A Novel Geometric Framework on Gram Matrix Trajectories for Human Behavior Understanding

TL;DR

This paper introduces a geometric framework for analyzing human landmark trajectories on a Riemannian manifold, enabling improved action, emotion, and facial expression recognition from 3D data and videos.

Contribution

It develops a novel space-time geometric representation of landmarks as trajectories on a Riemannian manifold, incorporating shape and covariance for enhanced human behavior analysis.

Findings

Achieves competitive results in action recognition benchmarks.

Effective in emotion and facial expression recognition tasks.

Provides a new geometric approach for trajectory comparison and classification.

Abstract

In this paper, we propose a novel space-time geometric representation of human landmark configurations and derive tools for comparison and classification. We model the temporal evolution of landmarks as parametrized trajectories on the Riemannian manifold of positive semidefinite matrices of fixed-rank. Our representation has the benefit to bring naturally a second desirable quantity when comparing shapes, the spatial covariance, in addition to the conventional affine-shape representation. We derived then geometric and computational tools for rate-invariant analysis and adaptive re-sampling of trajectories, grounding on the Riemannian geometry of the underlying manifold. Specifically, our approach involves three steps: (1) landmarks are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank to build time-parameterized trajectories; (2) a temporal warping is performed on the trajectories, providing a geometry-aware (dis-)similarity measure between them; (3) finally, a pairwise proximity function SVM is used to classify them, incorporating the (dis-)similarity measure into the kernel function. We show that such representation and metric achieve competitive results in applications as action recognition and emotion recognition from 3D skeletal data, and facial expression recognition from videos. Experiments have been conducted on several publicly available up-to-date benchmarks.