@inproceedings{8135,
  abstract     = {Discrete Morse theory has recently lead to new developments in the theory of random geometric complexes. This article surveys the methods and results obtained with this new approach, and discusses some of its shortcomings. It uses simulations to illustrate the results and to form conjectures, getting numerical estimates for combinatorial, topological, and geometric properties of weighted and unweighted Delaunay mosaics, their dual Voronoi tessellations, and the Alpha and Wrap complexes contained in the mosaics.},
  author       = {Edelsbrunner, Herbert and Nikitenko, Anton and Ölsböck, Katharina and Synak, Peter},
  booktitle    = {Topological Data Analysis},
  isbn         = {9783030434076},
  issn         = {21978549},
  pages        = {181--218},
  publisher    = {Springer Nature},
  title        = {{Radius functions on Poisson–Delaunay mosaics and related complexes experimentally}},
  doi          = {10.1007/978-3-030-43408-3_8},
  volume       = {15},
  year         = {2020},
}

@phdthesis{7460,
  abstract     = {Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.

For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.},
  author       = {Ölsböck, Katharina},
  issn         = {2663-337X},
  keywords     = {shape reconstruction, hole manipulation, ordered complexes, Alpha complex, Wrap complex, computational topology, Bregman geometry},
  pages        = {155},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{The hole system of triangulated shapes}},
  doi          = {10.15479/AT:ISTA:7460},
  year         = {2020},
}

@article{7666,
  abstract     = {Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.},
  author       = {Edelsbrunner, Herbert and Ölsböck, Katharina},
  issn         = {14320444},
  journal      = {Discrete and Computational Geometry},
  pages        = {759--775},
  publisher    = {Springer Nature},
  title        = {{Tri-partitions and bases of an ordered complex}},
  doi          = {10.1007/s00454-020-00188-x},
  volume       = {64},
  year         = {2020},
}

@article{6608,
  abstract     = {We use the canonical bases produced by the tri-partition algorithm in (Edelsbrunner and Ölsböck, 2018) to open and close holes in a polyhedral complex, K. In a concrete application, we consider the Delaunay mosaic of a finite set, we let K be an Alpha complex, and we use the persistence diagram of the distance function to guide the hole opening and closing operations. The dependences between the holes define a partial order on the cells in K that characterizes what can and what cannot be constructed using the operations. The relations in this partial order reveal structural information about the underlying filtration of complexes beyond what is expressed by the persistence diagram.},
  author       = {Edelsbrunner, Herbert and Ölsböck, Katharina},
  journal      = {Computer Aided Geometric Design},
  pages        = {1--15},
  publisher    = {Elsevier},
  title        = {{Holes and dependences in an ordered complex}},
  doi          = {10.1016/j.cagd.2019.06.003},
  volume       = {73},
  year         = {2019},
}