On the Design of, $SO$ (3)-Invariant Feature Extraction Functions for 3D Point Clouds

The Rigid Transformation Universal Manifold Em-bedding (RTUME) [1] is a covariant mapping of 3-D Point clouds to matrices such that the representation of the rigid transformation between two point cloud observations on an ob-ject is preserved between their corresponding RTUME matrices. Therefore, in the RTUME framework, the registration parameters between point cloud observations are estimated from the RTUME matrices, in a low-dimensional linear space. However, this framework requires an invariant function to be defined on the processed point clouds, such that each point is assigned with a feature vector, invariant to rigid transformations. To achieve this we present a novel approach for adapting existing feature extraction functions, originally designed for the estimation of point correspondences (thus extracting rich geometrical information), to the RTUME framework. The adaptation is performed by marginalizing the extracted feature vectors with respect to the action of SO(3), thus eliminating the dependency on rotations. The practical considerations of performing the marginalization are considered analytically and different approaches are evaluated experimentally. The improvement in achieving rotation invariance of the feature vectors and its effect on improving registration accuracy are experimentally validated.