In this paper, we investigate the properties of the Sliced Wasserstein Distance (SW) when employed as an objective functional. The SW metric has gained significant interest in the optimal transport and machine learning literature, due to its ability to capture intricate geometric properties of probability distributions while remaining computationally tractable, making it a valuable tool for various applications, including generative modeling and domain adaptation. Our study aims to provide a rigorous analysis of the critical points arising from the optimization of the SW objective. By computing explicit perturbations, we establish that stable critical points of SW cannot concentrate on segments. This stability analysis is crucial for understanding the behaviour of optimization algorithms for models trained using the SW objective. Furthermore, we investigate the properties of the SW objective, shedding light on the existence and convergence behavior of critical points. We illustrate our theoretical results through numerical experiments.

Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probabilitydistribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points whichminimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm in which Voronoi cells are replaced by Power cells, leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, which moreover admits spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a sort of Poliak-Łojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space and is shown to be bi-Hölder continuous. It enables the direct use of generic supervised and unsupervised learning algorithms on measure data consistently w.r.t. the Wasserstein geometry.

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of $\mathbb{R}^d$ and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e., on a union of simplices of arbitrary dimension between 2 and $d$. As in [F. Aurenhammer, F. Hoffman, and B. Aronov, Algorithmica, 20 (1998), pp. 61--76] we recast this optimal transport problem as the resolution of a nonlinear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this nonlinear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup, and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in ${R}^3$ to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing, or rigid point set registration on a mesh.

We present in this paper a generic and parameter-free algorithm to efficiently build a wide variety of optical components, such as mirrors or lenses, that satisfy some light energy constraints. In all of our problems, one is given a collimated or point light source and a desired illumination after reflection or refraction and the goal is to design the geometry of a mirror or lens which transports exactly the light emitted by the source onto the target. We first propose a general framework and show that eight different optical component design problems amount to solving a light energy conservation equation that involves the computation of visibility diagrams. We then show that these diagrams all have the same structure and can be obtained by intersecting a 3D Power diagram with a planar or spherical domain. This allows us to propose an efficient and fully generic algorithm capable to solve these eight optical component design problems. The support of the prescribed target illumination can be a set of directions or a set of points located at a finite distance. Our solutions satisfy design constraints such as convexity or concavity. We show the effectiveness of our algorithm on simulated and fabricated examples.

The Voronoi covariance measure (VCM) of a compact set $K$ of $\mathbb{R}^d$ is a tensor-valued measure that encodes geometrical information on $K$ and which is known to be resilient to Hausdorff noise but sensitive to outliers. In this paper, we generalize this notion to any distance-like function $\delta$ and define the $\delta$-VCM. Combining the VCM with the distance to a measure and also with the witnessed-$k$-distance, we get a provably good tool for normal estimation that is resilient to Hausdorff noise and to outliers. We present experiments showing the robustness of our approach for normal and curvature estimation and sharp feature detection.

We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error η, we provide an upper bounds on the number of probes that one has to perform in order to obtain an η-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem.

In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Ω(N2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem.

Consider a set P of N random points on the unit sphere of dimension d-1, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that after appropriate rescaling this halving polyhedron is Hausdorff close to the unit ball with high probability, as soon as the number of points grows like Omega(d log(d)). From this result, we deduce probabilistic lower bounds on the complexity of approximations of the distance to the empirical measure on the point set by distance-like functions.

Abstract We introduce a novel method for non‐rigid shape matching, designed to address the symmetric ambiguity problem present when matching shapes with intrinsic symmetries. Unlike the majority of existing methods which try to overcome this ambiguity by sampling a set of landmark correspondences, we address this problem directly by performing shape matching in an appropriate quotient space, where the symmetry has been identified and factored out. This allows us to both simplify the shape matching problem by matching between subspaces, and to return multiple solutions with equally good dense correspondences. Remarkably, both symmetry detection and shape matching are done without establishing any landmark correspondences between either points or parts of the shapes. This allows us to avoid an expensive combinatorial search present in most intrinsic symmetry detection and shape matching methods. We compare our technique with state‐of‐the‐art methods and show that superior performance can be achieved both when the symmetry on each shape is known and when it needs to be estimated.

Distance function to a compact set plays a central role in several areas of computational geometry. Methods that rely on it are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.

Abstract In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth‐mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.

We present an efficient and robust method for extracting curvature information, sharp features, and normal directions of a piecewise smooth surface from its point cloud sampling in a unified framework. Our method is integral in nature and uses convolved covariance matrices of Voronoi cells of the point cloud which makes it provably robust in the presence of noise. We show that these matrices contain information related to curvature in the smooth parts of the surface, and information about the directions and angles of sharp edges around the features of a piecewise-smooth surface. Our method is applicable in both two and three dimensions, and can be easily parallelized, making it possible to process arbitrarily large point clouds, which was a challenge for Voronoi-based methods. In addition, we describe a Monte-Carlo version of our method, which is applicable in any dimension. We illustrate the correctness of both principal curvature information and feature extraction in the presence of varying levels of noise and sampling density on a variety of models. As a sample application, we use our feature detection method to segment point cloud samplings of piecewise-smooth surfaces.

Abstract A common operation in many geometry processing algorithms consists of finding correspondences between pairs of shapes by finding structure‐preserving maps between them. A particularly useful case of such maps is isometries, which preserve geodesic distances between points on each shape. Although several algorithms have been proposed to find approximately isometric maps between a pair of shapes, the structure of the space of isometries is not well understood. In this paper, we show that under mild genericity conditions, a single correspondence can be used to recover an isometry defined on entire shapes, and thus the space of all isometries can be parameterized by one correspondence between a pair of points. Perhaps surprisingly, this result is general, and does not depend on the dimensionality or the genus, and is valid for compact manifolds in any dimension. Moreover, we show that both the initial correspondence and the isometry can be recovered efficiently in practice. This allows us to devise an algorithm to find intrinsic symmetries of shapes, match shapes undergoing isometric deformations, as well as match partial and incomplete models efficiently.

Abstract We address the problem of generating quality surface triangle meshes from 3D point clouds sampled on piecewise smooth surfaces. Using a feature detection process based on the covariance matrices of Voronoi cells, we first extract from the point cloud a set of sharp features. Our algorithm also runs on the input point cloud a reconstruction process, such as Poisson reconstruction, providing an implicit surface. A feature preserving variant of a Delaunay refinement process is then used to generate a mesh approximating the implicit surface and containing a faithful representation of the extracted sharp edges. Such a mesh provides an enhanced trade‐off between accuracy and mesh complexity. The whole process is robust to noise and made versatile through a small set of parameters which govern the mesh sizing, approximation error and shape of the elements. We demonstrate the effectiveness of our method on a variety of models including laser scanned datasets ranging from indoor to outdoor scenes.

Many algorithms for shape analysis and shape processing rely on accurate estimates of differential information such as normals and curvature. In most settings, however, care must be taken around non-smooth areas of the shape where these quantities are not easily defined. This problem is particularly prominent with point-cloud data, which are discontinuous everywhere. In this paper we present an efficient and robust method for extracting principal curvatures, sharp features and normal directions of a piecewise smooth surface from its point cloud sampling, with theoretical guarantees. Our method is integral in nature and uses convolved covariance matrices of Voronoi cells of the point cloud which makes it provably robust in the presence of noise. We show analytically that our method recovers correct principal curvatures and principal curvature directions in smooth parts of the shape, and correct feature directions and feature angles at the sharp edges of a piecewise smooth surface, with the error bounded by the Hausdorff distance between the point cloud and the underlying surface. Using the same analysis we provide theoretical guarantees for a modification of a previously proposed normal estimation technique. We illustrate the correctness of both principal curvature information and feature extraction in the presence of varying levels of noise and sampling density on a variety of models.