Team Geometrica

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Scientific Foundations
Application Domains
New Results
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Section: New Results

Topological and Geometric Inference

Proximity of Persistence Modules and their Diagrams

Participants : Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Steve Oudot.

This work has been done in collaboration with L. J. Guibas (Stanford University).

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 spaces. In this work, we present new stability results that do not suffer from the above restrictions. Furthermore, by working at an algebraic level directly, we make it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence [29] .

Analysis of Scalar Fields over Point Cloud Data and Clustering

Participants : Frédéric Chazal, Steve Oudot, Primoz Skraba.

This work has been done in collaboration with L. J. Guibas (Stanford University).

Given a real-valued function f defined over some metric space X, is it possible to recover some structural information about f from the sole information of its values at a finite set L of sample points on X, whose pairwise distances in X are given? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data and for the clustering of point cloud data sets [30] , [57] .

Gromov-Hausdorff Stable Signatures for Shapes using Persistence

Participants : Frédéric Chazal, David Cohen-Steiner, Steve Oudot.

This work has been done in collaboration with L. J. Guibas and F. Mémoli (Stanford University).

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 measures. Our signatures are well-suited for the study of unstructured point cloud data, which we illustrate through an application in shape classification [16] .

Stability of Curvature Measures

Participants : Frédéric Chazal, David Cohen-Steiner.

This work has been done in collaboration with A. Lieutier (Dassault Systèmes) and B. Thibert (Grenoble University).

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 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 $ \mu$ -reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature [18] .

Persistent Homology for Images, Kernels, and Cokernels

Participant : David Cohen-Steiner.

This work has been done in collaboration with H. Edelsbrunner, J. Harer, and D. Morozov (Duke University).

We extend the notion of persistent homology to sequences of images, kernels, and cokernels of maps induced by inclusion in a filtration of pairs of spaces. Specifically, we note that persistence in this context is well defined, we prove that the persistence diagrams are stable, and we explain how to compute them. We also show that image persistence diagrams allow to deal with the case where both the function and its domain are corrupted by noise, whereas classical persistence can only cope with function noise. In particular, we provide an efficient way to estimate the persistent homology of a function known only at a finite set of points [31] .

Robust Voronoi-based Feature and Curvature Estimation

Participant : Quentin Mérigot.

This work has been done in collaboration with L. Guibas and M. Ovsjanikov (Stanford).

We present an efficient and robust method for extracting principal curvatures, sharp features and normal directions of a piecewise smooth surface from a point cloud sampling, with theoretical guarantees. Our method is integral 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 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 [32] .

Numerical Methods for Surface Reconstruction from Point Sets

Participant : Pierre Alliez.

The increasing amount of uncertainty in measurement point sets has motivated a number of approximating surface reconstruction approaches which compute an implicit function such that one of its isosurfaces approximate well the input points. In this article [36] we discuss two numerical reconstruction methods. The first method computes an implicit function through solving for the Poisson problem from points which are enriched with oriented normals. The second method computes an implicit function through solving for a generalized eigenvalue problem and does not require orienting the data points beforehand. These approaches are positioned with respect to recent approaches geared toward an increased level of robustness to noise, sparse sampling and outliers.

Manifold Reconstruction using the Tangential Complex

Participants : Arijit Ghosh, Jean-Daniel Boissonnat.

We give a provably correct algorithm to reconstruct a k -dimensional manifold embedded in d -dimensional Euclidean space [52] . Input to our algorithm is a point sample coming from an unknown manifold. Unlike previous methods, we do not construct any subdivision of the embedding d -dimensional space. As a result, the running time of our algorithm depends only linearly on the extrinsic dimension d while it depends quadratically on the size of the input sample, and exponentially on the intrinsic dimension k . To the best of our knowledge, this is the first certified algorithm for manifold reconstruction whose complexity depends linearly on the ambient dimension. We also prove that for a dense enough sample the output of our algorithm is isotopic to the manifold and a close geometric approximation of the manifold.

Geometric Tomography with Guarantees

Participants : Pooran Memari, Jean-Daniel Boissonnat.

In collaboration with Omid Amini, CNRS-DMA, ENS.

We consider the problem of reconstructing an embedded compact 3-manifold (with boundary) in Im3 $\#8477 ^3$ from its cross-sections with a given set of cutting planes having arbitrary orientations. Under appropriate sampling conditions that are satisfied when the set of cutting planes is dense enough, we prove that the algorithm presented by Liu et al., as well as a related algorithm based on the Voronoi diagram of the cross-sections, preserves the homotopy type of the object [49] . Using the homotopy equivalence, we also show that the reconstructed object is homeomorphic to the original object. To the best of our knowledge, this is the first time that 3D shape reconstruction from cross-sections comes with such theoretical guarantees.

Surface Reconstruction from Multi-View Stereo

Participants : Nader Salman, Mariette Yvinec.

We describe an original method to reconstruct a 3D scene from a sequence of images. See Figure 5 . Our approach uses both the dense 3D point cloud extracted by multi-view stereovision and the calibrated images. It combines depth-maps construction in the image planes with surface reconstruction through restricted Delaunay triangulation. The method may handle very large scale outdoor scenes. Its accuracy has been tested on numerous scenes including the dense multi-view benchmark proposed by Strecha et al. Our results compare favorably with the current state of the art [34] , [33] .

Figure 5. On the left: two images of the Aiguille du midi dataset (©B.Vallet/IMAGINE); below: our low resolution reconstruction (270 000 triangles). On the right: our high resolution reconstruction (1 000 000 triangles). Note how most details pointed by yellow arrows are recovered in both resolutions.

Manifold Reconstruction in Arbitrary Dimensions using Witness Complexes

Participants : Jean-Daniel Boissonnat, Steve Oudot.

This work has been done in collaboration with Leonidas J. Guibas from Stanford University.

It is a now well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling assumptions. Unfortunately, these results do not extend to higher-dimensional manifolds, even under stronger sampling conditions. In this work, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between both complexes still hold on higher-dimensional manifolds. We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its time complexity is bounded by c(d)n2 , where n is the size of the input and c(d) is a constant depending solely on the dimension d of the ambient space.


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