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Persistent Homology and Its Applications

Show simple item record 2021-06-08T07:22:55Z 2021-06-08T07:22:55Z 2021-06-08
dc.title Persistent Homology and Its Applications en
ethesis.faculty Matemaattis-luonnontieteellinen tiedekunta fi
ethesis.faculty Faculty of Science en
ethesis.faculty Matematisk-naturvetenskapliga fakulteten sv
ethesis.faculty.URI Helsingin yliopisto fi University of Helsinki en Helsingfors universitet sv
dct.creator Karvonen, Elli
dct.issued 2021
dct.language.ISO639-2 eng
dct.abstract The topological data analysis studies the shape of a space at multiple scales. Its main tool is persistent homology, which is based on other homology theory, usually simplicial homology. Simplicial homology applies to finite data in real space, and thus it is mainly used in applications. This thesis aims to introduce the theories behind persistent homology and its application, image completion algorithm. Persistent homology is motivated by the question of which scale is the most essential to study data shape. A filtration contains all scales we want to explore, and thus it is an essential tool of persistent homology. The thesis focuses on forming a filtaration from a Delaunay triangulation and its subcomplexes, alpha-complexes. We will found that these provide sufficient tools to consider homology classes birth and deaths, but they are not particularly easy to use in practice. This observation motivates to define a regional complement of the dual alpha graph. We found that its components' and essential homology classes' birth and death times correspond. The algorithm utilize this observation to complete images. The results are good and mainly as could be expected. We discuss that algorithm has potential since it does need any training or other input parameters than data. However, future studies are needed to imply it, for example, in three-dimensional data. en
dct.subject topological data analysis
dct.subject simplicial homology
dct.subject persistent homology
dct.subject Delaunay triangulation
dct.subject alpha-complex
dct.language en
ethesis.isPublicationLicenseAccepted true
ethesis.language englanti fi
ethesis.language English en
ethesis.language engelska sv
ethesis.thesistype pro gradu -tutkielmat fi
ethesis.thesistype master's thesis en
ethesis.thesistype pro gradu-avhandlingar sv
dct.identifier.ethesis E-thesisID:ef7444ed-58e3-46bc-9383-d7bbd1c8147b
dct.identifier.urn URN:NBN:fi:hulib-202106082549
dc.type.dcmitype Text
ethesis.facultystudyline Matematiikka fi
ethesis.facultystudyline Mathematics en
ethesis.facultystudyline Matematik sv
ethesis.mastersdegreeprogram Matematiikan ja tilastotieteen maisteriohjelma fi
ethesis.mastersdegreeprogram Master's Programme in Mathematics and Statistics en
ethesis.mastersdegreeprogram Magisterprogrammet i matematik och statistik sv

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