Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

• Published in: Mathematics (Basel) 2024-07, Vol.12 (14), p.2164
• Main Authors: Liu, Zigang, El-Sousy, Fayez F. M., Larik, Nauman Ali, Quan, Huan, Ji, Tianyao
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; Classification; Complexity; Data analysis; Data points; Data structures; Deep learning; Discriminant analysis; Euclidean space; Exploitation; Fisher, Ronald Aylmer; Hilbert space; image set classification; Machine learning; Manifolds (mathematics); Matrices (mathematics); Methods; minimum Riemannian mean distance; Operators (mathematics); Performance evaluation; regularized local Fisher discriminant analysis; Riemannian manifold; symmetric positive definite matrices; tangent space
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math12142164

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.