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ANR GeMfaceT

GeMfaceT: A bridge between Geometric Measure and Discrete Surface Theories



Job Advertisement: ANR-funded postdoc position - 2 years in Paris-Saclay

Application to be sent by email up to the end of May: see call.

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Overview of the project

Continuous definitions (such as those of surface, regularity, dimension, curvatures ...) generally cannot be readily given a discrete counterpart. Moreover, this discrete counterpart is generally not unique and highly scale-dependent. There are multiple ways of developing a theory for discrete surfaces and the choice of an appropriate framework is directly related to the kind of discrete data we aim to process, and for which purpose i.e. the kind of surfaces we try to model. Regarding the kind of discrete data, two different situations occur: either they have been collected in an external context and come in some given form one has to deal with, or one has the freedom to decide which discretization is best suited for his issue. Let us mention some examples of discrete representations: triangulated surfaces, digital shapes, graph representations, level sets and diffuse interfaces etc. On the continuous side, the denomination "surface" encompasses a wide variety of objects ranging from usual 2-dimensional surfaces embedded in \( \mathbb{R}^3 \) to any dimension and co-dimension submanifold, abstract Riemannian and sub-Riemannian manifolds, rectifiable sets, tree-like and graphs structures, stratified spaces etc. In this project, we propose to focus on unstructured data in the sense that we do not have any underlying parametrization or topological information associated with our data, a typical example being bf point cloud data (e.g. obtained from scan acquisition) or different kinds of diffuse approximations (as in MRI for instance). Our motivation is twofold: first, a large range of data-types initially come without any parametrization information. Moreover, let us point out that the construction of such a parametrization (for instance the definition of a triangulation starting from a point cloud) is a challenging active topic in itself, and a better understanding of unstructured data is an essential pre-processing step. Of course, working with parametrized objects has a lot of practical advantages making them quite popular. Nonetheless, it has also a well-known drawback as it does not allow to handle changes of topology and self-crossings. Furthermore, with the ever increasing amount of collected data, there is an increasing part of those that are far from lying on a nice manifold. In such a case, discrete parametrized surfaces cannot represent them properly, and the problem does not lie at the discrete level but rather in choosing a manifold model. This observation alone would explain that we propose to focus on modelling surfaces in an extended non-manifold sense (e.g. allowing objects with singularities or varying dimension). It turns out that the need for a discrete surface theory allowing to handle non regular objects already arises when addressing well-known geometric variational problems as the Plateau problem (think of a soap film spanning a cubic frame).
Geometric measure theory offers a particularly well-suited framework for the study of such unstructured discrete surfaces. The long-standing Plateau problem has given birth to several different weakenings of the notion of surface. While their common purpose was to gain compactness while preserving mass/area continuity (or lower semi-continuity at least), they actually provide consistent settings for developing a theory of discrete surfaces. Indeed, currents and normal cycles have already allowed to design convergent curvature estimators [CSM06] as well as to tackle shape registration issues [DPTA09,GR16]. On the other hand, a varifold perspective for discrete surfaces has been proposed in [CT13,Bue15,BLM17], evidencing good approximation properties in in terms of the so-called Bounded Lipschitz distance (also called flat distance) and a flexible notion of discrete mean curvature with stability and convergence property with respect to Bounded Lipschitz distance as well. These properties have been subsequently extended to the whole second fundamental form [BLM22], without restriction on dimension or co-dimension and this theory is applicable to surfaces with singularities.
This project aims at pushing forward such interactions between classical geometric measure theory and practical issues in discrete geometry.

Variability of dimension

When analyzing data in large dimensions, one often looks for the optimal lower dimension to project the data without losing important information. However, the multiscale nature of the data often makes it difficult to clearly define dimension: we rather obtain a set of dimensions depending on the quality of the approximation that we are allowing on the data. Following [Pen17], an interesting idea to tackle this variability is the notion of flags, which are series of properly embedded subspaces. Flags generalize Grassmannians (linear subspaces of fixed dimensions) and moreover guarantee that higher dimensional subspaces include the lower dimensional ones so that approximations done at different levels remain consistent, like in Principal Component Analysis. It is then natural to consider flagfolds as distributions of fields of flags, defined as linear forms integrating them, similarly to the way varifolds integrate fields of Grassmannians. One of the challenges is to adapt these concepts in a consistent way, allowing for instance a regular transition from a \(3d \)-thin tube to a 1d-line, not only in the spatial variable but also in the Grassmannian. A subsequent issue would be to adapt weak curvatures to this level. This is an ongoing project with X. Pennec.

Variability of dimension

Taking advantage of the robustness of the notion of approximate mean curvature by our previous work [BLM17], we were able in [BR22] (with M. Rumpf) to realize a mean curvature flow resulting in singular well-known soap films, that is minimal surfaces spanning a given frame. The discrete flow automatically handles structural changes through the formation of singularities as the discrete mean curvature remains well-defined. We refer to [BR22] for an insight into the potentiality of such an approach. However, on the theoretical side, such a flow suffers from the loss of variational meaning: the varifold mean curvature arises as the first variation of the area functional and this is the starting point of Brakke construction. Unfortunately, it is unclear whether there is a variational interpretation to our discrete curvature and what would be the proper discrete area functional. We intend to push forward the theoretical analysis of such discrete curvature flows in order to obtain convergence results and improve numerical experiments. This is an ongoing project with G.P. Leonardi, S. Masnou and A. Sagueni. Furthermore, some singular evolutions result in an object with mixed dimensions, for instance the mean curvature flow starting from a dumbell. Hence this objective would naturally benefit from the achievements of the first one, leading to an even larger class of singular evolutions.

Anisotropic setting

We want to understand to what extent the previous approach can be adapted to the anisotropic perimeter case, described in [DPDRG16] for varifolds, modelling crystalline growth. The important point is to identify the crucial differences induced by the anisotropy in the area functional, both in the theoretical analysis and in the potential numerical simulations. Beyond the anisotropic perimeter question, we would like to tackle sub-Riemannian settings as the Grushin plane or Heisenberg group.

Discrete Brakke Flow

In most cases, we are only provided with sets of points in \( \mathbb{R}^n \) (for some \( n \)) and we need to infer weights and approximate tangents (not to mention dimension) in order to infer the varifold structure from the data. We intend to explore the statistical aspects of deterministic curvatures estimators developed in [BLM17] in the varifold framework, in the spirit of [AL19]. As those curvatures estimators rely on varifold structure and convergence has been established with respect to weak star topology of varifolds, it is necessary to investigate varifold reconstruction from the data in this statistical framework. It is a long-time standing question that has been given multiple relevant answers. Yet, to the best of our knowledge, there is no mass estimator proven to be convergent, considering integral varifolds for instance. A better understanding of this issue is probably also closely related to the proper discrete area functional mentioned above. We hope to obtain a better understanding of the different properties involved in the convergence of curvature estimators (normal noise versus tangential noise/non uniform sampling for instance).
The whole project aims at interfacing challenging concepts from Geometric Measure Theory with practical issues in the study of Discrete Surfaces and we keep in mind other concepts from geometric measure theory such as currents or flat chains to model discrete surfaces. Which approximation properties hold, for which topology and which class of discrete objects? What curvature estimator or second order differential operator (such as Laplace-Beltrami operator) can be consistently defined.

References

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[BLM22] B. Buet, G. P. Leonardi and S. Masnou, Weak and approximate curvatures of a measure: a varifold perspective, Nonlinear Analysis, 2022.

[BR22] B. Buet and M. Rumpf, Mean curvature motion of point cloud varifolds, ESAIM: M2AN, 2022.

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