Adaptive Smoothing V-Spline for Trajectory Reconstruction

Trajectory reconstruction is the process of inferring the path of a moving object between successive observation points. Widely separated points and measurement errors can cause traditional spline method to infer trajectories with sharp angles not typical of a moving object. Smoothing spline methods can efficiently build up a more smooth trajectory. In conventional smoothing splines, the objective function is augmented with a penalty term, which has a single parameter that controls the smoothness of reconstruction. Adaptive smoothing splines extend the single parameter to a function that can vary, hence the degree of smoothness can be different regions. A new method named the V-spline is proposed, which incorporates both location and velocity information but penalizes excessive accelerations. In the application of interest, the penalty term is also dependent on a known operational state of the object. The V-spline includes a parameter that controls the degree to which the velocity information is used in the reconstruction. In addition, the smoothing penalty adapts the observations are irregular in time. An extended cross-validation technique is used to find all spline parameters.