Multiscale SPD manifold learning for rehabilitation exercise evaluation

Abstract

Rehabilitation exercise assessment plays a crucial role in patient recovery, particularly for individuals recovering from injuries, surgeries, or illnesses affecting mobility. In this paper, we propose a novel approach for the assessment of skeleton-based data recorded from rehabilitation exercise, where we present the data as points on symmetric positive definite (SPD) manifold. Our method addresses the limitations of traditional Euclidean-based approaches by leveraging the SPD manifold's ability to preserve motion variations and spatial relationships. We propose a novel framework leveraging SPD manifold to preserve the intrinsic geometry of human motion and capture nonlinear variations in complex movements. By embedding motion data into SPD manifolds, we integrate unsupervised K-Nearest Neighbors (KNN) with Riemannian geometry for precise classification of correct and incorrect movements. We further develop a Tangent Space Linear SPD Support Vector Machine (SVM), optimized via stochastic gradient descent (SGD) in the tangent space at the identity matrix. Additionally, a tailored neural network architecture with multi-scale feature extraction enhances movement assessment by capturing hierarchical patterns in vectorized SPD data. Our specialized neural network, designed for vectorized SPD data, outperforms state-of-the-art methods on three benchmark datasets: Kimore, UI-PRMD, and EHE. In cross-subject evaluations, accuracy improves to 92.40% (UI-PRMD), 85.18% (Kimore), and 87.59% (EHE), with even greater improvements in random train-test splits. Although the proposed method involves a high parameter count while reducing computational complexity in terms of floating-point operations, literature suggests that certain concepts and objects recur across diverse mathematical domains, often carrying significant implications. Additionally, manifold transformations in data representation effectively capture the intrinsic geometric structure. Furthermore, our training process is faster than state-of-the-art methods, leading to quicker model convergence and reduced computational overhead without compromising accuracy. These results highlight the potential of SPD manifolds for accurate, reliable rehabilitation assessment.

Keywords: Multi-scale classification; Rehabilitation exercise assessment; SPD manifolds; Tangent space representation.

Figure 1

Overview of multiscale SPD manifold classification framework.

Algorithm 1

KNN Classification on the SPD Manifold

Algorithm 2

Tangent Space Linear SPD SVM

Figure 2

Neural network pipeline for SPD manifold data processing.

Figure 3

KIMORE dataset visualization.