  In this work, we propose computational models and algorithms for point cloud
registration with non-rigid transformation. First, point clouds sampled from
manifolds originally embedded in some Euclidean space $\mathbb{R}^D$ are
transformed to new point clouds embedded in $\mathbb{R}^n$ by
Laplace-Beltrami(LB) eigenmap using the $n$ leading eigenvalues and
corresponding eigenfunctions of LB operator defined intrinsically on the
manifolds. The LB eigenmap are invariant under isometric transformation of the
original manifolds. Then we design computational models and algorithms for
registration of the transformed point clouds in distribution/probability form
based on the optimal transport theory which provides both generality and
flexibility to handle general point clouds setting. Our methods use robust
sliced-Wasserstein distance, which is as the average of projected Wasserstein
distance along different directions, and incorporate a rigid transformation to
handle ambiguities introduced by the Laplace-Beltrami eigenmap. By going from
smaller $n$, which provides a quick and robust registration (based on coarse
scale features) as well as a good initial guess for finer scale registration,
to a larger $n$, our method provides an efficient, robust and accurate approach
for multi-scale non-rigid point cloud registration.
