Online Clustering of Moving Hyperplanes

We propose a recursive algorithm for clustering trajectories lying in multiple moving hyperplanes. Starting from a given or random initial condition, we use normalized gradient descent to update the coefficients of a time varying polynomial whose degree is the number of hyperplanes and whose derivatives at a trajectory give an estimate of the vector normal to the hyperplane containing that trajectory. As time proceeds, the estimates of the hyperplane normals are shown to track their true values in a stable fashion. The segmentation of the trajectories is then obtained by clustering their associated normal vectors. The final result is a simple recursive algorithm for segmenting a variable number of moving hyperplanes. We test our algorithm on the segmentation of dynamic scenes containing rigid motions and dynamic textures, e.g., a bird floating on water. Our method not only segments the bird motion from the surrounding water motion, but also determines patterns of motion in the scene (e.g., periodic motion) directly from the temporal evolution of the estimated polynomial coefficients. Our experiments also show that our method can deal with appearing and disappearing motions in the scene.