Nonnegative Definite EAP and ODF Estimation via a Unified Multi-Shell HARDI Reconstruction

International audienceIn High Angular Resolution Diffusion Imaging (HARDI), Orientation Distribution Function (ODF) and Ensemble Average Propagator (EAP) are two important Probability Density Functions (PDFs) which reflect the water diffusion and fiber orientations. Spherical Polar Fourier Imaging (SPFI) is a recent model-free multi-shell HARDI method which estimates both EAP and ODF from the diffusion signals with multiple b values. As physical PDFs, ODFs and EAPs are nonnegative definite respectively in their domains S^2 and R^3 . However, existing ODF / EAP estimation methods like SPFI seldom consider this natural constraint. Although some works considered the nonnegative constraint on the given discrete samples of ODF / EAP, the estimated ODF/EAP is not guaranteed to be nonnegative definite in the whole continuous domain. The Riemannian framework for ODFs and EAPs has been proposed via the square root parameterization based on pre-estimated ODFs and EAPs by other methods like SPFI. However, there is no work on how to estimate the square root of ODF / EAP called as the wavefuntion directly from diffusion signals. In this paper, based on the Riemannian framework for ODFs / EAPs and Spherical Polar Fourier (SPF) basis representation, we propose a unified model-free multi-shell HARDI method, named as Square Root Parameterized Estimation (SRPE), to simultaneously estimate both the wavefunction of EAPs and the nonnegative definite ODFs and EAPs from diffusion signals. The experiments on synthetic data and real data showed SRPE is more robust to noise and has better EAP reconstruction than SPFI, especially for EAP profiles at large radius

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

Estimation of Fiber Orientations Using Neighborhood Information

Data from diffusion magnetic resonance imaging (dMRI) can be used to reconstruct fiber tracts, for example, in muscle and white matter. Estimation of fiber orientations (FOs) is a crucial step in the reconstruction process and these estimates can be corrupted by noise. In this paper, a new method called Fiber Orientation Reconstruction using Neighborhood Information (FORNI) is described and shown to reduce the effects of noise and improve FO estimation performance by incorporating spatial consistency. FORNI uses a fixed tensor basis to model the diffusion weighted signals, which has the advantage of providing an explicit relationship between the basis vectors and the FOs. FO spatial coherence is encouraged using weighted l1-norm regularization terms, which contain the interaction of directional information between neighbor voxels. Data fidelity is encouraged using a squared error between the observed and reconstructed diffusion weighted signals. After appropriate weighting of these competing objectives, the resulting objective function is minimized using a block coordinate descent algorithm, and a straightforward parallelization strategy is used to speed up processing. Experiments were performed on a digital crossing phantom, ex vivo tongue dMRI data, and in vivo brain dMRI data for both qualitative and quantitative evaluation. The results demonstrate that FORNI improves the quality of FO estimation over other state of the art algorithms.Comment: Journal paper accepted in Medical Image Analysis. 35 pages and 16 figure

Spatially Regularizing High Angular Resolution Diffusion Imaging

Many recent high angular resolution diffusion imaging (HARDI) reconstruction techniques have been introduced to infer ensemble average propagator (EAP),describing the three-dimensional (3D) average diffusion process of water molecules or the angular structure information contained in EAP, orientation distribution function (ODF). Most of these methods perform reconstruction independently at each voxel, which essentially ignoring the functional nature of the HARDI data at different voxels in space. The aim of my thesis is to develop methods which can spatially and adaptively infer the EAP, or ODF of water diffusion in regions with complex fiber configurations. In Chapter 3, we propose a penalized multi-scale adaptive regression model (PMARM) framework to spatially and adaptively infer the ODF of water diffusion in regions with complex fiber configurations. We first represent DW-MRI signals using Spherical Harmonic (SH) basis, then apply PMARM on advanced statistical methods to calculate the coefficients of SH representation, from which ODF representation is calculated using Funk-Radon transformation. PMARM reconstructs the ODF at each voxel by adaptively borrowing the spatial information from the neighboring voxels. We show in the real and simulated data sets that PMARM can substantially reduce the noise level, while improving the ODF reconstruction. In Chapter 4, we propose a robust multi-scale adaptive and sequential smoothing (MASS) method framework to robustly, spatially and adaptively infer the EAP of water diffusion in regions with complex fiber configurations. We first calculate spherical polar Fourier basis representation of the DW-MRI signals, and then apply MASS adaptively and sequentially updating SPF representation by borrowing the spatial information from the neighboring voxels. We show in the real and simulated data sets that MASS can reduce the angle detection errors on fiber crossing area and provides more accurate reconstructions than standard voxel-wise methods and robust MASS performs very well with the presence of outliers. In Chapter 5, we extend multi-scale adaptive method framework to dictionary learning methods, and show that by adding smoothing technique, we can significantly improve the accuracy of EAP reconstruction and reduce the angle detection errors on fiber crossing, even in very low signal-to-noise ratio situation.Doctor of Philosoph

Non-Negative Spherical Deconvolution (NNSD) for Fiber Orientation Distribution Function Estimation

International audienceIn diffusion Magnetic Resonance Imaging (dMRI), Spherical Deconvolution (SD) is a commonly used approach for estimating the fiber Orientation Distribution Function (fODF). As a Probability Density Function (PDF) that characterizes the distribution of fiber orientations, the fODF is expected to be non-negative and to integrate to unity on the continuous unit sphere S2 . However, many existing approaches, despite using continuous representation such as Spherical Harmonics (SH), impose non-negativity only on discretized points of S2. Therefore, non-negativity is not guaranteed on the whole S2 Existing approaches are also known to .exhibit false positive fODF peaks, especially in regions with low anisotropy, causing an over-estimation of the number of fascicles that traverse each voxel. This paper proposes a novel approach, called Non-Negative SD (NNSD), to overcome the above limitations. NNSD offers the following advantages. First, NNSD is the first SH based method that guarantees non-negativity of the fODF throughout the unit sphere. Second, unlike approaches such as Maximum Entropy SD (MESD), Cartesian Tensor Fiber Orientation Distribution (CT-FOD), and discrete representation based SD (DR-SD) techniques, the SH representation allows closed form of spherical integration, efficient computation in a low dimensional space resided by the SH coefficients, and accurate peak detection on the continuous domain defined by the unit sphere. Third, NNSD is significantly less susceptible to producing false positive peaks in regions with low anisotropy. Evaluations of NNSD in comparison with Constrained SD (CSD), MESD, and DR-SD (implemented using L1-regularized least-squares with non-negative constraint), indicate that NNSD yields improved performance for both synthetic and real data. The performance gain is especially prominent for high resolution (1.25 mm)^3 data

Joint Spatial-Angular Sparse Coding, Compressed Sensing, and Dictionary Learning for Diffusion MRI

Neuroimaging provides a window into the inner workings of the human brain to diagnose and prevent neurological diseases and understand biological brain function, anatomy, and psychology. Diffusion Magnetic Resonance Imaging (dMRI) is an emerging medical imaging modality used to study the anatomical network of neurons in the brain, which form cohesive bundles, or fiber tracts, that connect various parts of the brain. Since about 73% of the brain is water, measuring the flow, or diffusion of water molecules in the presence of fiber bundles, allows researchers to estimate the orientation of fiber tracts and reconstruct the internal wiring of the brain, in vivo. Diffusion MRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis has been from a voxel-wise vantage point with added spatial regularization considered post-hoc. In contrast, we propose a new joint spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly. This has the ability to improve many aspects of dMRI processing and analysis and re-envision the core representation of dMRI data from a local perspective to a global one. In this thesis, we propose three main contributions which take advantage of such joint spatial-angular representations to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. First, we will show that we can achieve sparser representations of dMRI by utilizing a global spatial-angular dictionary instead of a purely voxel-wise angular dictionary. As dMRI data is very large in size, we provide a number of novel extensions to popular spare coding algorithms that perform efficient optimization on a global-scale by exploiting the separability of our dictionaries over the spatial and angular domains. Next, compressed sensing is used to accelerate signal acquisition based on an underlying sparse representation of the data. We will show that our proposed representation has the potential to push the limits of the current state of scanner acceleration within a new compressed sensing model for dMRI. Finally, sparsity can be further increased by learning dictionaries directly from datasets of interest. Prior dictionary learning for dMRI learn angular dictionaries alone. Our third contribution is to learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. Traditionally, the problem of dictionary learning is non-convex with no guarantees of finding a globally optimal solution. We derive the first theoretical results of global optimality for this class of dictionary learning problems. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any general signal types that can benefit from separable dictionaries. We hope the contributions in this thesis may be adopted in the larger signal processing, computer vision, and machine learning communities. dMRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Computationally speaking, researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis is from a voxel-wise, or angular, vantage point with post-hoc consideration of their local spatial neighborhoods. In contrast, we propose a new global spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly, to improve many aspects of dMRI processing and analysis, including the important need for accelerating the otherwise time-consuming acquisition of advanced dMRI protocols. In this thesis, we propose three main contributions which utilize our joint spatial-angular representation to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. We will show that sparser codes are possible by utilizing a global dictionary instead of a voxel-wise angular dictionary. This allows for a reduction of the number of measurements needed to reconstruct a dMRI signal to increase acceleration using compressed sensing. Finally, instead of learning angular dictionaries alone, we learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. In addition, this problem is non-convex and so we derive the first theories to guarantee convergence to a global minimum. As dMRI data is very large in size, we provide a number of novel extensions to popular algorithms that perform efficient optimization on a global-scale by exploiting the separability of our global dictionaries over the spatial and angular domains. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any separable dictionary setting which we hope may be adopted in the larger image processing, computer vision, and machine learning communities

Higher-Order Tensors and Differential Topology in Diffusion MRI Modeling and Visualization

Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) is a noninvasive method for creating three-dimensional scans of the human brain. It originated mostly in the 1970s and started its use in clinical applications in the 1980s. Due to its low risk and relatively high image quality it proved to be an indispensable tool for studying medical conditions as well as for general scientific research. For example, it allows to map fiber bundles, the major neuronal pathways through the brain. But all evaluation of scanned data depends on mathematical signal models that describe the raw signal output and map it to biologically more meaningful values. And here we find the most potential for improvement. In this thesis we first present a new multi-tensor kurtosis signal model for DW-MRI. That means it can detect multiple overlapping fiber bundles and map them to a set of tensors. Compared to other already widely used multi-tensor models, we also add higher order kurtosis terms to each fiber. This gives a more detailed quantification of fibers. These additional values can also be estimated by the Diffusion Kurtosis Imaging (DKI) method, but we show that these values are drastically affected by fiber crossings in DKI, whereas our model handles them as intrinsic properties of fiber bundles. This reduces the effects of fiber crossings and allows a more direct examination of fibers. Next, we take a closer look at spherical deconvolution. It can be seen as a generalization of multi-fiber signal models to a continuous distribution of fiber directions. To this approach we introduce a novel mathematical constraint. We show, that state-of-the-art methods for estimating the fiber distribution become more robust and gain accuracy when enforcing our constraint. Additionally, in the context of our own deconvolution scheme, it is algebraically equivalent to enforcing that the signal can be decomposed into fibers. This means, tractography and other methods that depend on identifying a discrete set of fiber directions greatly benefit from our constraint. Our third major contribution to DW-MRI deals with macroscopic structures of fiber bundle geometry. In recent years the question emerged, whether or not, crossing bundles form two-dimensional surfaces inside the brain. Although not completely obvious, there is a mathematical obstacle coming from differential topology, that prevents general tangential planes spanned by fiber directions at each point to be connected into consistent surfaces. Research into how well this constraint is fulfilled in our brain is hindered by the high precision and complexity needed by previous evaluation methods. This is why we present a drastically simpler method that negates the need for precisely finding fiber directions and instead only depends on the simple diffusion tensor method (DTI). We then use our new method to explore and improve streamsurface visualization.<br /

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018