10th Winter School on Computational Geometry

10th Winter School on Computational Geometry

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10th Winter School on Computational Geometry

Helmut Alt

COMPUTATIONAL ASPECTS OF PACKING PROBLEMS

Jean-Daniel Boissonnat

Delaunay triangulation of manifolds

  • 1. Delaunay triangulations in Euclidean and Laguerre geometry
    1.1. Euclidean case, Delaunay's theorem.
    1.2. Weighted points. Affine diagrams. Union of balls.
    1.3. Examples of affine diagrams.

  • 2. Good Delaunay triangulations and meshes
    2.1. Nets and combinatorial bounds. Delaunay refinement.
    2.2. Thickness, protection, stability of Delaunay triangulations.
    2.3. Perturbation algorithms. The Local Lovasz Lemma and heuristics.

  • 3. Triangulation of topological spaces
    3.1. Topological spaces, equivalences between topological spaces.
    3.2. (Abstract) simplicial complexes, nerve theorem.
    3.3. Geometric complexes and filtrations : Cech, Rips, alpha-complexes.
    3.4. Data structures for simplicial complexes.

  • 4. Reconstruction of submanifolds of Rd
    4.1. Sets of positive reach.
    4.2. Quality of approximation.
    4.3. Distance functions and homotopic shape reconstruction.
    4.4. The curse of dimensionality, the tangential Delaunay complex.

  • 5. Delaunay triangulation of Riemannian manifolds
    5.1. Anisotropic Delaunay triangulation.
    5.2. Delaunay triangulation of Riemannian manifolds.
    5.3. Statistical spaces. Fischer metric and Bregman divergence.
  • Delaunay triangulations of manifolds
  • Delaunay triangulation of manifolds 2 (Good triangulations and meshes)
  • Delaunay triangulation of manifolds 3 (Triangulation of topological spaces)
  • Delaunay triangulation of manifolds 4 (Shape reconstruction)
  • Delaunay triangulation of manifolds 5


Carola Wenk

Algorithms for Map Construction and Comparison

  • Map-Matching
    Frechet distance
    Frechet map-matching
    HMM map-matching

  • Map Construction I
    Density-based methods
    Intersection linking
    Frechet-based

  • Map Construction II
    Frechet-clustering
    Bundle map-construction

  • Map Comparison
    Hausdorff distance
    Path-based distances
    Local persistent homology-based distance
    Graph sampling-based distance
    Local signatures

  • Extensions
    Handling directed edges, multiple lanes, turn restrictions
    Map comparison for directed and weighted graphs
    Map updates and detecting change
    Frechet range queries
  • Map-Matching
  • Map Construction I
  • Map Construction II
  • Map Comparison
  • Extensions