Image-based target tracking using least-squares trajectory estimation without a priori knowledge

Accurate object tracking has become increasingly important with developments in Lidar and high-resolution camera systems for remote surveillance. Typical methods utilize Markov Chain state estimation paired with Kalman or particle filters. These methods, however, are susceptible to loss of track if an error is encountered or a loss of image occurs. We developed a set of algorithms built on more recent efforts that have been aimed at reconstructing the three-dimensional trajectory of the object. These methods typically require a priori information about the target being tracked when using a single camera. We attempt to accomplish trajectory reconstruction without information about the system beyond a known camera matrix. Using feature locations extracted from a series of images combined with the associated camera matrices, we generate a linear system of equations relating the image coordinates in two-dimensional projective space to the real-world three-dimensional coordinates. We use this system of equations to generate a least-squares polynomial fit for the data. The resulting system of equations successfully recovers polynomial equations of motion and outperforms Taylor approximations of equal degree outside of the convergent region. Introducing translational errors causes the error-minimizing angle combination to differ from the single most likely analytic angle combination. We employ a mean-approximating method to generate an adjusted angular value and demonstrate this represents an improvement in accuracy over the analytic solution.