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

Multi-scale Non-Rigid Point Cloud Registration Using Robust Sliced-Wasserstein Distance via Laplace-Beltrami Eigenmap