Steve Oudot

Manifold reconstruction in arbitrary dimensions using witness complexes

It is a well-established fact that the witness complex is closelyrelated to the restricted Delaunay triangulation in lowdimensions. Specifically, it has been proved that the witness complexcoincides with the restricted Delaunay triangulation on curves, and isstill a subset of it on surfaces, under mild samplingassumptions. Unfortunately, these results do not extend tohigher-dimensional manifolds, even under stronger samplingconditions. In this paper, we show how the sets of witnesses andlandmarks can be enriched, so that the nice relations that existbetween both complexes still hold on higher-dimensional manifolds. Wealso use our structural results to devise an algorithm thatreconstructs manifolds of any arbitrary dimension or co-dimension atdifferent scales. The algorithm combines a farthest-point refinementscheme with a vertex pumping strategy. It is very simple conceptually,and it does not require the input point sample W to be sparse. Itstime complexity is bounded by c(d) |W|2, where c(d) is a constantdepending solely on the dimension d of the ambient space.

Provably good sampling and meshing of Lipschitz surfaces

In the last decade, a great deal of work has been devoted to the elaboration of a sampling theory for smooth surfaces. The goal was to ensure a good reconstruction of a given surface S from a finite subset E of S. The sampling conditions proposed so far offer guarantees provided that E is sufficiently dense with respect to the local feature size of S, which can be true only if S is smooth since the local feature size vanishes at singular points.In this paper, we introduce a new measurable quantity, called the Lipschitz radius, which plays a role similar to that of the local feature size in the smooth setting, but which is well-defined and positive on a much larger class of shapes. Specifically, it characterizes the class of Lipschitz surfaces, which includes in particular all piecewise smooth surfaces such that the normal deviation is not too large around singular points.Our main result is that, if S is a Lipschitz surface and E is a sample of S such that any point of S is at distance less than a fraction of the Lipschitz radius of S, then we obtain similar guarantees as in the smooth setting. More precisely, we show that the Delaunay triangulation of E restricted to S is a 2-manifold isotopic to S lying at bounded Hausdorff distance from S, provided that its facets are not too skinny.We further extend this result to the case of loose samples. As an application, the Delaunay refinement algorithm we proved correct for smooth surfaces works as well and comes with similar guarantees when applied to Lipschitz surfaces.

Reconstruction using witness complexes

We present a novel reconstruction algorithm that, given an input point set sampled from an objectxa0S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and inxa03d. Specifically, we prove that both complexes are equal in 2d and closely related inxa03d, under some mild sampling assumptions.

Geodesic delaunay triangulations in bounded planar domains

We introduce a new feature size for bounded domains in the plane endowed with an intrinsic metric. Given a point <i>x</i> in a domain <i>X</i>, the <i>systolic feature size</i> of <i>X</i> at <i>x</i> measures half the length of the shortest loop through <i>x</i> that is not null-homotopic in <i>X</i>. 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 <i>X</i>, provided that a reliable approximation to the intrinsic metric of <i>X</i> is available. Under sufficient sampling conditions involving the systolic feature size, we show that the geodesic Delaunay triangulation <i>D</i>_<i>x</i>(<i>L</i>) of a finite sampling <i>L</i> is homotopy equivalent to <i>X</i>. Under similar conditions, <i>D</i>_<i>x</i>(<i>L</i>) is sandwiched between the geodesic witness complex <i>C</i>^<i>W</i>_<i>X</i>(<i>L</i>) and a relaxed version <i>C</i>^<i>W</i>_<i>X,ν</i>(<i>L</i>). In the conference version of the article, we took advantage of this fact and proved that the homology of <i>D</i>_<i>x</i>(<i>L</i>) (and hence the one of <i>X</i>) can be retrieved by computing the persistent homology between <i>C</i>^<i>W</i>_<i>X</i>(<i>L</i>) and <i>C</i>^<i>W</i>_<i>X,ν</i>(<i>L</i>). Here, we investigate further and show that the homology of <i>X</i> can also be recovered from the persistent homology associated with inclusions of type <i>C</i>^<i>W</i>_<i>X,ν</i>(<i>L</i>)↪<i>C</i>^<i>W</i>_{<i>X,ν′</i>}(<i>L</i>), under some conditions on the parameters ν≤ν′. Similar results are obtained for Vietoris-Rips complexes in the intrinsic metric. The 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 bounded planar domain <i>X</i>, selecting a landmark set of sufficient density, and computing the homology of <i>X</i> using geodesic witness complexes or Rips complexes.