Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction

ELECTROMAGNETIC TRACKER CHARACTERIZATION AND OPTIMAL TOOL DESIGN (WITH APPLICATIONS TO ENT SURGERY)

Electromagnetic tracking systems prove to have great potential for serving as the tracking component of image guided surgery (IGS) systems. However, despite their major advantage over other trackers in that they do not require line-of-sight to the sensors, their use has been limited primarily due to their inherent measurement distortion problem. Presented here are methods of mapping the measurement field distortion and results describing the distortion present in various environments. Further, a framework for calibration and characterization of the tracking system’s systematic error is presented. The error maps are used to generate polynomial models of the distortion that can be used to dynamically compensate for measurement errors. The other core theme of this work is related to optimal design of electromagnetically tracked tools; presented here are mathematical tools for analytically predicting error propagation and optimally configuring sensors on a tool. A software simulator, using a model of the magnetic field distortion, is used to further design and test these tools in a simulation of actual measurement environments before ever even being built. These tools are used to design and test a set of electromagnetically tracked instruments, specifically for ENT surgical applications

Virtual Morphology and Evolutionary Morphometrics in the new millenium.

EDITED VOLUME; abstract N/

Human motion analysis

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Graph Embedding with Data Uncertainty

spectral-based subspace learning is a common data preprocessing step in many machine learning pipelines. The main aim is to learn a meaningful low dimensional embedding of the data. However, most subspace learning methods do not take into consideration possible measurement inaccuracies or artifacts that can lead to data with high uncertainty. Thus, learning directly from raw data can be misleading and can negatively impact the accuracy. In this paper, we propose to model artifacts in training data using probability distributions; each data point is represented by a Gaussian distribution centered at the original data point and having a variance modeling its uncertainty. We reformulate the Graph Embedding framework to make it suitable for learning from distributions and we study as special cases the Linear Discriminant Analysis and the Marginal Fisher Analysis techniques. Furthermore, we propose two schemes for modeling data uncertainty based on pair-wise distances in an unsupervised and a supervised contexts.Comment: 20 pages, 4 figure

Statistical analysis of diffusion tensor imaging

This thesis considers the statistical analysis of diffusion tensor imaging (DTI). DTI is an advanced magnetic resonance imaging (MRI) method that provides a unique insight into biological microstructure \textit{in vivo} by directionally describing the water molecular diffusion. We firstly develop a Bayesian multi-tensor model with reparameterisation for capturing water diffusion at voxels with one or more distinct fibre orientations. Our model substantially alleviates the non-identifiability issue present in the standard multi-tensor model. A Markov chain Monte Carlo (MCMC) algorithm is then developed to study the uncertainty of the model parameters based on the posterior distribution. We apply the Bayesian method to Monte Carlo (MC) simulated datasets as well as a healthy human brain dataset. A region containing crossing fibre bundles is investigated using our multi-tensor model with automatic model selection. A diffusion tensor, a covariance matrix related to the molecular displacement at a particular voxel in the brain, is in the non-Euclidean space of 3x3 positive semidefinite symmetric matrices. We define the sample mean of tensor data to be the Fréchet mean. We carry out the non-Euclidean statistical analysis of diffusion tensor data. The primary focus is on the use of Procrustes size-and-shape space. Comparisons are made with other non-Euclidean techniques, including the log-Euclidean, Riemannian, Cholesky, root Euclidean and power Euclidean methods. The weighted generalised Procrustes analysis has been developed to efficiently interpolate and smooth an arbitrary number of tensors with the flexibility of controlling individual contributions. A new anisotropy measure, Procrustes Anisotropy is defined and compared with other widely used anisotropy measures. All methods are illustrated through synthetic examples as well as white matter tractography of a healthy human brain. Finally, we use Giné’s statistic to design uniformly distributed diffusion gradient direction schemes with different numbers of directions. MC simulation studies are carried out to compare effects of Giné’s and widely used Jones' schemes on tensor estimation. We conclude by discussing potential areas for further research

Graph matching using position coordinates and local features for image analysis

Encontrar las correspondencias entre dos imágenes es un problema crucial en el campo de la visión por ordenador i el reconocimiento de patrones. Es relevante para un amplio rango de propósitos des de aplicaciones de reconocimiento de objetos en las áreas de biometría, análisis de documentos i análisis de formas hasta aplicaciones relacionadas con la geometría desde múltiples puntos de vista tales cómo la recuperación de la pose, estructura desde el movimiento y localización y mapeo. La mayoría de las técnicas existentes enfocan este problema o bien usando características locales en la imagen o bien usando métodos de registro de conjuntos de puntos (o bien una mezcla de ambos). En las primeras, un conjunto disperso de características es primeramente extraído de las imágenes y luego caracterizado en la forma de vectores descriptores usando evidencias locales de la imagen. Las características son asociadas según la similitud entre sus descriptores. En las segundas, los conjuntos de características son considerados cómo conjuntos de puntos los cuales son asociados usando técnicas de optimización no lineal. Estos son procedimientos iterativos que estiman los parámetros de correspondencia y de alineamiento en pasos alternados. Los grafos son representaciones que contemplan relaciones binarias entre las características. Tener en cuenta relaciones binarias al problema de la correspondencia a menudo lleva al llamado problema del emparejamiento de grafos. Existe cierta cantidad de métodos en la literatura destinados a encontrar soluciones aproximadas a diferentes instancias del problema de emparejamiento de grafos, que en la mayoría de casos es del tipo "NP-hard". El cuerpo de trabajo principal de esta tesis está dedicado a formular ambos problemas de asociación de características de imagen y registro de conjunto de puntos como instancias del problema de emparejamiento de grafos. En todos los casos proponemos algoritmos aproximados para solucionar estos problemas y nos comparamos con un número de métodos existentes pertenecientes a diferentes áreas como eliminadores de "outliers", métodos de registro de conjuntos de puntos y otros métodos de emparejamiento de grafos. Los experimentos muestran que en la mayoría de casos los métodos propuestos superan al resto. En ocasiones los métodos propuestos o bien comparten el mejor rendimiento con algún método competidor o bien obtienen resultados ligeramente peores. En estos casos, los métodos propuestos normalmente presentan tiempos computacionales inferiores.Trobar les correspondències entre dues imatges és un problema crucial en el camp de la visió per ordinador i el reconeixement de patrons. És rellevant per un ampli ventall de propòsits des d’aplicacions de reconeixement d’objectes en les àrees de biometria, anàlisi de documents i anàlisi de formes fins aplicacions relacionades amb geometria des de múltiples punts de vista tals com recuperació de pose, estructura des del moviment i localització i mapeig. La majoria de les tècniques existents enfoquen aquest problema o bé usant característiques locals a la imatge o bé usant mètodes de registre de conjunts de punts (o bé una mescla d’ambdós). En les primeres, un conjunt dispers de característiques és primerament extret de les imatges i després caracteritzat en la forma de vectors descriptors usant evidències locals de la imatge. Les característiques son associades segons la similitud entre els seus descriptors. En les segones, els conjunts de característiques son considerats com conjunts de punts els quals son associats usant tècniques d’optimització no lineal. Aquests son procediments iteratius que estimen els paràmetres de correspondència i d’alineament en passos alternats. Els grafs son representacions que contemplen relacions binaries entre les característiques. Tenir en compte relacions binàries al problema de la correspondència sovint porta a l’anomenat problema de l’emparellament de grafs. Existeix certa quantitat de mètodes a la literatura destinats a trobar solucions aproximades a diferents instàncies del problema d’emparellament de grafs, el qual en la majoria de casos és del tipus “NP-hard”. Una part del nostre treball està dedicat a investigar els beneficis de les mesures de ``bins'' creuats per a la comparació de característiques locals de les imatges. La resta està dedicat a formular ambdós problemes d’associació de característiques d’imatge i registre de conjunt de punts com a instàncies del problema d’emparellament de grafs. En tots els casos proposem algoritmes aproximats per solucionar aquests problemes i ens comparem amb un nombre de mètodes existents pertanyents a diferents àrees com eliminadors d’“outliers”, mètodes de registre de conjunts de punts i altres mètodes d’emparellament de grafs. Els experiments mostren que en la majoria de casos els mètodes proposats superen a la resta. En ocasions els mètodes proposats o bé comparteixen el millor rendiment amb algun mètode competidor o bé obtenen resultats lleugerament pitjors. En aquests casos, els mètodes proposats normalment presenten temps computacionals inferiors

Integrering av multivariate data i systembiologi

Owing to the rapid rate of development in the field of systems biology researchers have faced many new challenges with regard to handling the large amount of generated data sets originating from different –omics techniques, integrating and analyzing them and finally interpreting the results in a meaningful way. Different statistical methods have been implemented in the field of systems biology. The use of chemometrics approaches for the integration and analysis of systems biology data has recently increased. Different chemometrics methods are potentially available for integrating –omics data and detecting variable and sample patterns. An important challenge is to decide which method to use for the analysis of –omics data sets and how to pre-process the data sets for this purpose. Special attention needs to be given to the validity of the detected patterns. In this study we have been working on developing multi-block methods for integrating different types of systems biology data and investigating the co-variation patterns among the measured variables. A special focus was given to the validation of the results of the multi-block methods CPCA and MBPLSR. Different types of graphical tools were introduced for the purpose of validation. We have also developed pre-processing techniques that could explicitly be used for lipidomics data sets. A framework was built for pre-processing, integrating, analyzing and interpreting the lipidomics data sets. The framework was then used for the analysis of a lipidomics data set from a human intervention study. Working on the development of the validation tools required an understanding of the concept of DFs consumption during the multi-block modeling. Therefore, we ran simulation studies where we investigated the number of DFs that were consumed during the modeling processes of PCA and CPCA. Another important issue for applying multi-block methods is the choice of the deflation method. Hence, we studied different deflation strategies available for Multi-block PCA and investigated their interpretational aspects.På grunn av rask utvikling innen systembiologi har forskere møtt mange nye utfordringer med hensyn til håndtering av store datamengder, som genereres med forskjellige -omics teknikker. Det er en stor utfordring både å integrere, analysere og til slutt tolke resultatene på en meningsfull måte. Ulike statistiske metoder har blitt implementert for analyse av systembiologi data. Bruk av kjemometri for integrering og analyse av biologiske data har økt mye den siste tiden. I utgangspunktet finnes det flere metoder fra kjemometri som kan brukes for å integrere data fra forskjellige –omics teknikker og for å oppdage grupperinger av objekter og variabler. En stor utfordring er å bestemme hvilken metode som skal brukes til analyse av -omics datasett og hvordan pre-prosessere datasettene. Det er også viktig å validere de grupperingene som har blitt oppdaget. I denne studien har vi jobbet med å utvikle multiblokk metoder for å integrere ulike typer data fra systembiologi og å undersøke samvariasjon blant de målte variablene. Det har spesielt vært fokus på validering av resultatene av multiblokkmetoder som CPCA og MBPLSR. Ulike typer verktøy ble innført for å sikre valideringen. Vi har utviklet pre-prosessering teknikker som kan brukes spesielt til lipidomics datasett. Vi har bygget et rammeverk for pre-prosessering, integrering, analysering og tolkning av lipidomics datasett. Metoden er blitt brukt til å analysere et lipidomics datasett fra et human intervensjonsstudie. Utvikling av validerings metoder krever en forståelse av bruk av antall frihetsgrader under modelleringen. Det har derfor blitt gjennomført simuleringsstudier hvor vi undersøkte antallet frihetsgrader som ble brukt under modellering med PCA og CPCA. Et annet viktig tema når man bruker multiblokk metoder er valget av deflasjonsmetoden. Det er blitt studert ulike deflasjonsstrategier som er tilgjengelige for multiblokk PCA og undersøkt deres tolkningsaspekter.NordForsk ; Foundation for Research Levy on Agricultural Products in Norwa