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

TL;DR

This work tackles motion recognition under contextual variations by introducing Bi-Invariant Local Trajectory-Shape Similarity (BILTS), a bi-invariant, bounded, and third-order descriptor of rigid-body trajectory shape. The authors derive both a continuous-time and a discretized version, leveraging a functional bi-invariant frame built from the Instantaneous Screw Axis and a two-step Taylor expansion of screw twists, and compute a scalar distance via a weighted Frobenius norm. They develop an extended QR-based algorithm (eQR) to extract a twice upper-triangular descriptor, and add regularization (BILTS+) to improve robustness near singularities; they validate against DLA and SYN datasets, achieving state-of-the-art recognition rates (mean 95.3%, with 100% in SYN for BILTS+). The approach demonstrates strong robustness to both world- and body-frame variations, and shows promise for gesture recognition, trajectory segmentation, and force-trajectory analysis in real-time and adaptive contexts. Overall, BILTS advances invariant trajectory similarity by combining bi-invariance, locality, and boundedness, enabling reliable recognition across diverse contexts without heavy reliance on frame calibration.

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.

Figures (11)

• Figure 1: Conceptual figure to visualize the invariant descriptor approach. For a given rigid-body motion, trajectory coordinates (of which only one is visualized above) can vary substantially depending on the selected world reference frame $\{w_1\}$ or $\{w_2\}$ and body reference frame $\{b_1\}$ or $\{b_2\}$. This makes motion similarity measurement based on trajectory coordinates highly frame dependent. To mitigate this dependency, we propose to extract invariant trajectory features which remain unchanged under reference frame transformations. Comparing these invariant features yields a more robust foundation for motion similarity measurement across varying reference frames.
• Figure 2: Visualization of the spatial screw twist $_w \mathbf{t} {}$, the body-fixed screw twist $_b \mathbf{t} {}$, and the Instantaneous Screw Axis (ISA) of the rigid-body motion. Rotational velocity vectors are indicated by double-headed arrows, while translational velocity vectors are represented by single-headed arrows. The translational velocity parallel to the ISA, $\boldsymbol{v}^{ISA}$, and the location of the point on the ISA closest to the origin of the reference frame, ${_w}\boldsymbol{p}_\perp$ or ${_b}\boldsymbol{p}_\perp$, can be calculated from the screw twists $_w \mathbf{t} {}$ or $_b \mathbf{t} {}$, respectively. Corresponding calculation formulas are provided at the lower right of the figure.
• Figure 3: Visualization of the right-invariant shape descriptor consisting of three spatial screw twists expressed in the spatial frame $\{w\}$. These twists are right-invariant, i.e. they are independent on the location and orientation of the body frame $\{b\}$.
• Figure 4: Construction of the functional bi-invariant frame $\{f\}$.
• Figure 5: Illustration of the bi-invariant shape descriptor consisting of the same three twists in Figure \ref{['fig:taylor1']}, but now expressed in the functional bi-invariant frame $\{f\}$. These twists are additionally left-invariant, hence they are also independent on the location and orientation of the spatial frame $\{w\}$.