Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition

Area of Science:

• Computer Vision
• Machine Learning
• Geometric Deep Learning

Background:

• Parametric models for image and video data often exhibit linear subspace structures.
• Recognizing patterns in image and video sets requires robust modeling of these subspace properties.

Purpose of the Study:

• To unify parametric models for videos and image sets using Grassmann and Stiefel manifolds.
• To develop geometric tools for subspace-based recognition and statistical modeling.

Main Methods:

• Describing linear dynamic models and unordered image sets as finite-dimensional linear subspaces.
• Casting inference over subspaces as a problem on the Grassmann manifold.
• Utilizing Riemannian geometry, including metrics and geodesics, for statistical modeling and classification.

Main Results:

• Demonstrated that subspace inference can be effectively performed on the Grassmann manifold.
• Developed intrinsic and extrinsic statistical methods for maximum-likelihood classification.
• Derived unsupervised clustering algorithms based on manifold geometry.

Conclusions:

• The proposed geometric approach significantly improves performance in diverse vision applications.
• This framework offers a unified perspective for subspace-based modeling in computer vision.
• The methods are effective for activity recognition, face recognition, object recognition, and video clustering.