Identity maps and their extensions on parameter spaces: Applications to anomaly detection in video

We propose an algorithm for detecting anomalies in video sequences. In order to build an appropriate model, video of nominal activity is utilized to construct an anomaly free representation of the data. The resulting model produces alarm notifications when anomalous activity is observed. The approach involves characterizing segments of video as subspaces and invoking the geometric framework of Grassmann manifolds, i.e., the space of k-dimensional subspaces of n-dimensional space, Gr(k, n). With subspaces treated as points on Gr(k, n) together with a suitably chosen Grassmannian metric, one can exploit novel aspects of the geometry of the data for the purpose of anomaly detection. This mathematical framework is used to extend the Multivariate State Estimation Technique to the context of Grassmann manifolds. We present an application to the ETHZ Living Room Data Set for detecting anomalous activities.