Mixtures of von Mises Distributions for People Trajectory Shape Analysis

People trajectory analysis is a recurrent task inmany pattern recognition applications, such as surveillance,behavior analysis, video annotation, and many others. In thispaper we propose a new framework for analyzing trajectoryshape, invariant to spatial shifts of the people motion in thescene. In order to cope with the noise and the uncertainty ofthe trajectory samples, we propose to describe the trajectoriesas a sequence of angles modelled by distributions of circularstatistics, i.e. a mixture of von Mises (MovM) distributions.To deal with MovM, we define a new specific EM algorithmfor estimating the parameters and derive a closed form of theBhattacharyya distance between single vM pdfs. Trajectories arethen modelled with a sequence of symbols, corresponding tothe most suitable distribution in the mixture, and comparedeach other after a global alignment procedure to cope withtrajectories of different lengths. The trajectories in the trainingset are clustered according with their shape similarity in an offlinephase, and testing trajectories are then classified with aspecific on-line EM, based on sufficient statistics. The approachis particularly suitable for classifying people trajectories in videosurveillance, searching for abnormal (i.e. infrequent) paths. Testson synthetic and real data are provided with also a completecomparison with other circular statistical and alignment methods.