We present a simple, fast, and smooth scheme to approximate Algebraic Point Set Surfaces using non-compact kernels, which is particularly suited for filtering and reconstructing point sets presenting large missing parts. Our key idea is to consider a moving level-of-detail of the input point set which is adaptive w.r.t. to the evaluation location, just such as the samples weights are output sensitive in the traditional moving least squares scheme. We also introduce an adaptive progressive octree refinement scheme, driven by the resulting implicit surface, to properly capture the modeled geometry even far away from the input samples. Similarly to typical compactly-supported approximations, our operator runs in logarithmic time while defining high quality surfaces even on challenging inputs for which only global optimizations achieve reasonable results. We demonstrate our technique on a variety of point sets featuring geometric noise as well as large holes.

The squared Wasserstein distance is a natural quantity to compare probability distributions in a non-parametric setting. This quantity is usually estimated with the plug-in estimator, defined via a discrete optimal transport problem which can be solved to $\epsilon$-accuracy by adding an entropic regularization of order $\epsilon$ and using for instance Sinkhorn's algorithm. In this work, we propose instead to estimate it with the Sinkhorn divergence, which is also built on entropic regularization but includes debiasing terms. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels, of order $\epsilon^{1/2}$, which leads to improved computational complexity bounds and a strong speedup in practice. Our theoretical analysis covers the case of both randomly sampled densities and deterministic discretizations on uniform grids. We also propose and analyze an estimator based on Richardson extrapolation of the Sinkhorn divergence which enjoys improved statistical and computational efficiency guarantees, under a condition on the regularity of the approximation error, which is in particular satisfied for Gaussian densities. We finally demonstrate the efficiency of the proposed estimators with numerical experiments.

In this work we introduce a new dissimilarity measure for shape registration using the notion of normal cycles, a concept from geometric measure theory which allows us to generalize curvature for nonsmooth subsets of the Euclidean space. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. This approach is closely similar to previous works based on currents and varifolds [M. Vaillant and J. Glaunès, Surface matching via currents, in Information Processing in Medical Imaging, G. E. Christensen and M. Sonka, eds., Lecture Notes in Comput. Sci. 3565, Springer, Berlin, 2005, pp. 381--392; N. Charon and A. Trouvé, SIAM J. Imaging Sci., 6 (2013), pp. 2547--2580]. We derive the computational setting for discrete curves in $\mathbb{R}^3$, using the large deformation diffeomorphic metric mapping framework as the model for deformations. We present synthetic and real data experiments and compare them with the currents and varifolds approaches.