Line Constrained Estimation With Applications to Target Tracking: Exploiting Sparsity and Low-Rank

Trajectory estimation of moving targets is examined; in particular, quasi-linear trajectories are considered. Background subtraction methods, exploiting low-rank backgrounds, and sparse features of interest are extended to incorporate linear constraints. The line constraint is enforced via a rotation that yields an additional low rank condition. The proposed method is applied to single object tracking in video, wherein the trajectory can be parameterized as a line. The optimization is solved via the augmented Lagrange multiplier method. An average performance improvement of 4 dB is observed over previous background subtraction methods for estimating the position and velocity of the target. Furthermore, about a 6.2 dB gain is seen over previous target tracking methods that do not exploit the linear nature of the trajectory. The Cramér-Rao bound (CRB) for background subtraction with a linear constraint is derived and numerical results show that the proposed method achieves near optimal performance via comparison to the CRB. An aggregated error is shown to converge to zero and a boundedness analysis is conducted which suggests that the iterative algorithm is convergent as confirmed by simulations. Finally, the proposed technique is applied to real video data and is shown to be effective in estimating quasi-linear trajectories.