Registration of algebraic varieties using Riemannian optimization
Florentin Goyens Coralia Cartis Stéphane Chrétien
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.