@phdthesis{18667,
  abstract     = {Many chemical and physical properties of materials are determined by the material’s shape,
for example the size of its pores and the width of its tunnels. This makes materials science
a prime application area for geometrical and topological methods. Nevertheless many
methods in topological data analysis have not been satisfyingly extended to the needs of
materials science. This thesis provides new methods and new mathematical theorems
targeted at those specific needs by answering four different research questions. While the
motivation for each of the research questions arises from materials science, the methods
are versatile and can be applied in different areas as well. 

The first research question is concerned with image data, for example a three-dimensional
computed tomography (CT) scan of a material, like sand or stone. There are two commonly
used topologies for digital images and depending on the application either of them might be
required. However, software for computing the topological data analysis method persistence
homology, usually supports only one of the two topologies. We answer the question how to
compute persistent homology of an image with respect to one of the two topologies using
software that is intended for the other topology. 

The second research question is concerned with image data as well, and asks how much
of the topological information of an image is lost when the resolution is coarsened. As
computer tomography scanners are more expensive the higher the resolution, it is an
important question in materials science to know which resolution is enough to get satisfying
persistent homology. We give theoretical bounds on the information loss based on different
geometrical properties of the object to be scanned. In addition, we conduct experiments on
sand and stone CT image data. 

The third research question is motivated by comparing crystalline materials efficiently. As
the atoms within a crystal repeat periodically, crystalline materials are either modeled by
unmanageable infinite periodic point sets, or by one of their fundamental domains, which is
unstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with
appropriate properties such that comparing the crystals can be eased by comparing the
fingerprints instead. We define the density fingerprint and prove the necessary properties. 

The fourth research question is motivated by studying the hole-structure or connectedness,
i.e. persistent homology or merge trees, of crystalline materials. A common way to deal
with periodicity is to take a fundamental domain and identify opposite boundaries to form a
torus. However, computing persistent homology or merge trees on that torus loses some
of the information materials scientists are interested in and is additionally not stable under
certain noise. We therefore decorate the merge tree stemming from the torus with additional
information describing the density and growth rate of the periodic copies of a component
within a growing spherical window. We prove all desired properties, like stability and efficient
computability.},
  author       = {Heiss, Teresa},
  isbn         = {978-3-99078-052-7},
  issn         = {2663-337X},
  keywords     = {persistent homology, topological data analysis, periodic, crystalline materials, images, fingerprint},
  pages        = {111},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{New methods for applying topological data analysis to materials science}},
  doi          = {10.15479/at:ista:18667},
  year         = {2024},
}