Multiscale Nonrigid Point Cloud Registration Using Rotation-Invariant Sliced-Wasserstein Distance via Laplace--Beltrami Eigenmap

In this work, we propose computational models and algorithms for point cloud registration with nonrigid transformation. First, point clouds sampled from manifolds originally embedded in some Euclidean space are transformed to new point clouds embedded in $\mathbb{R}^n$ by the Laplace--Beltrami (LB) eigenmap, which is invariant under isometric transformation, using the first $n$ leading eigenvalues and corresponding eigenfunctions of the LB operator. Then we develop computational models and algorithms for registration of the transformed point clouds in a distribution/probability sense based on optimal transport, which provides both generality and flexibility for point cloud registration. In particular, we propose to use a rotation-invariant sliced-Wasserstein distance to achieve computation efficiency and handle ambiguities introduced by LB eigenmaps. By going from smaller $n$, which provides a quick and robust registration in coarse scale as well as a good initial guess for registration in finer scale, to a larger $n$, our method provides an efficient and robust multiscale nonrigid point cloud registration.