Section: New Results

Modeling in Diffusion MRI

A computational diffusion MRI and parametric dictionary learning framework for modeling the diffusion signal and its features

Participants : Sylvain Merlet, Emmanuel Caruyer [SBIA, University of Pennsylvania Medical School,USA] , Aurobrata Ghosh, Rachid Deriche.

In this work, we first propose an original and efficient computational framework to model continuous diffusion MRI (dMRI) signals and analytically recover important diffusion features such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF). Then, we develop an efficient parametric dictionary learning algorithm and exploit the sparse property of a well-designed dictionary to recover the diffusion signal and its features with a reduced number of measurements. The properties and potentials of the technique are demonstrated using various simulations on synthetic data and on human brain data acquired from 7T and 3T scanners. It is shown that the technique can clearly recover the dMRI signal and its features with a much better accuracy compared to state-of-the-art approaches, even with a small and reduced number of measurements. In particular, we can accurately recover the ODF in regions of multiple fiber crossing, which could open new perspectives for some dMRI applications such as fiber tractography.

This work has been published in Medical Image Analysis  [21] . It is part of Merlet's PhD thesis defended on Sept. 11th, 2013  [11] .

Continuous diffusion signal, EAP and ODF estimation via compressive sensing in diffusion MRI

Participants : Sylvain Merlet, Rachid Deriche.

In this work, we exploit the ability of Compressed Sensing (CS) to recover the whole 3D Diffusion MRI (dMRI) signal from a limited number of samples while efficiently recovering important diffusion features such as the Ensemble Average Propagator (EAP) and the Orientation Distribution Function (ODF). Some attempts to use CS in estimating diffusion signals have been done recently. However, this was mainly an experimental insight of CS capabilities in dMRI and the CS theory has not been fully exploited. In this work, we also propose to study the impact of the sparsity, the incoherence and the RIP property on the reconstruction of diffusion signals. We show that an efficient use of the CS theory enables to drastically reduce the number of measurements commonly used in dMRI acquisitions. Only 20–30 measurements, optimally spread on several b-value shells, are shown to be necessary, which is less than previous attempts to recover the diffusion signal using CS. This opens an attractive perspective to measure the diffusion signals in white matter within a reduced acquisition time and shows that CS holds great promise and opens new and exciting perspectives in diffusion MRI (dMRI).

This work has been published in Medical Image Analysis [22] . It is part of Merlet's PhD thesis defended on Sept. 11th, 2013  [11] .

Constrained diffusion kurtosis imaging using ternary quartics & MLE

Participants : Aurobrata Ghosh, Tristan Milne, Rachid Deriche.

Diffusion kurtosis imaging (DKI) is a recent improvement over diffusion tensor imaging that characterizes tissue by quantifying non-gaussian diffusion using a 3D fourth-order kurtosis tensor. DKI needs to consider three constraints to be physically relevant. Further, it can be improved by considering the Rician signal noise model. A DKI estimation method is proposed that considers all three constraints correctly, accounts for the signal noise and incorporates efficient gradient-based optimization to improve over existing methods.

In this work, the ternary quartic parameterization is utilized to elegantly impose the positivity of the kurtosis tensor implicitly. Sequential quadratic programming with analytical gradients is employed to solve nonlinear constrained optimization efficiently. Finally, a maximum likelihood estimator based on Rician distribution is considered to account for signal noise.

Extensive experiments conducted on synthetic data verify a MATLAB implementation by showing dramatically improved performance in terms of estimation time and quality. Experiments on in vivo cerebral data confirm that in practice the proposed method can obtain improved results.

This work has been published in [18] .

Compressive Sensing DSI

Participants : Sylvain Merlet, Michael Paquette [Sherbrooke Connectivity Imaging Laboratory, Computer Science Departement, Université de Sherbrooke, Québec, Canada] , Maxime Descoteaux [Sherbrooke Connectivity Imaging Laboratory, Computer Science Departement, Université de Sherbrooke, Québec, Canada] , Rachid Deriche.

Compressive Sensing (CS) offers an efficient way to decrease the number of measurements required in Diffusion Spectrum Imaging (DSI). This method aims to reconstruct the Ensemble Average Propagator (EAP) and, for the purpose of this contest, we compute the numerical Orientation Distribution Function (ODF) by integrating the EAP over a solid angle. In this abstract, we briefly describe three important points underlying the CS technique in order to accelerate DSI, namely the sparsity, the Restricted Isometry Property (RIP) and the L1 reconstruction scheme. Due to the high b-values required in the sampling protocol, our approach enters the heavyweight sampling category. Nevertheless, only 64 measurements are used for the reconstruction.

This work has been published in [31] . It is part of Merlet's PhD thesis defended on Sept. 11th, 2013  [11] .

4th Order symmetric tensors and positive ADC modelling

Participants : Aurobrata Ghosh, Rachid Deriche.

High Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn't guarantee positivity. In this work, we address the problem of estimating 4th order tensors with positive diffusion profiles. Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert's theorem on non-negative ternary quartics. In this work, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint.

This work has been published in [37] .

Higher-Order tensors in diffusion imaging: A survey

Participants : Thomas Schultz [MPI for Intelligent Systems, Tubingen, Germany] , Andrea Fuster [Eindhoven University of Technology, The Netherlands] , Aurobrata Ghosh, Luc Florack [Eindhoven University of Technology, The Netherlands] , Rachid Deriche, Lek-Heng Lim [University of Chicago, USA] .

Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathematical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging. This survey gives a careful introduction to the foundations of higher-order tensor algebra, and explains how some concepts from linear algebra generalize to the higher-order case. From the application side, it reviews a variety of distinct higher-order tensor models that arise in the context of diffusion imaging, such as higher-order diffusion tensors, q-ball or fiber Orientation Distribution Functions (ODFs), and fourth-order covariance and kurtosis tensors. By bridging the gap between mathematical foundations and application, it provides an introduction that is suitable for practitioners and applied mathematicians alike, and propels the field by stimulating further exchange between the two.

This work has been published in [39] .

Regularized spherical polar fourier diffusion MRI with optimal dictionary learning

Participants : Jian Cheng [University of North Carolina at Chapel Hill,USA] , Tianzi Jiang [LIAMA, China] , Rachid Deriche, Shen Dinggang [University of North Carolina at Chapel Hill,USA] , Yap Pew-Thian [University of North Carolina at Chapel Hill,USA] .

One 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 work 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.

This work has been published in [27] .

Fiber orientation distribution from non-negative sparse recovery

Participants : Aurobrata Ghosh, Thinhinane Megherbi [USTHB, Algeria] , Linda Oulebsir-Boumghar [USTHB, Algeria] , Rachid Deriche.

We revisit the theory of spherical deconvolution and propose a new fiber orientation distribution (FOD) model that can efficiently reconstruct extremely narrow fiber-crossings from limited number of acquisitions. First, we show how to physically model fiber-orientations as rank-1 tensors. Then, we parameterize the FODs with tensors that are decomposable into non-negative sums of rank-1 tensors and finally, we propose a non-negative sparse recovery scheme to estimate FODs of any tensor order from limited acquisitions. Our method features three important advantages: (1) it estimates non-negative FODs, (2) it estimates the number of fiber-compartments, which need not be predefined and (3) it computes the fiber-directions directly, rendering maxima detection superfluous. We test for various SNRs on synthetic, phantom and real data and find our method accurate and robust to signal-noise: fibers crossing up to 23° are recovered from just 21 acquisitions. This opens new and exciting perspectives in diffusion MRI (dMRI), where our improved characterization of the FOD can be of great help for applications such as tractography.

This work has been published in [29] .

A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI

Participants : Aurobrata Ghosh, Elias Tsigaridas [PolSys Project-Team, Inria, Paris Rocquencourt, France] , Bernard Mourrain [Galaad Project-Team, Inria, Sophia Antipolis, Méditerranée, France] , Rachid Deriche.

This work has been partially supported by the ANR project NucleiPark and the France-Parkinson Association.

Antipodally symmetric spherical functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the spherical function. In this work we propose a method to automatically compute all the extrema of a spherical function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a spherical function can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the spherical function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision.

To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a spherical function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this work can be applied to any spherical function described in either the SH basis, ST basis or the HP basis.

This work has been published in  [19] .

ODF maxima computation using hill climbing algorithm

Participants : Thinhinane Megherbi [USTHB, Algeria] , Makhlouf Laouchedi [EMP, Algeria] , Houssem Khabatti [EMP, Algeria] , Linda Oulebsir-Boumghar [USTHB, Algeria] , Ishak Serrat [EMP, Algeria] , Vincent Perlbarg [LIF, UMRS 678, INSERM, UPMC - Paris 6] , Rachid Deriche.

Diffusion MRI (dMRI) is the only technique to probe in-vivo and non-invasively fiber structure of white matter. Diffusion was first modeled using the classical Second Order Diffusion Tensor model. However, this model is limited in regions of multiple fiber crossings and this has motivated the development of many approaches to extract crossing fibers. Methods like Diffusion Spectrum Imaging (DSI), High Angular Resolution Diffusion Imaging (HARDI) and the High Order Tensor techniques have been proposed to reconstruct specific functions like the Orientation Distribution Function (ODF) whose maxima do correspond to the directions of the multiple fibers.

In this work, we are interested to extract all the crossing fibers characterized as the maxima of the Orientation Distribution Function (ODF). A Hill Climbing algorithm based approach has been developed and implemented to efficiently and accurately extract all the fibers. Promising experimental results obtained with synthetic and real data illustrate the potential of the technique.

This work has been submitted to ISBI'2014 and accepted for presentation and publication.

On SHORE and SPF bases

Participants : Elodie Pozzi, Gonzalo Sanguinetti, Rachid Deriche.

The 3D Simple Harmonic Oscillation Reconstruction and Estimation (SHORE) basis and the Spherical Polar Fourier (SPF) basis were introduced recently to represent the dMRI signal in the full 3D Q-space. SPF presents some continuity problems at the origin which led to our development of the modified SPF basis we introduced to overcome this issue. These bases can be written with radial and angular functions. The radial part of the decomposition is a family of orthogonal functions (the Gauss-Laguerre functions) and the angular component are the spherical harmonic functions. Even though they look similar, they have different properties. The first objective of this work has been to analyse and clarify the differences between those bases. This has been accomplished by describing the spanned spaces. The second goal has been to classify the bases according to their continuity and differentiability and thus draw a more focused comparison between.

This on-going work will be submitted to a journal.