Geometric Properties of Grassmannians and Color Image Set Recognition Using Quaternionic Grassmannians
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Authors
Wang, Xiang Xiang
Issue Date
2025
Type
Dissertation
Language
en_US
Keywords
Alternative Title
Abstract
Riemannian manifolds provide a basic framework for understanding curved spaces and are widely used in geometry, physics, machine learning, and computer vision. Among them, Grassmannians, which are spaces of linear subspaces, have nonnegative curvature and are useful in data analysis and signal processing. However, their geometric properties are not as well studied as those of the manifold of positive definite matrices, which has clear hyperbolic geometric properties. This dissertation explores both theoretical and practical aspects of Grassmannians and their quaternionic extensions. First, we establish key geometric inequalities on Grassmannians, including a semi-parallelogram law, a law of cosines, and geodesic triangle inequalities. These results give a better understanding of the geometric structure of Grassmannians as Riemannian manifolds and provide useful mathematical tools. Building on this, we propose a new way to recognize color image sets using quaternionic Grassmannians. Traditional methods usually treat color channels separately, which ignores the relationships between them. By using quaternions, which naturally store color information in a compact and meaningful way, we define a new distance measure for quaternionic Grassmannians and prove a triangle inequality for this metric. This allows for a more effective way to recognize color image sets while preserving their structure. To test our method, we run experiments on a benchmark dataset and show that the quaternionic Grassmannian-based approach performs better than traditional methods in color image set classification. These results demonstrate the potential of using Grassmannian geometry and quaternion algebra in computer vision.