BILTS: A Bi-Invariant Similarity Measure for Robust Object Trajectory Recognition under Reference Frame Variations

Abstract: When similar object motions are performed in diverse contexts but are meant to be recognized under a single classification, these contextual variations act as disturbances that negatively affect accurate motion recognition. In this paper, we focus on contextual variations caused by reference frame variations. To robustly deal with these variations, similarity measures have been introduced that compare object motion trajectories in a context-invariant manner. However, most are highly sensitive to noise near singularities, where the measure is not uniquely defined, and lack bi-invariance (invariance to both world and body frame variations). To address these issues, we propose the novel \textit{Bi-Invariant Local Trajectory-Shape Similarity} (BILTS) measure. Compared to other measures, the BILTS measure uniquely offers bi-invariance, boundedness, and third-order shape identity. Aimed at practical implementations, we devised a discretized and regularized version of the BILTS measure which shows exceptional robustness to singularities. This is demonstrated through rigorous recognition experiments using multiple datasets. On average, BILTS attained the highest recognition ratio and least sensitivity to contextual variations compared to other invariant object motion similarity measures. We believe that the BILTS measure is a valuable tool for recognizing motions performed in diverse contexts and has potential in other applications, including the recognition, segmentation, and adaptation of both motion and force trajectories.