---
OA_place: publisher
_id: '18667'
abstract:
- lang: eng
  text: "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."
acknowledgement: "I was supported by the European Research Council (ERC) Horizon 2020
  project\r\n“Alpha Shape Theory Extended” No. 788183 and by the Pöttinger Scholarship.
  In addition,\r\nI am very thankful for having been able to attend the second Workshop
  for Women in\r\nComputational Topology in July 2019, funded by the Mathematical
  Sciences Institute at\r\nANU, the US National Science Foundation through the award
  CCF-1841455, the Australian\r\nMathematical Sciences Institute and the Association
  for Women in Mathematics. Two of the\r\nprojects presented in this thesis started
  there. One of them reached completion thanks to\r\nfunding from the MSRI Summer
  Research in Mathematics program awarded to me and my\r\ncollaborators in 2020."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Teresa
  full_name: Heiss, Teresa
  id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
  last_name: Heiss
  orcid: 0000-0002-1780-2689
citation:
  ama: Heiss T. New methods for applying topological data analysis to materials science.
    2024. doi:<a href="https://doi.org/10.15479/at:ista:18667">10.15479/at:ista:18667</a>
  apa: Heiss, T. (2024). <i>New methods for applying topological data analysis to
    materials science</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/at:ista:18667">https://doi.org/10.15479/at:ista:18667</a>
  chicago: Heiss, Teresa. “New Methods for Applying Topological Data Analysis to Materials
    Science.” Institute of Science and Technology Austria, 2024. <a href="https://doi.org/10.15479/at:ista:18667">https://doi.org/10.15479/at:ista:18667</a>.
  ieee: T. Heiss, “New methods for applying topological data analysis to materials
    science,” Institute of Science and Technology Austria, 2024.
  ista: Heiss T. 2024. New methods for applying topological data analysis to materials
    science. Institute of Science and Technology Austria.
  mla: Heiss, Teresa. <i>New Methods for Applying Topological Data Analysis to Materials
    Science</i>. Institute of Science and Technology Austria, 2024, doi:<a href="https://doi.org/10.15479/at:ista:18667">10.15479/at:ista:18667</a>.
  short: T. Heiss, New Methods for Applying Topological Data Analysis to Materials
    Science, Institute of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-12-17T16:17:55Z
date_published: 2024-12-17T00:00:00Z
date_updated: 2026-04-07T12:54:10Z
day: '17'
ddc:
- '514'
- '516'
- '004'
degree_awarded: PhD
department:
- _id: GradSch
- _id: HeEd
doi: 10.15479/at:ista:18667
ec_funded: 1
file:
- access_level: open_access
  checksum: 247bb057aed2fba1cd4711917aaa2d77
  content_type: application/pdf
  creator: theiss
  date_created: 2024-12-19T10:24:46Z
  date_updated: 2024-12-19T10:24:46Z
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  checksum: 9648b45c07a008ee11a07f99856a139d
  content_type: application/zip
  creator: theiss
  date_created: 2024-12-19T10:24:50Z
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  file_name: PhD_Thesis.zip
  file_size: 17197731
  relation: source_file
file_date_updated: 2024-12-19T10:24:50Z
has_accepted_license: '1'
keyword:
- persistent homology
- topological data analysis
- periodic
- crystalline materials
- images
- fingerprint
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: '111'
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '788183'
  name: Alpha Shape Theory Extended
publication_identifier:
  isbn:
  - 978-3-99078-052-7
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '10828'
    relation: part_of_dissertation
    status: public
  - id: '11440'
    relation: part_of_dissertation
    status: public
  - id: '18673'
    relation: part_of_dissertation
    status: public
  - id: '9345'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
title: New methods for applying topological data analysis to materials science
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...