Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

The aim of this summer school is to bring together researchers from diverse backgrounds on the common problem of discrete curvature and make an update on the many achievements of the last decade.

In recent years, new concepts have emerged in different areas of mathematics and computer science, all related to the notion of discrete curvature. The notion of discrete curvature being not uniquely defined, unlike in the case of differential geometry, it is still an open subject of investigation. We note in particular the following approaches:

1) The optimal transportation point of view, via (i) the definition of so-called coarse Ricci curvature of Ollivier on metric measure spaces, obtained by comparing the W1 Wasserstein distance metric, which has been extended on graphs by Lin, Lu & Yau, Bauer, Jost & Liu. This quantity has very good approximation properties of the continuous manifolds but is quite difficult to calculate (though not impossible). Or (ii) the approach of curvature-dimension inequalities, now well known in the probabilistic framework, applied to the discrete case, including the works of Bonciocat & Sturm. And (iii) finally the recent work of Maas on Markov kernels.

2) Close but different, discrete curvature can be defined by measures of curvature (intrinsic and extrinsic) such as proposed by Morvan, Cohen-Steiner et al. in the spirit of geometric measure theory (via the normal currents). These also boast good convergence properties to the continuous, while keeping a computable link with more conventional notions based on Gaussian curvature on vertices and mean curvature carried by the edges. Note that this approach provides a definition of the principal directions in the discrete, which is an important issue in 2D image processing.

3) The theory of discrete integrable systems developed by the Berlin School (Pinkall, Bobenko, Polthier et al.). More generally, we study special polyhedral surfaces (e.g. admitting quadrilateral or circular meshes). Those are the discrete analogues of specific parameterizations of surfaces or of particular classes of surfaces, such as minimal, constant mean or gaussian curvature. This analogy is justified by the convergence and the existence of similar structural properties (transformations of the space of solutions). In this approach, the geometric properties of the moduli space characterizes the discrete differential geometry.

4) The discrete calculus, already explored long ago, and is now seen in terms of discrete differential geometry and not just numerical computation, i.e. seeking to preserve the structural properties of the continuum. Curvature then arises from the PDEs satisfied by the discrete operators (typically with Bochner Laplacian. It is the work of Forman, and Desbrun, Hirani, Kanso, Leok, Marsden, Schröder & Tong, as well as Polimeni & Grady. Again there are still many open questions, as well as the real-life test.

5) Finally, it should be noted various ad hoc approaches and definitions of the discrete curvature in various situations, with their properties and computational advantages. These are often the approaches used in applications.

The motivation for defining such objects are both theoretical and practical:
- Make a link between theory and practice: a good theoretical foundation ought to ensure wide applicability of concepts and formulas, with possibly qualitatively better tools. Conversely, applications are sources of challenges and ideas and a good testbed for these very ideas.
- Obtain metric quantities such as volume or diameter (e.g. Myers theorem), topology (Myers again and Gauss-Bonnet, as well as more specific properties of graphs).
- Understand the different flows (mean curvature flow and related), which possess "smoothing" properties for polyhedral surfaces (used in denoising sampled surfaces).
- Recover geometric invariants of surfaces: the principal curvatures, bending lines, feature lines, ...and obtain special parameterizations for more natural image processing.

- On the applied side, the curvature is a key concept in pattern recognition, in the sense that areas of high curvature are important for vision. An important example is the model introduced by Mumford with the elastica, which has found applications both in image restoration, image interpolation that by filtering, segmentation, etc.. Other fields where discrete curvature intervenes are image recognition, CAD and architecture.

We believe it is crucial for all contributors in these various fields to meet one another, discuss their views and share their knowledge. The target audience includes both senior researchers who will find a new point of view on issues they know and/or put into perspective their contributions, as well as young researchers and PhD students interested in this growing domain. The speakers will come from academic research, applied research and private companies.

This topic is one of the main axes developed by Labex Bézout (Université Paris-Est) and as such benefits from its support.

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