Grassmannian Dimensionality Reduction Using Triplet Margin Loss for UME Classification of 3D Point Clouds

We consider the problem of classifying 3-D objects undergoing rigid transformations. It has been shown that the rigid transformation universal manifold embedding (RTUME) provides a mapping from the orbit of observations on some object to a single low-dimensional linear subspace of Euclidean space. This linear subspace is invariant to the geometric transformations. In the classification problem the RTUME subspace extracted from an experimental observation is tested against a set of subspaces representing the different object manifolds, in search for the nearest class. We elaborate on the design problem of the RTUME operator in the case where the point cloud sampled from the object is sparse, noisy, and non-uniformly sampled. By introducing metric learning and negative-mining techniques into the framework of Grassmannian dimensionality reduction for universal manifold embedding, we improve classification performance for these challenging sampling conditions.