Mixtures of von Mises Distributions for People Trajectory Shape Analysis

People trajectory analysis is a recurrent task in many pattern recognition applications, such as surveillance, behavior analysis, video annotation, and many others. In this paper, we propose a new framework for analyzing trajectory shape, invariant to spatial shifts of the people motion in the scene. In order to cope with the noise and the uncertainty of the trajectory samples, we propose to describe the trajectories as a sequence of angles modeled by distributions of circular statistics, i.e., a mixture of von Mises (MovM) distributions. To deal with MovM, we define a new specific expectation-maximization (EM) algorithm for estimating the parameters and derive a closed form of the Bhattacharyya distance between single von Mises pdfs. Trajectories are then modeled with a sequence of symbols, corresponding to the most suitable distribution in the mixture, and compared each other after a global alignment procedure to cope with trajectories of different lengths. The trajectories in the training set are clustered according to their shape similarity in an off-line phase, and testing trajectories are then classified with a specific on-line EM, based on sufficient statistics. The approach is particularly suitable for classifying people trajectories in video surveillance, searching for abnormal (i.e., infrequent) paths. Tests on synthetic and real data are provided with also a complete comparison with other circular statistical and alignment methods.