Topological trajectory clustering with relative persistent homology

Cloud Robotics techniques based on Learning from Demonstrations suggest promising alternatives to manual programming of robots and autonomous vehicles. One challenge is that demonstrated trajectories may vary dramatically: it can be very difficult, if not impossible, for a system to learn control policies unless the trajectories are clustered into meaningful consistent subsets. Metric clustering methods, based on a distance measure, require quadratic time to compute a pairwise distance matrix and do not naturally distinguish topologically distinct trajectories. This paper presents an algorithm for topological clustering based on relative persistent homology, which, for a fixed underlying simplicial representation and discretization of trajectories, requires only linear time in the number of trajectories. The algorithm incorporates global constraints formalized in terms of the topology of sublevel or superlevel sets of a function and can be extended to incorporate probabilistic motion models. In experiments with real automobile and ship GPS trajectories as well as pedestrian trajectories extracted from video, the algorithm clusters trajectories into meaningful consistent subsets and, as we show in an experiment with ship trajectories, results in a faster and more efficient clustering than a metric clustering by Fréchet distance.