title: Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling
creator: Sitenko, Dmitrij
subject: 500
subject: 500 Natural sciences and mathematics
subject: 510
subject: 510 Mathematics
description: In this thesis, we focus on the image labeling problem which is the task of performing unique  pixel-wise label decisions to simplify the image while reducing its redundant information. We  build upon a recently introduced geometric approach for data labeling by assignment flows  [  APSS17  ] that comprises a smooth dynamical system for data processing on weighted graphs.  Hereby we pursue two lines of research that give new application and theoretically-oriented  insights on the underlying segmentation task.  We demonstrate using the example of Optical Coherence Tomography (OCT), which is the  mostly used non-invasive acquisition method of large volumetric scans of human retinal tis-  sues, how incorporation of constraints on the geometry of statistical manifold results in a novel  purely data driven  geometric  approach for order-constrained segmentation of volumetric data  in any metric space. In particular, making diagnostic analysis for human eye diseases requires  decisive information in form of exact measurement of retinal layer thicknesses that has be done  for each patient separately resulting in an demanding and time consuming task. To ease the  clinical diagnosis we will introduce a fully automated segmentation algorithm that comes up  with a high segmentation accuracy and a high level of built-in-parallelism. As opposed to many  established retinal layer segmentation methods, we use only local information as input without  incorporation of additional global shape priors. Instead, we achieve physiological order of reti-  nal cell layers and membranes including a new formulation of ordered pair of distributions in an  smoothed energy term. This systematically avoids bias pertaining to global shape and is hence  suited for the detection of anatomical changes of retinal tissue structure. To access the perfor-  mance of our approach we compare two different choices of features on a data set of manually  annotated  3  D OCT volumes of healthy human retina and evaluate our method against state of  the art in automatic retinal layer segmentation as well as to manually annotated ground truth  data using different metrics.  We generalize the recent work [  SS21  ] on a variational perspective on assignment flows and  introduce a novel nonlocal partial difference equation (G-PDE) for labeling metric data on graphs.  The G-PDE is derived as nonlocal reparametrization of the assignment flow approach that was  introduced in  J. Math. Imaging & Vision  58(2), 2017. Due to this parameterization, solving the  G-PDE numerically is shown to be equivalent to computing the Riemannian gradient flow with re-  spect to a nonconvex potential. We devise an entropy-regularized difference-of-convex-functions  (DC) decomposition of this potential and show that the basic geometric Euler scheme for inte-  grating the assignment flow is equivalent to solving the G-PDE by an established DC program-  ming scheme. Moreover, the viewpoint of geometric integration reveals a basic way to exploit  higher-order information of the vector field that drives the assignment flow, in order to devise a  novel accelerated DC programming scheme. A detailed convergence analysis of both numerical  schemes is provided and illustrated by numerical experiments.
date: 2023
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserverhttps://archiv.ub.uni-heidelberg.de/volltextserver/33387/1/Thesis.pdf
identifier: DOI:10.11588/heidok.00033387
identifier: urn:nbn:de:bsz:16-heidok-333878
identifier:   Sitenko, Dmitrij  (2023) Nonlocal Graph-PDEs and Riemannian Gradient Flows for Image Labeling.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/33387/
rights: info:eu-repo/semantics/openAccess
rights: http://archiv.ub.uni-heidelberg.de/volltextserver/help/license_urhg.html
language: eng