Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI

International audienceIn diffusion magnetic resonance imaging (dMRI), the Ensemble Average Propagator (EAP), also known as the propagator, describes completely the water molecule diffusion in the brain white matter without any prior knowledge about the tissue shape. In this paper, we describe a new and efficient method to accurately reconstruct the EAP in terms of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients, and efficiently combines it to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Our work provides an efficient analytical solution to estimate, from few measurements, the diffusion propagator at any radius. We also provide a new analytical solution to extract an important feature characterising the tissue microstructure: the Orientation Distribution Function (ODF). We illustrate and prove the effectiveness of our method in reconstructing the propagator and the ODF on both noisy multiple q-shell synthetic and phantom data

Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging

International audienceIn diffusion MRI (dMRI) domain, many High Angular Resolution Diffusion Imaging (HARDI) methods were proposed to estimate Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF). They normally need many samples, which limits their applications. Some Compressive Sensing (CS) based methods were proposed to estimate ODF in Q-Ball Imaging (QBI) from limited samples. However EAP estimation is much more difficult than ODF in QBI. Recently Spherical Polar Fourier Imaging (SPFI) was proposed to represent diffusion signal using Spherical Polar Fourier (SPF) basis without specific assumption on diffusion signals and analytically obtain EAP and ODF via the Fourier dual SPF (dSPF) basis from arbitrarily sampled signal. Normally the coefficients of SPF basis are estimated via Least Square with weighted L2 norm regularization (L2-SPFI). However, L2-SPFI needs a truncated basis to avoid overfitting, which brings some estimation errors. By considering the Fourier relationship between EAP and signal and the Fourier basis pair provided in SPFI, we propose a novel EAP estimation method, named L1-SPFI, to estimate EAP from limited samples using CS technique, and favorably compare it to the classical L2-SPFI method. L1-SPFI estimates the coefficients in SPFI using least square with weighted L1 norm regularization. The weights are designed to enhance the sparsity. L1-SPFI significantly accelerates the ordinary CS based Fourier reconstruction method. This is performed by using SPF basis pair in CS estimation process which avoids the numerical Fourier transform in each iteration step. By considering high order basis in L1 optimization, L1-SPFI improves EAP reconstruction especially for the angular resolution. The proposed L1-SPFI was validated by synthetic, phantom and real data. The CS EAP and ODF estimations are discussed in detail and we show that recovering the angular information from CS EAP requires much less samples than exact CS EAP reconstruction. Various experiments on synthetic, phantom and real data validate the fact that SPF basis can sparsely represent DW-MRI signals and L1-SPFI largely improves the CS EAP reconstruction especially the angular resolution

Regularized Spherical Polar Fourier Diffusion MRI with Optimal Dictionary Learning

International audienceOne important problem in diffusion MRI (dMRI) is to recover the diffusion weighted signal from only a limited number of samples in q-space. An ideal framework for solving this problem is Compressed Sensing (CS), which takes advantage of the signal's sparseness or compressibility, allowing the entire signal to be reconstructed from relatively few measurements. CS theory requires a suitable dictionary that sparsely represents the signal. To date in dMRI there are two kinds of Dictionary Learning (DL) methods: 1) discrete representation based DL (DR-DL), and 2) continuous representation based DL (CR-DL). Due to the discretization in q-space, DR-DL suffers from the numerical errors in interpolation and regridding. By considering a continuous representation using Spherical Polar Fourier (SPF) basis, this paper proposes a novel CR-DL based Spherical Polar Fourier Imaging, called DL-SPFI, to recover the diffusion signal as well as the Ensemble Average Propagator (EAP) in continuous 3D space with closed form. DL-SPFI learns an optimal dictionary from the space of Gaussian diffusion signals. Then the learned dictionary is adaptively applied for different voxels in a weighted LASSO framework to robustly recover the di ffusion signal and the EAP. Compared with the start-of-the-art CR-DL method by Merlet et al. and DRDL by Bilgic et al., DL-SPFI has several advantages. First, the learned dictionary, which is proved to be optimal in the space of Gaussian diffusion signal, can be applied adaptively for different voxels. To our knowledge, this is the first work to learn a voxel-adaptive dictionary. The importance of this will be shown theoretically and empirically in the context of EAP estimation. Second, based on the theoretical analysis of SPF basis, we devise an efficient learning process in a small subspace of SPF coefficients, not directly in q-space as done by Merlet et al.. Third, DL-SPFI also devises different regularization for different atoms in the learned dictionary for robust estimation, by considering the structural prior in the space of signal exemplars. We evaluate DL-SPFI in comparison to L1-norm regularized SPFI (L1-SPFI) with fixed SPF basis, and the DR-DL by Bilgic et al. The experiments on synthetic data and real data demonstrate that the learned dictionary is sparser than SPF basis and yields lower reconstruction error than Bilgic's method, even though only simple synthetic Gaussian signals were used for training in DL-SPFI in contrast to real data used by Bilgic et al

The analysis and application of dynamic MRI contrasts to grape berry biology

Magnetic resonance imaging (MRI) is a powerful, non-invasive imaging tool. When MRI is employed in the study biological systems, the acquired images reflect different aspects of system morphology and/or physiology. This thesis explores the application of relaxation and diffusion MRI to the study of different biological aspects of the fruit of the common grape vine, Vitis vinifera L., a highly valued botanical species. The results of this investigation have put forth a number of contributions to this area of research. The studies within this thesis began with a necessary validation for the application of diffusion MRI techniques to the grape berry using simulated cellular geometries to determine how broad plant cells could potentially influence the accurate reconstruction of the grape berry morphology. The result of this validation will also prove useful for other wide geometry applications wider than 10 μm. Relaxation and diffusion MRI was also used to study changes to berry morphology resulting from berry development and ripening. This study provided a novel perspective on grape berry development and demonstrated that diffusion anisotropy patterns correlated with the microstructure of the major pericarp tissues of grape berries, including the exocarp, outer and inner mesocarp, seed interior, as well as microstructural variations across grape berry development. This study also provided further evidence that the inner mesocarp striation patterns observed in the spin-spin relaxation weighted images of previous studies arise due to variations in cell width across the striation bands. Diffusion MRI was employed to investigate the morphological and physiological changes to occur within grape berries during fruit split, a costly source of fruit loss in vineyards. This study revealed water uptake through splits in the berry epidermis will result in the loss of parenchyma cell vitality about these wounds. The amount of water left standing on the surface of split grape berries may hence be an important determinant of the cellular response of the fruit to this trauma, and the subsequent establishment of adventitious fruit pathogens. Additionally, paramagnetically enhanced spin-lattice relaxation MRI was used to undertake a novel examination of the diffusive transport of manganese across the berry pericarp. The results of this study shows that the transport of manganese is within the berry xylem influences manganese exiting of ‘downstream’ of the pedicel, and that cellular membranes affect the spatial distribution of manganese across the berry pericarp. Manganese proved to be an excellent tracer for these experiments, and future investigations making use of paramagnetically enhanced relaxation MRI, perhaps employing other paramagnetic materials such as iron or copper, could prove to be valuable in determining how botanical species transport and store these materials within sink organs

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

Spherical Polar Fourier EAP and ODF Reconstruction via Compressed Sensing in Diffusion MRI

International audienceIn diffusion magnetic resonance imaging (dMRI), the Ensemble Average Propagator (EAP), also known as the propagator, describes completely the water molecule diffusion in the brain white matter without any prior knowledge about the tissue shape. In this paper, we describe a new and efficient method to accurately reconstruct the EAP in terms of the Spherical Polar Fourier (SPF) basis from very few diffusion weighted magnetic resonance images (DW-MRI). This approach nicely exploits the duality between SPF and a closely related basis in which one can respectively represent the EAP and the diffusion signal using the same coefficients, and efficiently combines it to the recent acquisition and reconstruction technique called Compressed Sensing (CS). Our work provides an efficient analytical solution to estimate, from few measurements, the diffusion propagator at any radius. We also provide a new analytical solution to extract an important feature characterising the tissue microstructure: the Orientation Distribution Function (ODF). We illustrate and prove the effectiveness of our method in reconstructing the propagator and the ODF on both noisy multiple q-shell synthetic and phantom data

Theoretical Analysis and Practical Insights on EAP Estimation via a Unified HARDI Framework

International audienceSince Diffusion Tensor Imaging (DTI) cannot describe complex non-Gaussian diffusion process, many techniques, called as single shell High Angular Resolution Diffusion Imaging (sHARDI) methods, reconstruct the Ensemble Average Propagator (EAP) or its feature Orientation Distribution Function (ODF) from diffusion weighted signals only in single shell. Q-Ball Imaging (QBI) and Diffusion Orientation Transform (DOT) are two famous sHARDI methods. However, these sHARDI methods have some intrinsic modeling errors or need some unreal assumptions. Moreover they are hard to deal with signals from different q-shells. Most recently several novel multiple shell HARDI (mHARDI) methods, including Diffusion Propagator Imaging (DPI), Spherical Polar Fourier Imaging (SPFI) and Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE), were proposed to analytically estimate EAP or ODF from multiple shell (or arbitrarily sampled) signals. These three methods all represent diffusion signal with some basis functions in spherical coordinate and use plane wave formula to analytically solve the Fourier transform. To our knowledge, there is no theoretical analysis and practical comparison among these sHARDI and mHARDI methods. In this paper, we propose a unified computational framework, named Analytical Fourier Transform in Spherical Coordinate (AFT-SC), to perform such theoretical analysis and practical comparison among all these five state-of-the-art diffusion MRI methods. We compare these five methods in both theoretical and experimental aspects. With respect to the theoretical aspect, some criteria are proposed for evaluation and some differences together with some similarities among the methods are highlighted. Regarding the experimental aspect, all the methods are compared in synthetic, phantom and real data. The shortcomings and advantages of each method are highlighted from which SPFI appears to be among the best because it uses an orthonormal basis that completely separates the spherical and radial information

Compressive Sensing Ensemble Average Propagator Estimation via L1 Spherical Polar Fourier Imaging

International audienceIn diffusion MRI (dMRI) domain, many High Angular Resolution Diffusion Imaging (HARDI) methods were proposed to estimate Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF). They normally need many samples, which limits their applications. Some Compressive Sensing (CS) based methods were proposed to estimate ODF in Q-Ball Imaging (QBI) from limited samples. However EAP estimation is much more difficult than ODF in QBI. Recently Spherical Polar Fourier Imaging (SPFI) was proposed to represent diffusion signal using Spherical Polar Fourier (SPF) basis without specific assumption on diffusion signals and analytically obtain EAP and ODF via the Fourier dual SPF (dSPF) basis from arbitrarily sampled signal. Normally the coefficients of SPF basis are estimated via Least Square with weighted L2 norm regularization (L2-SPFI). However, L2-SPFI needs a truncated basis to avoid overfitting, which brings some estimation errors. By considering the Fourier relationship between EAP and signal and the Fourier basis pair provided in SPFI, we propose a novel EAP estimation method, named L1-SPFI, to estimate EAP from limited samples using CS technique, and favorably compare it to the classical L2-SPFI method. L1-SPFI estimates the coefficients in SPFI using least square with weighted L1 norm regularization. The weights are designed to enhance the sparsity. L1-SPFI significantly accelerates the ordinary CS based Fourier reconstruction method. This is performed by using SPF basis pair in CS estimation process which avoids the numerical Fourier transform in each iteration step. By considering high order basis in L1 optimization, L1-SPFI improves EAP reconstruction especially for the angular resolution. The proposed L1-SPFI was validated by synthetic, phantom and real data. The CS EAP and ODF estimations are discussed in detail and we show that recovering the angular information from CS EAP requires much less samples than exact CS EAP reconstruction. Various experiments on synthetic, phantom and real data validate the fact that SPF basis can sparsely represent DW-MRI signals and L1-SPFI largely improves the CS EAP reconstruction especially the angular resolution

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