Mesh-RTUME: Universal Manifold Embedding for Estimating 3D Rigid Transformations of Surfaces

We consider the problems of estimating the underlying transformation and the detection of 3-D objects undergoing rigid transformations. It has been shown that the Rigid Transformation Universal Manifold Embedding (RTUME) provides a mapping from the set of all possible observations on some object to a transformation covariant matrix representation, such that its column space is invariant to the geometric transformation. In this paper, we re-derive and adapt the RTUME for the case where the observations are in the form of meshed surfaces. We prove that by evaluating the integrals that define the RTUME operator as surface integrals on the mesh representation of the observed surface, the invariance and covariance properties of the RTUME matrix representation hold, similarly to the case of point cloud observations. It is shown that the RTUME matrix representation can be efficiently evaluated using a barycentric coordinate representation of the observed surface mesh representation. The proposed Mesh-RTUME is shown to outperform the RTUME representation, evaluated from the point cloud representation of the surface, both in transformation estimation and in keypoint detection.