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

A preliminary study on the effect of motion correction on HARDI reconstruction

pre-printPost-acquisition motion correction is widely performed in diffusion-weighted imaging (DWI) to guarantee voxel-wise correspondence between DWIs. Whereas this is primarily motivated to save as many scans as possible if corrupted by motion, users do not fully understand the consequences of different types of interpolation schemes on the final analysis. Nonetheless, interpolation might increase the partial volume effect while not preserving the volume of the diffusion profile, whereas excluding poor DWIs may affect the ability to resolve crossing fibers especially with small separation angles. In this paper, we investigate the effect of interpolating diffusion measurements as well as the elimination of bad directions on the reconstructed fiber orientation diffusion functions and on the estimated fiber orientations. We demonstrate such an effect on synthetic and real HARDI datasets. Our experiments demonstrate that the effect of interpolation is more significant with small fibers separation angles where the exclusion of motion-corrupted directions decreases the ability to resolve such crossing fibers

Détection des croisements de fibre en IRM de diffusion par décomposition de tenseur : Approche analytique

National audienceL'IRM de diffusion (IRMd) est l'unique modalité qui permet d'explorer les structures neuronales de la substance blanche in-vivo et de manière non-invasive. La diffusion a d'abord été modélisée par le modèle du tenseur de diffusion du second ordre (DTI). Toutefois, ce modèle trouve rapidement ses limites dans les zones, nombreuses, où les fibres de la matière blanche se croisent. Pour surmonter cette limite et reconstruire les croisements de fibres, différentes approches ont été proposées telles que: l'imagerie à résonance magnétique (IRM) à haute résolution angulaire (HARDI) et les tenseurs d'ordre supérieur (HOT) ; ces méthodes permettent de reconstruire des fonctions telle que la fonction de distribution d'orientation de fibre (FOD) dont les maxima s'alignent sur les orientations des fibres multiples. Dans ce travail, on se propose d'extraire les directions des fibres caractérisées par les maxima de la fonction FOD. Pour cela, une approche analytique de décomposition de tenseur symétrique a été implémentée et efficacement adaptée pour extraire les directions des fibres avec précision. Différents résultats obtenus sur des données synthétiques et réelles illustrent l'efficacité de la méthode

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations

Evaluation des méthodes d'extraction des orientations locales des faisceaux de fibres par analyse quantitative de la connectivité

International audienceRecovering of the fiber orientations in each voxel constitutes an important step for the fiber tracking algorithms. In fact, the reliability of the resulted connectivity depends on how well the local fiber orientations were extracted. Based on the tractography results we evaluated and compared different methods of fiber orientations extraction. Thus, we analyzed quantitatively the resulted connectivity by using the Tractometer tool. This later allows by measuring a number of metrics to quantify the connections reliability and the tractography performance. All the methods of fiber orientations extraction were evaluated on two types of tractography algorithms, deterministic and probabilistic algorithms. Furthermore, all of these methods have been executed on two types of data, high angular resolution data acquired with 60 gradient directions and low angular resolution data, acquired with 30 gradient directions. These two types of data were corrupted with a Ricien noise of ratio SNR=20, 10. In this article, we present the results obtained by our validation and comparison work.La détection des orientations locales des faisceaux de fibres constitue une étape importante pour les algorithmes de suivi de fibres (tractographie).En effet, la fiabilité de la connectivité résultante dépend de la qualité de l'extraction de ces orientations locales de fibres. Sur la base des résultats produits au niveau de la tractographie, nous avons évalué et comparé un ensemble d'algorithmes d'extraction des orientations des faisceaux de fibres. Une analyse quantitative de la connectivité a été ainsi réalisée en utilisant un outil appelé le Tractometer. Cet outil permet grâce à un certain nombre de métriques de quantifier la fiabilité des connexions reconstruites et aussi la performance des algorithmes de suivi de fibre utilisés. Toutes les méthodes d'extraction des orientations de fibres mises en oeuvre ont été évaluées sur la base des résultats de deux types d'algorithmes de tractographie, déterministe et probabiliste. De plus, l'ensemble de ces méthodes ont été exécutées sur deux types de données de diffusion, des données à haute résolution angulaire de 60 directions de gradient et des données à basse résolution angulaire de 30 directions de gradient, ces deux jeux de données ont été corrompus par un bruit Ricien d'un rapport SNR de 20 puis de 10. Dans cet article, nous présentons les résultats obtenus par ce travail de validation et de comparaison

Discontinuous Fiber Composites, Volume II

Discontinuous fiber-reinforced polymers have gained importance in transportation industries due to their outstanding material properties, lower manufacturing costs and superior lightweight characteristics. One of the most attractive attributes of discontinuous fiber-reinforced composites is the ease with which they can be manufactured in large numbers, using injection and compression molding processes. The main aim of this Special Issue is to collect various investigations focused on the processing of discontinuous fiber-reinforced composites and the effect that processing has on fiber orientation, fiber length and fiber density distributions throughout the final product. Papers presenting investigations on the effect that fiber configurations have on the mechanical properties of the final composite products and materials were welcome in the Special Issue. Researchers who model and simulate processes involving discontinuous fiber composites as well as those performing experimental studies involving these composites were welcomed to submit papers. The authors were encouraged to present new models, constitutive laws, and measuring and monitoring techniques to provide a complete framework on these groundbreaking materials and to facilitate their use in different engineering applications

New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3 \rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.Comment: Revised and restructure