TY - BOOK

T1 - Stochastic Modelling on Manifolds

AU - Kühnel, Line

PY - 2018

Y1 - 2018

N2 - The constant improvement of data collection techniques increases the complexity of observed data objects. Cameras and scanning technologies make it possible to retrieve detailed images of everything from microscopic structures of a cell to 3D images of anatomical objects. No matter if data are curve outlines of a shape, medical images, or a collection of landmark points for an object, such complex data structures lack vector space properties and will challenge the well-known statistical theory for data in Euclidean space. All in all, new generalised statistical methods have to be developed for analysing non-linear data samples.This thesis presents methods for introducing uncertainty estimation in generalised statistical methods for analysing distributions of manifold-valued data objects. As closed-form expressions of probability density functions for distributions on manifolds are hard to obtain, the presented methods use stochastic theory to describe uncertainty and variation in manifold-valued data samples.A regression model is defined which by transportation of an Euclidean semi-martingale models the relation between Euclidean covariates and a manifold-valued response. Moreover, two methods are presented for describing the uncertainty in image data.The first model separates uncertainty in images into a warp effect and a spatial intensity effect via displacement fields on a discretised lattice. The second method estimates uncertainty by stochastic deformations of images. The stochastic deformations are modelled by a stochastic flow of diffeomorphisms based on the LDDMM framework. As a second part of the thesis, a software package primarily developed for Deep Learning tasks is used to perform concise implementation of non-linear statistical methods.

AB - The constant improvement of data collection techniques increases the complexity of observed data objects. Cameras and scanning technologies make it possible to retrieve detailed images of everything from microscopic structures of a cell to 3D images of anatomical objects. No matter if data are curve outlines of a shape, medical images, or a collection of landmark points for an object, such complex data structures lack vector space properties and will challenge the well-known statistical theory for data in Euclidean space. All in all, new generalised statistical methods have to be developed for analysing non-linear data samples.This thesis presents methods for introducing uncertainty estimation in generalised statistical methods for analysing distributions of manifold-valued data objects. As closed-form expressions of probability density functions for distributions on manifolds are hard to obtain, the presented methods use stochastic theory to describe uncertainty and variation in manifold-valued data samples.A regression model is defined which by transportation of an Euclidean semi-martingale models the relation between Euclidean covariates and a manifold-valued response. Moreover, two methods are presented for describing the uncertainty in image data.The first model separates uncertainty in images into a warp effect and a spatial intensity effect via displacement fields on a discretised lattice. The second method estimates uncertainty by stochastic deformations of images. The stochastic deformations are modelled by a stochastic flow of diffeomorphisms based on the LDDMM framework. As a second part of the thesis, a software package primarily developed for Deep Learning tasks is used to perform concise implementation of non-linear statistical methods.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/1pioq0f/alma99123304136505763

M3 - Ph.D. thesis

BT - Stochastic Modelling on Manifolds

PB - Department of Computer Science, Faculty of Science, University of Copenhagen

ER -