Point clouds are unordered sets of coordinates in 3D with no functional relation imposed on them. The Rigid Transformation Universal Manifold Embedding (RTUME) is a mapping of volumetric or surface measurements on a 3D object to matrices, such that when two observations on the same object are related by a rigid transformation, this relation is preserved between their corresponding RTUME matrices, thus providing linear and robust solution to the registration and detection problems. To make the RTUME framework of 3D object detection and registration applicable for processing point cloud observations, there is a need to define a function that assigns each point in the cloud with a value (feature vector), invariant to the action of the transformation group. Since existing feature extraction functions do not achieve the desired level of invariance to rigid transformations, to the variability of sampling patterns, and to model mismatches, we present a novel approach for designing dense feature extraction functions, compatible with the requirements of the RTUME framework. One possible implementation of the approach is to adapt existing feature extracting functions, whether learned or analytic, designed for the estimation of point correspondences, to the RTUME framework. The novel feature-extracting function design employs integration over $SO(3)$ to marginalize the pose dependency of extracted features, followed by projecting features between point clouds using nearest neighbor projection to overcome other sources of model mismatch. In addition, the non-linear functions that define the RTUME mapping are optimized using an MLP model, trained to minimize the RTUME registration errors. The overall RTUME registration performance is evaluated using standard registration benchmarks, and is shown to outperform existing SOTA methods.

We present a closed form solution to the problem of registration and detection of dense 3-D point clouds undergoing unknown rigid deformations. The solution is obtained by adapting the general framework of the universal manifold embedding (UME) to the case where the deformations the object may undergo are rigid. The UME nonlinearly maps functions (e.g., images, 3D models) related by geometric transformations of coordinates to the same linear subspace of some Euclidean space. Therefore registration, matching and classification are solved as linear problems in a lower dimensional space. In this paper we extend the UME framework to the special case where it is a-priori known that the geometric transformations are rigid (e.g. pose change of a 3-D rigid object). We further demonstrate the applicability of the methodology for the registration of 3-D point clouds. In the case where point correspondences are unknown, the majority of existing methods for registering 3-D point clouds are based on iteratively finding a transformation which minimizes some distance between the object and a model. The method proposed in this paper is notably different as registration is performed using a closed form solution that employs the UME low dimensional representation of the shapes to be registered.