A Grassmannian Graph Approach to Affine Invariant Feature Matching

In this work, we present a novel, theoretical approach to address one of the longstanding problems in computer vision: 2D and 3D affine invariant feature matching. Our proposed Grassmannian Graph (GrassGraph) framework employs a two stage procedure that is capable of robustly recovering correspondences between two unorganized, affinely related feature (point) sets. In the ideal case, the first stage maps the feature sets to an affine invariant Grassmannian representation, where the features are mapped into the same subspace. It turns out that coordinate representations extracted from the Grassmannian differ by an arbitrary orthonormal matrix. In the second stage, by approximating the Laplace-Beltrami operator (LBO) on these coordinates, this extra orthonormal factor is nullified, providing true affine invariant coordinates which we then utilize to recover correspondences via simple mutual nearest neighbor relations. Our validation benchmarks use large number of experimental trials performed on 2D and 3D datasets. Experimental results show that the proposed GrassGraph method successfully recovers large affine transformations.