A Grassmannian Graph Approach to Affine Invariant Feature Matching.

Area of Science:

• Computer Vision
• Geometric Deep Learning
• Computational Geometry

Background:

• Feature matching is crucial in computer vision for tasks like object recognition and 3D reconstruction.
• Existing methods struggle with affine transformations, which alter shape and orientation.
• Robustly matching features under varying scales and rotations remains a significant challenge.

Purpose of the Study:

• To develop a novel theoretical framework for 2D and 3D affine invariant feature matching.
• To introduce the Grassmannian Graph (GrassGraph) method for recovering correspondences between affinely related point sets.
• To achieve robust feature matching independent of large affine transformations.

Main Methods:

• A two-stage procedure involving mapping feature sets to an affine invariant Grassmannian representation.
• Approximation of the Laplace-Beltrami operator (LBO) on Grassmannian coordinates to nullify orthonormal factors.
• Utilizing derived affine invariant coordinates for correspondence recovery via mutual nearest neighbor relations.

Main Results:

• The GrassGraph framework demonstrates robust recovery of correspondences between unorganized, affinely related feature sets.
• Experimental validation on 2D and 3D datasets confirms the method's effectiveness.
• The approach successfully recovers correspondences under large affine transformations.

Conclusions:

• The proposed GrassGraph method offers a novel and effective solution for affine invariant feature matching in computer vision.
• The theoretical framework provides a robust way to handle 2D and 3D feature correspondences under significant geometric variations.
• This work advances the state-of-the-art in feature matching by addressing a long-standing problem in the field.