Team Geometrica

Overall Objectives
Scientific Foundations
Application Domains
New Results
Contracts and Grants with Industry
Other Grants and Activities

Section: New Results

Keywords : Computational topology, surface reconstruction, implicit surfaces, point set surfaces, surface learning, geometric probing, geometric inference.

Topological and geometric inference

Shape reconstruction from unorganized cross-sections

Participants : Jean-Daniel Boissonnat, Pooran Memari.

In this work, we consider the problem of reconstructing 2-dimensional geometric shapes from unorganized 1-dimensional cross-sections and provide theoretical guarantees. We study the problem in its full generality following the approach of Boissonnat and Memari we pursued last year, for the analogous 3D problem. We propose a new variant of this method and provide sampling conditions to guarantee that the output of the algorithm has the same topology as the original object and is close to it (for the Hausdorff distance). Although the problem of reconstructing 3D shapes from point clouds has received a lot of attention, there were no similar results for the problem of reconstructing shapes from planar cross-sections  [20] (See Figure  5 ).

Figure 5.

Top-left: unknown object. Top-right: cross sections.

Bottom-left: input of our algorithm. Bottom-right: reconstructed result (in blue).

Figure 6. An example of application to shape segmentation on a 2D domain. The segmentation function is the (normalized) diameter of the set of nearest boundary points. The barcode shows six long intervals, corresponding to the palm of the hand and to the five fingers. The results before and after merging non-persistence clusters are shown respectively to the left and to the right of the barcode.

A new framework for topological persistence

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

In collaboration with L. Guibas (Stanford University) and M. Glisse (Grenoble).

The concept of topological persistence introduced by H. Edelsbrunner et al. in 2000 is a rather general tool providing an efficient way to encode the qualitative and quantitative behavior of real-valued functions defined over topological spaces. Since its introduction, this encoding, known as the persistence diagram or barcode, has been extensively studied, specifically in topological data analysis where its stability properties allow to infer robust topological information on the studied data sets. Motivated by problems coming from topological data analysis (mainly the one considered in section 6.2.4 ), we have extended the notions of persistence and persistence diagrams to a larger setting than the one classically considered. We have proven new stability results for the persistence diagrams that lead to new applications in topological and geometric data analysis [42] (see Figure  6 ).

Persistence based algorithms for topological inference

Participants : Frédéric Chazal, Steve Oudot.

Manifold reconstruction has been extensively studied among the computational geometry community for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space Im1 $\#8477 ^d$ . However, the complexities of these methods scale up exponentially with d , which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds.

We have introduced a novel approach [28] that stands in-between reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, our algorithm combines two paradigms: greedy refinement, and topological persistence. It builds a set of landmarks iteratively, while maintaining nested pairs of complexes, whose images in Im1 $\#8477 ^d$ lie close to the data, and whose persistent homology eventually coincides with the one of the underlying shape. When the data points are sufficiently densely sampled from a smooth m -submanifold of Im1 $\#8477 ^d$ , our method retrieves the homology of the submanifold in time at most c( m) n5 , where n is the size of the input and c( m) is a constant depending solely on m . It can also provably well handle a wide range of compact subsets of Im1 $\#8477 ^d$ , though with higher complexities.

Topological analysis of scalar fields defined over point cloud data

Participants : Frédéric Chazal, Steve Oudot.

In collaboration with L. Guibas and P. Skraba (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 subset L of sample points, whose pairwise distances in X are given? We have provided a positive answer to this question [43] . More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we have introduced 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. These algorithms are simple and easy to implement, have reasonable complexities, and come with theoretical guarantees. To illustrate the generality and practicality of the approach, we also obtained experimental results in various applications, ranging from clustering to sensor networks.

Homology inference in the context of sensor networks

Participant : Steve Oudot.

In collaboration with L. Guibas (Stanford University) J. Gao (Stony Brook), and Y. Wang (Stony Brook).

In this work, we investigate the problem of inferring the homology of the domain underlying a sensor network from the sole knowledge of the connectivity between sensors. This problem has recevied a lot of attention in the recent years, and a number of partial solutions have been developed. We propose a complete and provably-good solution to the problem for the special case where the domain underlying the sensors is planar [33] .

We first introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point x in a domain X , the systolic feature size of X at x measures half the length of the shortest loop through x that is not null-homotopic in X . The resort to an intrinsic metric makes the systolic feature size rather insensitive to the local geometry of the domain, in contrast with its predecessors (local feature size, weak feature size, homology feature size). This reduces the number of samples required to capture the topology of X .

Under reasonable sampling conditions, we show that the geodesic Delaunay triangulation DX( L) of a finite sampling L of a bounded planar domain X is homotopy equivalent to X . Moreover, under similar conditions, DX( L) is sandwiched between the geodesic witness complex WX( L) and a relaxed version Im2 ${W_X^\#955 {(L)}}$ . Taking advantage of this fact, we prove that the homology of DX( L) (and hence the one of X ) can be retrieved by computing the persistent homology between WX( L) and Im2 ${W_X^\#955 {(L)}}$ .

We then investigate further and show that the homology of X can also be recovered from the persistent homology associated with inclusions of type Im3 ${W_X^\#955 {(L)}\#8618 W_X^\#955 ^'{(L)}}$ , under some conditions on the relaxation parameters $ \lambda$$ \le$$ \lambda$' . Similar results are obtained for Vietoris-Rips complexes as well. Our proofs draw some connections with recent advances on the front of homology inference from point cloud data, but also with several well-known concepts of Riemannian (and even metric) geometry.

On the algorithmic front, we propose algorithms for estimating the systolic feature size of a sampled planar domain X , selecting a landmark set of sufficient density, building its geodesic Delaunay triangulation, and computing the homology of X using geodesic witness complexes or Rips complexes. We also perform some experimental simulations that corroborate our theoretical results.

Extending persistence using Poincaré and Lefschetz duality

Participant : David Cohen-Steiner.

In collaboration with H. Edelsbrunner and J. Harer (Duke University).

Figure 7. Example of an extended persistence diagram.

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 Im4 $\#8477 ^3$ . In this paper, we extend persistence to essential homology classes (see figure 7 for an example of extended persistence diagram), present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to non-manifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds [17] .

Computing geometry aware handle and tunnel loops in 3D models

Participant : David Cohen-Steiner.

In collaboration with T.K. Dey, K. Li (Ohio State University) and J. Sun (Stanford University).

Figure 8. Homology generators selected by our algorithm.

Many applications such as topology repair, model editing, surface parametrization, 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 considering geometry is easy but may not be very useful. In this paper, we strike a balance by computing topologically correct loops that are also geometrically relevant (see figure 8 for sample results). Our algorithm is a novel application of the concepts from topological persistence introduced recently in computational topology. The usability of the computed loops is demonstrated with some examples in feature identification and topology simplification [18] .


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