On Non-Linear Trajectory Tracking via Variety-Based Background Subtraction Models

Trajectory tracking wherein the trajectories can be described by polynomials in the spatial coordinate space is considered. A background subtraction model is extended to include an algebraic variety-based low rank constraint that captures the trajectory. The optimization is solved via the Iteratively Reweighted Least Squares method, adapted to the new signal model. The convergence of the Variety-based Background Subtraction Iterative Reweighted Least Squares (VBSI) algorithm is investigated, where a boundedness analysis is conducted. It is proved that the sequence of estimates generated by VBSI is bounded; furthermore, any limit point of that sequence is a stationary point of the optimization problem. An aggregated error function is shown to converge to zero. A bound on the estimation error is derived that characterizes the relationship between performance and system parameters. Simulation results show that the proposed algorithm achieves excellent recovery of the key matrices in both noisy and noiseless scenarios, and typically converges in 20 iterations. Furthermore, a performance improvement of 2.4 dB with regards to the Mean Square Error is observed over prior methods. Finally, the proposed technique is applied to real video data and shown to be effective in estimating non-linear trajectories.