Registration of algebraic varieties using Riemannian optimization

We consider the point cloud registration problem, the task of finding a transformation between two point clouds that represent the same object but are expressed in different coordinate systems. Our approach is not based on a point-to-point correspondence, matching every point in the source point cloud to a point in the target point cloud. Instead, we assume and leverage a low-dimensional nonlinear geometric structure of the data. Firstly, we approximate each point cloud by an algebraic variety (a set defined by finitely many polynomial equations). This is done by solving an optimization problem on the Grassmann manifold, using a connection between algebraic varieties and polynomial bases. Secondly, we solve an optimization problem on the orthogonal group to find the transformation (rotation $+$ translation) which makes the two algebraic varieties overlap. We use second-order Riemannian optimization methods for the solution of both steps. Numerical experiments on real and synthetic data are provided, with encouraging results. Our approach is particularly useful when the two point clouds describe different parts of an objects (which may not even be overlapping), on the condition that the surface of the object may be well approximated by a set of polynomial equations. The first procedure -- the approximation -- is of independent interest, as it can be used for denoising data that belongs to an algebraic variety. We provide statistical guarantees for the estimation error of the denoising using Stein's unbiased estimator.