Estimating Rigid Transformations of Noisy Point Clouds Using the Universal Manifold Embedding

We present a closed form solution to the problem of registration of fully overlapping 3D point clouds undergoing unknown rigid transformations, as well as for detection and registration of sub-parts undergoing unknown rigid transformations. The solution is obtained by adapting the general framework of the universal manifold embedding (UME) to the case where the transformations the object may undergo are rigid. The UME nonlinearly maps functions related by certain types of geometric transformations of coordinates to the same linear subspace of some Euclidean space while retaining the information required to recover the transformation. Therefore registration, matching and classification can be solved as linear problems in a low dimensional linear 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. While a variety of methods exist for point cloud registration, the method proposed in this paper is notably different as registration is achieved by a closed form solution that employs the UME low dimensional representation of the shapes to be registered.