Generalized Wishart processes for interpolation over diffusion tensor fields

Diffusion Magnetic Resonance Imaging (dMRI) is a non-invasive tool for watching the microstructure of fibrous nerve and muscle tissue. From dMRI, it is possible to estimate 2-rank diffusion tensors imaging (DTI) fields, that are widely used in clinical applications: tissue segmentation, fiber tractography, brain atlas construction, brain conductivity models, among others. Due to hardware limitations of MRI scanners, DTI has the difficult compromise between spatial resolution and signal noise ratio (SNR) during acquisition. For this reason, the data are often acquired with very low resolution. To enhance DTI data resolution, interpolation provides an interesting software solution. The aim of this work is to develop a methodology for DTI interpolation that enhance the spatial resolution of DTI fields. We assume that a DTI field follows a recently introduced stochastic process known as a generalized Wishart process (GWP), which we use as a prior over the diffusion tensor field. For posterior inference, we use Markov Chain Monte Carlo methods. We perform experiments in toy and real data. Results of GWP outperform other methods in the literature, when compared in different validation protocols

High rank tensor and spherical harmonic models for diffusion MRI processing

Diffusion tensor imaging (DTI) is a non-invasive quantitative method of characterizing tissue micro-structure. Diffusion imaging attempts to characterize the manner by which the water molecules within a particular location move within a given amount of time. Measurement of the diffusion tensor (D) within a voxel allows a macroscopic voxel-averaged description of fiber structure, orientation and fully quantitative evaluation of the microstructural features of healthy and diseased tissue.;The rank two tensor model is incapable of resolving multiple fiber orientations within an individual voxel. This shortcoming of single tensor model stems from the fact that the tensor possesses only a single orientational maximum. Several authors reported this non-mono-exponential behavior for the diffusion-induced attenuation in brain tissue in water and N-Acetyl Aspartate (NAA) signals, that is why the Multi-Tensor, Higher Rank Tensor and Orientation Distribution Function (ODF) were introduced.;Using the higher rank tensor, we will propose a scheme for tensor field interpolation which is inspired by subdivision surfaces in computer graphics. The method applies to Cartesian tensors of all ranks and imposes smoothness on the interpolated field by constraining the divergence and curl of the tensor field. Results demonstrate that the subdivision scheme can better preserve anisotropicity and interpolate rotations than some other interpolation methods. As one of the most important applications of DTI, fiber tractography was implemented to study the shape geometry changes. Based on the divergence and curl measurement, we will introduce new scalar measures that are sensitive to behaviors such as fiber bending and fanning.;Based on the ODF analysis, a new anisotropy measure that has the ability to describe multi-fiber heterogeneity while remaining rotationally invariant, will be introduced, which is a problem with many other anisotropy measures defined using the ODF. The performance of this novel measure is demonstrated for data with varying Signal to Noise Ratio (SNR), and different material characteristics

Multi-valued geodesic tractography for diffusion weighted imaging

Diffusion-Weighted Imaging (DWI) is a Magnetic Resonance(MR) technique that measures water diffusion characteristics in tissue for a given direction. The diffusion profile in a specific location can be obtained by combining the DWI measurements of different directions. The diffusion profile gives information about the underlying fibrous structure, e.g., in human brain white matter, based on the assumption that water molecules are moving less freely perpendicularly to the fibrous structure. From the DW-MRI measurements often a positive definite second-order tensor is defined, the so-called diffusion tensor (DT). Neuroscientists have begun using diffusion tensor images (DTI) to study a host of various disorders and neurodegenerative diseases including Parkinson, Alzheimer and Huntington. The techniques for reconstructing the fiber tracts based on diffusion profiles are known as tractography or fiber tracking. There are several ways to extract fibers from the raw diffusion data. In this thesis, we explain and apply geodesic-based tractography techniques specifically, where the assumption is that fibers follow the most efficient diffusion propagation paths. A Riemannian manifold is defined using as metric the inverse of the diffusion tensor. A shortest path in this manifold is one with the strongest diffusion along this path. Therefore geodesics (i.e., shortest paths) on this manifold follow the most efficient diffusion paths. The geodesics are often computed from the stationary Hamilton-Jacobi equation (HJ). One characteristic of solving the HJ equation is that it gives only the single-valued viscosity solution corresponding to the minimizer of the length functional. It is also well known that the solution of the HJ equation can develop discontinuities in the gradient space, i.e., cusps. Cusps occur when the correct solution should become multi-valued. HJ methods are not able to handle this situation. To solve this, we developed a multi-valued solution algorithm for geodesic tractography in a metric space defined by given by diffusion tensor imaging data. The algorithm can capture all possible geodesics arriving at a single voxel instead of only computing the first arrival. Our algorithm gives the possibility of applying different cost functions in a fast post-processing. Moreover, the algorithm can be used for capturing possible multi-path connections between two points. In this thesis, we first focus on the mathematical and numerical model for analytic and synthetic fields in twodimensional domains. Later, we present the algorithm in three-dimensions with examples of synthetic and brain data. Despite the simplicity of the DTI model, the tractography techniques using DT are shown to be very promising to reveal the structure of brain white matter. However, DTI assumes that each voxel contains fibers with only one main orientation and it is known that brain white matter has multiple fiber orientations, which can be arbitrary many in arbitrary directions. Recently, High Angular Resolution Diffusion Imaging (HARDI) acquisition and its modeling techniques have been developed to overcome this limitation. As a next contribution we propose an extension of the multi-valued geodesic algorithm to HARDI data. First we introduce the mathematical model for more complex geometries using Finsler geometry. Next, we propose, justify and exploit the numerical methods for computing the multi-valued solution of these equations

Geodesic tractography segmentation for directional medical image analysis

Acknowledgements page removed per author's request, 01/06/2014.Geodesic Tractography Segmentation is the two component approach presented in this thesis for the analysis of imagery in oriented domains, with emphasis on the application to diffusion-weighted magnetic resonance imagery (DW-MRI). The computeraided analysis of DW-MRI data presents a new set of problems and opportunities for the application of mathematical and computer vision techniques. The goal is to develop a set of tools that enable clinicians to better understand DW-MRI data and ultimately shed new light on biological processes. This thesis presents a few techniques and tools which may be used to automatically find and segment major neural fiber bundles from DW-MRI data. For each technique, we provide a brief overview of the advantages and limitations of our approach relative to other available approaches.Ph.D.Committee Chair: Tannenbaum, Allen; Committee Member: Barnes, Christopher F.; Committee Member: Niethammer, Marc; Committee Member: Shamma, Jeff; Committee Member: Vela, Patrici

Nonlinear Data: Theory and Algorithms

Techniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications

Incompressible immiscible multiphase flows in porous media: a variational approach

We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schem\`e a la [R. Jordan, D. Kinder-lehrer \& F. Otto, SIAM J. Math. Anal, 29(1):1--17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure

A survey of the Schr\"odinger problem and some of its connections with optimal transport

This article is aimed at presenting the Schr\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\"odinger problem. We also give a survey of the related literature. In addition, some new results are proved.Comment: To appear in Discrete \& Continuous Dynamical Systems - Series A. Special issue on optimal transpor

Estimation and Processing of Ensemble Average Propagator and Its Features in Diffusion MRI

Diffusion MRI (dMRI) is the unique technique to infer the microstructure of the white matter in vivo and noninvasively, by modeling the diffusion of water molecules. Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF) are two important Probability Density Functions (PDFs) which reflect the water diffusion. Estimation and processing of EAP and ODF is the central problem in dMRI, and is also the first step for tractography. Diffusion Tensor Imaging (DTI) is the most widely used estimation method which assumes EAP as a Gaussian distribution parameterized by a tensor. Riemannian framework for tensors has been proposed successfully in tensor estimation and processing. However, since the Gaussian EAP assumption is oversimplified, DTI can not reflect complex microstructure like fiber crossing. High Angular Resolution Diffusion Imaging (HARDI) is a category of methods proposed to avoid the limitations of DTI. Most HARDI methods like Q-Ball Imaging (QBI) need some assumptions and only can handle the data from single shell (single $b$ value), which are called as single shell HARDI (sHARDI) methods. However, with the development of scanners and acquisition methods, multiple shell data becomes more and more practical and popular. This thesis focuses on the estimation and processing methods in multiple shell HARDI (mHARDI) which can handle the diffusion data from arbitrary sampling scheme. There are many original contributions in this thesis. -First, we develop the analytical Spherical Polar Fourier Imaging (SPFI), which represents the signal using SPF basis and obtains EAP and its various features including ODFs and some scalar indices like Generalized Fractional Anisotropy (GFA) from analytical linear transforms. In the implementation of SPFI, we present two ways for scale estimation and propose to consider the prior $E(0)=1$ in estimation process. -Second, a novel Analytical Fourier Transform in Spherical Coordinate (AFT-SC) framework is proposed to incorporate many sHARDI and mHARDI methods, explore their relation and devise new analytical EAP/ODF estimation methods. -Third, we present some important criteria to compare different HARDI methods and illustrate their advantages and limitations. -Fourth, we propose a novel diffeomorphism invariant Riemannian framework for ODF and EAP processing, which is a natural generalization of previous Riemannian framework for tensors, and can be used for general PDF computing by representing the square root of the PDF called wavefunction with orthonormal basis. In this Riemannian framework, the exponential map, logarithmic map and geodesic have closed forms, the weighted Riemannian mean and median uniquely exist and can be estimated from an efficient gradient descent. Log-Euclidean framework and Affine-Euclidean framework are developed for fast data processing. -Fifth, we theoretically and experimentally compare the Euclidean metric and Riemannian metric for tensors, ODFs and EAPs. -Finally, we propose the Geodesic Anisotropy (GA) to measure the anisotropy of EAPs, Square Root Parameterized Estimation (SRPE) for nonnegative definite ODF/EAP estimation, weighted Riemannian mean/median for ODF/EAP interpolation, smoothing, atlas estimation. The concept of \emph{reasonable mean value interpolation} is presented for interpolation of general PDF data.L'IRM de diffusion est a ce jour la seule technique a meme d'observer in vivo et de fac¸on non-invasive les structures fines de la mati'ere blanche, en modelisant la diffusion des molecules d'eau. Le propagateur moyen (EAP pour Ensemble average Propagator en anglais) et la fonction de distribution d'orientation (ODF pour Orientation Distribution Function en anglais) sont les deux fonctions de probabilites d'int'erˆet pour caracteriser la diffusion des molecules d'eau. Le probleme central en IRM de diffusion est la reconstruction et le traitement de ces fonctions (EAP et ODF); c'est aussi le point de depart pour la tractographie des fibres de la mati'ere blanche. Le formalisme du tenseur de diffusion (DTI pour Diffusion Tensor Imaging en anglais) est le modele le plus couramment utilise, et se base sur une hypothese de diffusion gaussienne. Il existe un cadre riemannien qui permet d'estimer et de traiter correctement les images de tenseur de diffusion. Cependant, l'hypothese d'une diffusion gaussienne est une simplification, qui ne permet pas de d'écrire les cas ou la structure microscopique sous-jacente est complexe, tels que les croisements de faisceaux de fibres. L'imagerie 'a haute resolution angulaire (HARDI pour High Angular Resolution Diffusion Imaging en anglais) est un ensemble de methodes qui permettent de contourner les limites du modele tensoriel. La plupart des m'ethodes HARDI 'a ce jour, telles que l'imagerie spherique de l'espace de Fourier (QBI pour Q-Ball Imaging en anglais) se basent sur des hypoth'eses reductrices, et prennent en compte des acquisitions qui ne se font que sur une seule sphere dans l'espace de Fourier (sHARDI pour single-shell HARDI en anglais), c'est-a-dire une seule valeur du coefficient de ponderation b. Cependant, avec le developpement des scanners IRM et des techniques d'acquisition, il devient plus facile d'acquerir des donn'ees sur plusieurs sph'eres concentriques. Cette th'ese porte sur les methodes d'estimation et de traitement de donnees sur plusieurs spheres (mHARDI pour multiple-shell HARDI en anglais), et de facon generale sur les methodes de reconstruction independantes du schema d'echantillonnage. Cette these presente plusieurs contributions originales. En premier lieu, nous developpons l'imagerie par transformee de Fourier en coordonnees spheriques (SPFI pour Spherical Polar Fourier Imaging en anglais), qui se base sur une representation du signal dans une base de fonctions a parties radiale et angulaire separables (SPF basis pour Spherical Polar Fourier en anglais). Nous obtenons, de fac¸on analytique et par transformations lineaires, l'EAP ainsi que ses caracteristiques importantes : l'ODF, et des indices scalaires tels que l'anisotropie fractionnelle generalisee (GFA pour Generalized Fractional Anisotropy en anglais). En ce qui concerne l'implementation de SPFI, nous presentons deux methodes pour determiner le facteur d'echelle, et nous prenons en compte le fait que E(0) = 1 dans l'estimation. En second lieu, nous presentons un nouveau cadre pour une transformee de Fourier analytique en coordonnees spheriques (AFT-SC pour Analytical Fourier Transform in Spherical Coordinate en anglais), ce qui permet de considerer aussi bien les methodes mHARDI que sHARDI, d'explorer les relations entre ces methodes, et de developper de nouvelles techniques d'estimation de l'EAP et de l'ODF. Nous presentons en troisieme lieu d'importants crit'eres de comparaison des differentes methodes HARDI, ce qui permet de mettre en lumiere leurs avantages et leurs limites. Dans une quatrieme partie, nous proposons un nouveau cadre riemannien invariant par diffeomorphisme pour le traitement de l'EAP et de l'ODF. Ce cadre est une generalisation de la m'ethode riemannienne precedemment appliquee au tenseur de diffusion. Il peut etre utilise pour l'estimation d'une fonction de probabilite representee par sa racine carree, appelee fonction d'onde, dans une base de fonctions orthonormale. Dans ce cadre riemannien, les applications exponentielle et logarithmique, ainsi que les geodesiques ont une forme analytique. La moyenne riemannienne ponderee ainsi que la mediane existent et sont uniques, et peuvent etre calculees de facon efficace par descente de gradient. Nous developpons egalement un cadre log-euclidien et un cadre affine-euclidien pour un traitement rapide des donnees. En cinquieme partie, nous comparons, theoriquement et sur un plan exp'erimental, les metriques euclidiennes et riemanniennes pour les tenseurs, l'ODF et l'EAP. Finalement, nous proposons l'anisotropie geodesique (GA pour Geodesic Anisotropy en anglais) pour mesurer l'anisotropie de l'EAP; une parametrisation par la racine carrée (SRPE pour Square-Root Parameterized Estimation en anglais) pour l'estimation d'un EAP et d'une ODF positifs; la mediane et la moyenne riemanniennes ponderees pour l'interpolation, le lissage et la construction d'atlas bas'es sur l'ODF et de l'EAP. Nous introduisons la notion de valeur moyenne raisonnable pour l'interpolation de fonction de probabilites en general