Directional Statistics Approach Based on Instantaneous Rotational Parameters of Tri-axial Trajectories for Footstep Detection

Abstract

Polarization of tri-axial signals is defined using instantaneous rotational characteristics of the three-dimensional (3D) trajectory. We propose a rotational model to parameterize the time evolution of the 3D trajectory as a sequence of scaled rotations. Using this model, the velocity-to-rotation transform is defined to estimate the eigenangle, eigenaxis and orientation quaternion that quantify the instantaneous rotational parameters of the trajectory. These rotational parameters correspond to p-dimensional directional random vectors (DRVs). We propose two approaches to discriminate between the presence and absence of an elliptically polarized trajectory generated by human footsteps. In the first approach, we fit a von Mises---Fisher probability density function to the DRVs and estimate the concentration parameter. In the second approach, we employ the Kullback---Leibler divergence between the estimated nonparametric hyperspherical probability densities. The detection performance of the proposed metrics is shown to achieve an accuracy of $$97\%$$97% compared to existing approaches of $$82\%$$82% for footstep signals.