Abstract The estimation of differential quantities on oriented point cloud is a classical step for many geometry processing tasks in computer graphics and vision. Even if many solutions exist to estimate such quantities, they usually fail at satisfying both a stable estimation with theoretical guarantee, and the efficiency of the associated algorithm. Relying on the notion of corrected curvature measures [LRT22, LRTC20] designed for surfaces, the method introduced in this paper meets both requirements. Given a point of interest and a few nearest neighbours, our method estimates the whole curvature tensor information by generating random triangles within these neighbours and normalising the corrected curvature measures by the corrected area measure. We provide a stability theorem showing that our pointwise curvatures are accurate and convergent, provided the noise in position and normal information has a variance smaller than the radius of neighbourhood. Experiments and comparisons with the state‐of‐the‐art confirm that our approach is more accurate and much faster than alternatives. The method is fully parallelizable, requires only one nearest neighbour request per point of computation, and is trivial to implement.

Abstract A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ3 × 𝕊2 by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed‐form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale‐space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state‐of‐the‐art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.

In this paper we propose an optimization-based framework to multiple object matching. The framework takes maps computed between pairs of objects as input, and outputs maps that are consistent among all pairs of objects. The central idea of our approach is to divide the input object collection into overlapping sub-collections and enforce map consistency among each sub-collection. This leads to a distributed formulation, which is scalable to large-scale datasets. We also present an equivalence condition between this decoupled scheme and the original scheme. Experiments on both synthetic and real-world datasets show that our framework is competitive against state-of-the-art multi-object matching techniques.

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.

Many real-world tasks for autonomous agents benefit from understanding dynamic inter-object interactions. Detecting, analyzing and differentiating between the various ways that an object can be interacted with provides implicit information about its function. This can help train autonomous agents to handle objects and understand unknown scenes. We describe a general mathematical framework to analyze and classify interactions, defined as dynamic motions performed by an active object onto a passive one. We factorize interactions via motion features computed in the spatio-temporal domain, and encoded into a global, object-centric signature. Equipped with a similarity measure to compare such signatures, we showcase classification of interactions with a single object. We also propose a novel acquisition setup combining RGBD sensing with a virtual reality (VR) display, to capture interactions with purely virtual objects.

Abstract We consider the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar. When the shapes are quite different, a point‐to‐point map is not always appropriate, so our focus in this paper is a method to build a set of correspondences between shape regions or parts. The proposed approach exploits a variety of feature functions on the shapes and makes use of the key observation that points in matching parts have similar ranks in the sorting of the corresponding feature values. Our algorithm proceeds in two steps. We first build an affinity matrix between points on the two shapes, based on feature rank similarity over many feature functions. We then define a notion of stability of a pair of regions, with respect to this affinity matrix, obtained as a fixed point of a nonlinear operator. Our method yields a family of corresponding maximally stable regions between the two shapes that can be used to define shape parts. We observe that this is an instance of the biclustering problem and that it is related to solving a constrained maximal eigenvalue problem. We provide an algorithm to solve this problem that mimics the power method. We show the robustness of its output to noisy input features as well its convergence properties. The obtained part correspondences are shown to be almost perfect matches in the isometric case, and also semantically appropriate even in non‐isometric cases. We provide numerous examples and applications of this technique, for example to sharpening correspondences in traditional shape matching algorithms.

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.

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.

Abstract We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the Gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive μ‐reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive μ‐reach can thus be approximated by the curvature measures of the offset of a point‐cloud sample.

It has been observed for a long time that the operation consisting of offsetting a solid by a quantity r and then offsetting its complement by d < r produces, in some cases, a new solid with the same topology but with a smooth boundary. While this fact has been widely used in Computer Aided Geometric Design or in the field of image processing, we provide here for the first time a tight and robust condition that guarantees the smoothness of the new solid and gives a lower bound on its reach (distance to the medial axis). This condition is based on the general properties of the distance function to a compact set and relies on the recently introduced critical function and μ-reach.

BACKGROUND: Phylogenetic approaches are commonly used to predict which amino acid residues are critical to the function of a given protein. However, such approaches display inherent limitations, such as the requirement for identification of multiple homologues of the protein under consideration. Therefore, complementary or alternative approaches for the prediction of critical residues would be desirable. Network analyses have been used in the modelling of many complex biological systems, but only very recently have they been used to predict critical residues from a protein's three-dimensional structure. Here we compare a couple of phylogenetic approaches to several different network-based methods for the prediction of critical residues, and show that a combination of one phylogenetic method and one network-based method is superior to other methods previously employed. RESULTS: We associate a network with each member of a set of proteins for which the three-dimensional structure is known and the critical residues have been previously determined experimentally. We show that several network-based centrality measurements (connectivity, 2-connectivity, closeness centrality, betweenness and cluster coefficient) accurately detect residues critical for the protein's function. Phylogenetic approaches render predictions as reliable as the network-based measurements, although, interestingly, the two general approaches tend to predict different sets of critical residues. Hence we propose a hybrid method that is composed of one network-based calculation--the closeness centrality--and one phylogenetic approach--the Conseq server. This hybrid approach predicts critical residues more accurately than the other methods tested here. CONCLUSION: We show that network analysis can be used to improve the prediction of amino acids critical for protein function, when utilized in combination with phylogenetic approaches. It is proposed that such improvement is due to the complementary nature of these approaches: network-based methods tend to predict as critical those residues that are highly connected and internal (i.e., non-surface), although some surface residues are indeed identified as critical by network analyses; whereas residues chosen by phylogenetic approaches display a lower overall probability of being surface inaccessible.