---
res:
  bibo_abstract:
  - "Many chemical and physical properties of materials are determined by the material’s
    shape,\r\nfor example the size of its pores and the width of its tunnels. This
    makes materials science\r\na prime application area for geometrical and topological
    methods. Nevertheless many\r\nmethods in topological data analysis have not been
    satisfyingly extended to the needs of\r\nmaterials science. This thesis provides
    new methods and new mathematical theorems\r\ntargeted at those specific needs
    by answering four different research questions. While the\r\nmotivation for each
    of the research questions arises from materials science, the methods\r\nare versatile
    and can be applied in different areas as well. \r\n\r\nThe first research question
    is concerned with image data, for example a three-dimensional\r\ncomputed tomography
    (CT) scan of a material, like sand or stone. There are two commonly\r\nused topologies
    for digital images and depending on the application either of them might be\r\nrequired.
    However, software for computing the topological data analysis method persistence\r\nhomology,
    usually supports only one of the two topologies. We answer the question how to\r\ncompute
    persistent homology of an image with respect to one of the two topologies using\r\nsoftware
    that is intended for the other topology. \r\n\r\nThe second research question
    is concerned with image data as well, and asks how much\r\nof the topological
    information of an image is lost when the resolution is coarsened. As\r\ncomputer
    tomography scanners are more expensive the higher the resolution, it is an\r\nimportant
    question in materials science to know which resolution is enough to get satisfying\r\npersistent
    homology. We give theoretical bounds on the information loss based on different\r\ngeometrical
    properties of the object to be scanned. In addition, we conduct experiments on\r\nsand
    and stone CT image data. \r\n\r\nThe third research question is motivated by comparing
    crystalline materials efficiently. As\r\nthe atoms within a crystal repeat periodically,
    crystalline materials are either modeled by\r\nunmanageable infinite periodic
    point sets, or by one of their fundamental domains, which is\r\nunstable under
    perturbation. Therefore a fingerprint of crystalline materials is needed, with\r\nappropriate
    properties such that comparing the crystals can be eased by comparing the\r\nfingerprints
    instead. We define the density fingerprint and prove the necessary properties.
    \r\n\r\nThe fourth research question is motivated by studying the hole-structure
    or connectedness,\r\ni.e. persistent homology or merge trees, of crystalline materials.
    A common way to deal\r\nwith periodicity is to take a fundamental domain and identify
    opposite boundaries to form a\r\ntorus. However, computing persistent homology
    or merge trees on that torus loses some\r\nof the information materials scientists
    are interested in and is additionally not stable under\r\ncertain noise. We therefore
    decorate the merge tree stemming from the torus with additional\r\ninformation
    describing the density and growth rate of the periodic copies of a component\r\nwithin
    a growing spherical window. We prove all desired properties, like stability and
    efficient\r\ncomputability.@eng"
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Teresa
      foaf_name: Heiss, Teresa
      foaf_surname: Heiss
      foaf_workInfoHomepage: http://www.librecat.org/personId=4879BB4E-F248-11E8-B48F-1D18A9856A87
    orcid: 0000-0002-1780-2689
  bibo_doi: 10.15479/at:ista:18667
  dct_date: 2024^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2663-337X
  - http://id.crossref.org/issn/978-3-99078-052-7
  dct_language: eng
  dct_publisher: Institute of Science and Technology Austria@
  dct_subject:
  - persistent homology
  - topological data analysis
  - periodic
  - crystalline materials
  - images
  - fingerprint
  dct_title: New methods for applying topological data analysis to materials science@
...