eprintid: 34628
rev_number: 16
eprint_status: archive
userid: 8033
dir: disk0/00/03/46/28
datestamp: 2024-03-26 15:27:06
lastmod: 2024-03-27 10:00:48
status_changed: 2024-03-26 15:27:06
type: doctoralThesis
metadata_visibility: show
creators_name: Mäder, Tim
title: Stratified Topological Data Analysis: Theory, Methods and Applications in Image Patch Spaces
subjects: ddc-510
divisions: i-110400
adv_faculty: af-11
cterms_swd: Mathematik
cterms_swd: Topologie
cterms_swd: Algorithmus
abstract: Singular spaces have gained prominence in diverse applications, generating interest in investigating stratified structures using methods within the field of topological data analysis (TDA). While persistent homology is a fundamental concept in TDA, its limitations in distinguishing stratified spaces prompt the need for alternative invariants. To address this need, we introduce the concept of persistent stratified homotopy
types. We establish a series of properties analogous to those of the ordinary persistent homotopy type, such as stability, computability, and inference, which are crucial for TDA applications. In tackling the challenge of detecting singularities in data, we present methods to approximate stratifications from non-stratified data, sampled in the vicinity of sufficiently well-behaved two-strata Whitney stratified spaces. Our findings enable the formulation of a sampling theorem for approximately inferring (persistent) stratified homotopy types of suitably well-behaved two strata Whitney stratified spaces. Bridging theory and practical implementation, we introduce two distinct approaches to measure singularity in data. Leveraging insights from local persistent homology studies and employing the Hausdorff distance serve this purpose. Moreover, we delve into the topology of image patch spaces, providing novel insights and reassessing existing models, especially with regard to potential stratified
structures.
date: 2024
id_scheme: DOI
id_number: 10.11588/heidok.00034628
ppn_swb: 1884473733
own_urn: urn:nbn:de:bsz:16-heidok-346289
date_accepted: 2024-03-20
advisor: HASH(0x558eaa745f38)
language: eng
bibsort: MADERTIMSTRATIFIED
full_text_status: public
place_of_pub: Heidelberg
citation:   Mäder, Tim  (2024) Stratified Topological Data Analysis: Theory, Methods and Applications in Image Patch Spaces.  [Dissertation]     
document_url: https://archiv.ub.uni-heidelberg.de/volltextserver/34628/1/Doktorarbeit_Tim_M%C3%A4der.pdf