BMS/SFB Summer School: Discrete Differential Geometry
Vladimir Bazhanov (Australian National University): Introduction to Yang-Baxter equation and statistical mechanics
Daniel Cremers (TU München)
1) Geometric Reconstruction from Images
The geometric reconstruction of the observed 3D world from a collection of images is among the central challenge in the field of computer vision. In my presentation, I will review some of the central concepts with an emphasis on the reconstruction of spatially dense surfaces. In particular, I will discuss convex formulations of multiple view reconstruction, super-resolution texture estimation and the particular case of single view reconstruction. Furthermore, I will present a number of recent developments on 3D modeling with RGB-D cameras.
2) Novel Algorithms for 3D Shape Analysis
With novel algorithms for 3D reconstruction from standard cameras, range scanners or RGB-D cameras an increasing number of digitized three-dimensional objects is becoming available. In my presentation, I will introduce novel algorithms to analyze three-dimensional shapes. In the first part, I will introduce a novel algorithm to compute a dense elastic matching of shapes. More specifically, we compute a dense correspondence which minimizes the elastic thin-shell energy by means of large-scale linear programming relaxations. In the second part, I will introduce the Wave Kernel Signature as a novel feature descriptor for shape analysis. It is defined as the time-averaged probability to detect quantum mechanical particles at respective points of the shape. In numerous experiments, I will show that it is well suited for various challenges of shape analysis ranging from correspondence finding and matching to shape decomposition.
3) Convex Relaxations for Image Segmentation
I will review the Mumford-Shah functional, some special cases and generalizations including the piecewise constant limit, anisotropic versions and vectorial formulations. I will discuss how respective functionals can be relaxed to convex problems which can be solved efficiently using provably convergent primal-dual algorithms. Applications to computer vision problems such as segmentation, denoising and semantic labeling demonstrate that the convex relaxations allow to compute near-optimal solutions which are independent of initialization.
Vladimir Fock (Université de Strasbourg et CNRS): Integrable systems and discrete Dirac operator
Gitta Kutyniok (TU Berlin): Applied Harmonic Analysis meets Geometry
Feng Luo (Rutgers): An introduction to Teichmüller theory from the triangulation point of view
We will give a quick introduction to Teichmüller space of surfaces with boundary using triangulations. The topics to be covered are:
1. Basic hyperbolic geometry
2. Teichmüller space of surfaces
3. Penner's coordinate and and Thurston's coordinate of Teichmüller spaces
4. Variational principles associated to the triangulated surfaces
5. Poisson structures on Teichmüller spaces
Ulrich Pinkall (TU Berlin): Conformal deformations of surfaces
Those functions that arise as the curvature function κ of some closed plane curve of length L form a submanifold M of codimension three in the vector space of all L-periodic functions. M can be considered as the "shape space" for plane curves. Here, we provide a similar approach for surfaces in space. For example, by the uniformization theorem, every smooth topological sphere in space admits a conformal parametrization f: S2→ R3. The rolepreviously played by κ is now taken by the so-called mean curvature half density H|df| induced on S2 via f. Thosehalf-densities on S2 that arise as mean curvature half-densities of surfaces now form a hypersurface M in the euclidean vector space of all half-densities. Working on M instead of dealing directly with the immersions f dramatically simplifies many operations on surfaces that previously either were not tractable or required hard analysis. One striking example is provided by the gradient flow of the Willmore functional ∫H2.Other applications arise in Computer Graphics, where it is sometimes more useful to modify the curvature of a surface than to try to handle the point positions directly.
Wolfgang K. Schief (U of New South Wales): Projective differential geometry of surfaces: integrable structure and discretization
Yuri Suris (TU Berlin): The Lagrangian theory of discrete integrable systems
Serge Tabachnikov (Pennsylvania State University): Pentagram map, the first 21 years
The pentagram map is a geometric iteration on polygons in the projective plane: given a polygon, one draws its short (skip one vertex) diagonals and takes their consecutive intersection points as the vertices of a new polygon. This map commutes with projective transformations and thereby defines a map on the moduli space of projective equivalence classes of polygons. Introduced by Richard Schwartz more than 20 years ago, the pentagram map has attracted much attention in the recent years. In my talks I shall touch upon various aspects of this rich subject. The pentagram map is a completely integrable discrete dynamical system; I shall explain what this means and outline the proof. The pentagram map has a natural continuous limit, as the number of sides of the polygons goes to infinity. This limit is identified with the Boussinesq equation, one of the best studied completely integrable PDEs of soliton type. I shall describe a new family of completely integrable discrete dynamical systems, including the pentagram map, and explain their connection to the theory of cluster algebras, a new and rapidly developing area with numerous connections to diverse fields of mathematics. I shall also comment on the deep relation between the geometry of the moduli space of projective equivalence classes of polygons and certain combinatorial objects, such as the frieze patterns and SL(k)-tilings. In particular, I shall prove the classical Conway-Coxeter theorem relating freeze patterns with triangulations of polygons. As a by-product, I shall present a number of new configuration theorems of projective geometry, akin to the Pappus and Pascal theorems (in fact, one of these theorems is not proved yet!)
Max Wardetzky (U Göttingen): A unified approach to smooth and discrete curvatures using Cartan's moving frames