of points in the parameter space for each pair of dimensions. In order to compare the original and generated sequences of points using histograms, particular care has to be taken on the selection of the bin width used to compute those histograms. Indeed, a too large bin width will act as a smoothing effect on the original data while a too small bin width will result in an over-fitting of the data. In order to solve this problem, and for reproducibility of the comparison method, we used Scott's rule to select the bin size [16]. The bin size for each dimension is given by the formula:

$$h=3.5 \sigma N^{-\frac{1}{3}}$$ (4)

where $N$ is the total number of points in the sequence and $\sigma$ is the standard deviation of the data computed with respect to the selected dimension.

The two dimensional histograms of a reference and a generated set of points are then compared using the mean of the Bhattacharyya overlap computed for each pair of dimensions. The standard error can also be computed to represent our confidence in the result. This measure represents how close the point distributions of the generated and the reference sequences are. A value of $1$ corresponds to a perfect match of the distributions. A value of $0$ corresponds to two totally different distributions.

The facial behaviour models are assessed with respect to the original video sequence used to create those models. The distributions of points in the appearance parameter space should match with the original one if the model performs its task correctly.