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63

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Introduction

**Skills and Expertise**

## Publications

Publications (63)

Given an input 3D geometry such as a triangle soup or a point set, we address the problem of generating a watertight and orientable surface triangle mesh that strictly encloses the input. The output mesh is obtained by greedily refining and carving a 3D Delaunay triangulation on an offset surface of the input, while carving with empty balls of radi...

In this paper, we provide a numerical tool to study a material’s coherence from a set of 2D Lagrangian trajectories sampling a dynamical system, i.e., from the motion of passive tracers. We show that eigenvectors of the Burau representation of a topological braid derived from the trajectories have levelsets corresponding to components of the Nielse...

The spectrum of a network or graph $G=(V,E)$ with adjacency matrix $A$, consists of the eigenvalues of the normalized Laplacian $L= I - D^{-1/2} A D^{-1/2}$. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of...

Geometric inference deals with the problem of recovering the geometry and topology of a compact subset K of \(\mathbb{R}^{d}\) from an approximation by a finite set P. This problem has seen several important developments in the previous decade. Many of the proposed constructions share a common feature: they estimate the geometry of the underlying c...

We introduce an efficient computational method for generating dense and low distortion maps between two arbitrary surfaces of same genus. Instead of relying on semantic correspondences or surface parameterization, we directly optimize a variance-minimizing transport plan between two input surfaces that defines an as-conformal-as-possible inter-surf...

We introduce in this paper an algorithm that generates from an input tolerance volume a surface triangle mesh guaranteed to be within the tolerance, intersection free and topologically correct. A pliant meshing algorithm is used to capture the topology and discover the anisotropy in the input tolerance volume in order to generate a concise output....

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined
with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a
function of their genus. More precisely, we show the existence of a universal constants C such that the k-th
eigenvalue λ_k of the normalized Laplacian...

We describe a framework for robust shape reconstruction from raw point sets, based on optimal transportation between measures, where the input point sets are seen as distribution of masses. In addition to robustness to defect-laden point sets, hampered with noise and outliers, our approach can reconstruct smooth closed shapes as well as piecewise s...

We propose a noise‐adaptive shape reconstruction method specialized to smooth, closed shapes. Our algorithm takes as input a defect‐laden point set with variable noise and outliers, and comprises three main steps. First, we compute a novel noise‐adaptive distance function to the inferred shape, which relies on the assumption that the inferred shape...

Notre approche consiste à considérer le nuage de points en entrée comme une mesure discrète (une distribution de masses), et à construire une approximation par une mesure continue (et constante par morceaux)sur les faces d’un complexe simplicial. La distance entre les deux mesures est calculée par une approximation du transport optimal obtenue par...

We introduce a robust and feature-capturing surface reconstruction and simplification method that turns an input point set into a low triangle-count simplicial complex. Our approach starts with a (possibly non-manifold) simplicial complex filtered from a 3D Delaunay triangulation of the input points. This initial approximation is iteratively simpli...

Reconstructing a 3D shape from sample points is a central problem faced in medical applications, reverse engineering, natural sciences, cultural heritage projects, etc. While these applications motivated intense research on 3D surface reconstruction, the problem of reconstructing more general shapes hardly received any attention. This paper develop...

Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal
of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset
from the approximating point cloud data. It appears that the study of distance functions allows one to ad...

We propose a robust 2D shape reconstruction and simplification algorithm which takes as input a defect‐laden point set with noise and outliers. We introduce an optimal‐transport driven approach where the input point set, considered as a sum of Dirac measures, is approximated by a simplicial complex considered as a sum of uniform measures on 0‐ and...

Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the k-nearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong appr...

In many practical situations, the object of study is only known through a finite set of possibly noisy sample points. A natural question is then to recover the geometry and the topology of the unknown object from this information. The most classical example is probably surface reconstruction, where the points

We propose a modular framework for robust 3D reconstruction from unorganized, unoriented, noisy, and outlierridden geometric data. We gain robustness and scalability over previous methods through an unsigned distance approximation to the input data followed by a global stochastic signing of the function. An isosurface reconstruction is finally dedu...

We study the boundary measures of compact subsets of the d-dimensional Euclidean space, which are closely related to Federer’s curvature measures. We show that they can be computed efficiently for point clouds and suggest that these measures can be used for geometric
inference. The main contribution of this work is the proof of a quantitative stabi...

We prove two stability results for Lipschitz functions on triangulable, compact metric spaces and consider applications of
both to problems in systems biology. Given two functions, the first result is formulated in terms of the Wasserstein distance
between their persistence diagrams and the second in terms of their total persistence.
KeywordsConti...

Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (Betti numbers, curvatures,...) of this subset from the approximating point cloud data. In recent years, it appeared that the study of distance functio...

This work adresses the problem of the approximation of the normals of the offsets of general compact sets in euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the $\mu$-reach of the compact set, a recently int...

We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov-Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with me...

This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring that the zero-set of a smooth function and the one of a piecewise linear approximation of it are isotopic. Then, we deduce from this criterion an implicit surface meshing algorithm certifying that the output mesh is isotopic t...

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of
shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual
homology classes. For the sublevelset filtration of a surface...

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 $\mu$-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdo...

Motivated by the measurement of local homology and of functions on noisy domains, we extend the notion of persistent homology to sequences of kernels, images, and cokernels of maps induced by inclusions in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams...

Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable s...

Many applications such as topology repair, model editing, surface parameterization, and feature recognition benefit from computing loops on surfaces that wrap around their 'handles' and 'tunnels'. Computing such loops while optimizing their geometric lengths is difficult. On the other hand, computing such loops without con-sidering geometry is easy...

We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a suff...

We introduce the boundary measure at scale r of a compact subset of the n-dimensional Euclidean space. We show how it can be computed for point clouds and suggest these measures can be used for feature detection. The main contribution of this work is the proof a quantitative stability theorem for boundary measures using tools of convex analysis and...

It has been observed for a long time that the operation consisting of offseting a solid by a quantity $r$ and then offseting 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 p...

We introduce an algorithm for reconstructing watertight surfaces from unoriented point sets. Using the Voronoi diagram of the input point set, we deduce a tensor field whose principal axes and eccentricities locally represent respectively the most likely direction of the normal to the surface, and the confidence in this direction estimation. An imp...

Using the theory of normal cycles, we associate with each geometric subset of a Riemannian manifold a —tensor-valued— curvature measure, which we call its second fundamental measure. This measure provides a finer description of the geometry of singular sets than the standard curvature measures. Moreover, we deal with approximation of curvature meas...

Meshing algorithms can be roughly characterized as (i) continuation-based methods, that grow a mesh following the surface, and (ii) mesh-based methods, which build some sort of three-dimensional scaffolding around the surface. Although continuation-based methods are often used in practice, it is not easy to achieve correctness guarantees for them....

We introduce a framework for quadrangle meshing of discrete manifolds. Based on discrete differential forms, our method hinges on extending the discrete Laplacian operator (used extensively in modeling and animation) to allow for line singularities and singularities with fractional indices. When assembled into a singularity graph, these line singul...

Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its per- tinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in (10) computes the pairs from an ordering of the simpl...

We introduce a parameterized notion of feature size that interpolates between the minimum of the local feature size and the
recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to
noisy samplings of general compact sets in euclidean space. These conditions are sufficient to ensure th...

Point clouds and meshes are ubiquitous in computational geometry and its applications. These subsets of Euclidean space represent in general smooth objects with or without singularities. It is then natural to study their geometry by mimicking the differential geometry techniques adapted for smooth surfaces. The aim of the following pages is to list...

In this paper, we give a very simple and purely topological condition for two surfaces to be isotopic. This work is motivated by the problem of surface approximation. Applications to implicit surfaces are given, as well as connections with the well-known concepts of skeleton and local feature size.

In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions \italic{and} connectivity. As this energy is known to be the ${\cal L}^1$ distance between an isotropic quadra...

ABSTRACT In this paper, we bound the difference between the total mean cur- vatures of two closed surfaces in R, in terms of their total absolute curvatures and the Fr´ echet distance between,the volumes,they en- close. The proof relies on a combination,of methods,from algebraic topology and integral geometry. We also bound,the difference be- tween...

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We
prove that under mild assumptions on the function, the persistence diagram is stable: small changes in the function imply
only small changes in the diagram. We apply this result to estimating the homology of sets in a metric sp...

The tandem algorithm combines the marching cube algorithm for surface extraction and the edge contraction algorithm for surface simplification in lock-step to avoid the costly intermediate step of storing the entire extracted surface triangulation. Beyond this basic strategy, we introduce refinements to prevent artifacts in the resulting triangulat...

Achieving efficiency in mesh processing often demands that overly verbose 3D datasets be reduced to more concise, yet faithful representations. Despite numerous applications ranging from geometry compression to reverse engineering, concisely capturing the geometry of a surface remains a tedious task. In this paper, we present both theoretical and p...

We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC.The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algo...

In this paper, we propose a novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry. In particular, we use curvature directions to drive the remeshing process, mimicking the lines that artists themselves would use when creating 3D models from scratch. After extracting and...

In this paper, we present a new greedy algorithm for surface reconstruction from unorganized point sets. Starting from a seed facet, a piecewise linear surface is grown by adding Delaunay triangles one by one. The most plausible triangles are added first and in such a way as to prevent the appearance of topological singularities. The output is thus...

A rapidly growing number of applications requires to deal with three-dimensional objects on a computer. These objects are usually represented by triangulated surfaces. This thesis addresses three problems one encounters when dealing with such surfaces. We first give an algorithm which builds a volumic Delaunay triangulation containing a given trian...

We address the problem of isosurface meshing with topological guaranties. Assuming the criticalpoints of the considered function are given, we give a certified algorithm for this problem. Thisseems to be the first one in the literature.

In this paper, we propose a novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry. In particular, we use curvature directions to drive the remeshing process, mimicking the lines that artists themselves would use when creating 3D models from scratch. After extracting and...

We give a general Riemannian framework to the study of approximation of curvature measures, using the theory of the normal cycle. Moreover, we introduce a differential form which allows to define a new type of curvature measure encoding the second fundamental form of a hypersurface, and to extend this notion to geometric compact subsets of a Rieman...

This report deals with the approximation of a smooth surface M by a triangulated mesh T. We give an explicit bound on the difference of the curvature measures of M and the curvature measures of T, when T is close to M. The result is obtained by applying the theory of the normal cycle.

We address the problem of curvature estimation from sampled smooth surfaces. Building upon the theory of normal cycles, we derive a definition of the curvature tensor for polyhedral surfaces. This definition consists in a very simple and new formula. When applied to a polyhedral approximation of a smooth surface, it yields an efficient and reliable...

This report deals with approximations of geometric data defined on a hypersurf- ace of the Euclidean space E^n. Using geometric measure theory, we evaluate an upper bound on the flat norm of the difference of the normal cycle of a compact subset of E^n whose boundary is a smooth (closed oriented embedded) hypersurface, and the normal cycle of a com...

## Projects

Project (1)

The goal of the project is to define a suitable general framework to study both smooth and discrete submanifolds.