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Section: New Results

Computational Diffusion MRI

Improving dMRI Signal and Acquisitions

Diffusion MRI Signal Reconstruction with Continuity Constraint and Optimal Regularization

Participants : Emmanuel Caruyer, Rachid Deriche.

In diffusion MRI, the reconstruction of the full Ensemble Average Propagator (EAP) provides new insights in the diffusion process and the underlying microstructure. The reconstruction of the signal in the whole Q-space is still extremely challenging however. It requires very long acquisition protocols, and robust reconstruction to cope with the very low SNR at large b-values. Several reconstruction methods were proposed recently, among which the Spherical Polar Fourier (SPF) expansion, a promising basis for signal reconstruction. Yet the reconstruction in SPF is still subject to noise and discontinuity of the reconstruction. In this work, we present a method for the reconstruction of the diffusion attenuation in the whole Q-space, with a special focus on continuity and optimal regularization. We derive a modified Spherical Polar Fourier (mSPF) basis, orthonormal and compatible with SPF, for the reconstruction of a signal with continuity constraint. We also derive the expression of a Laplace regularization operator in the basis, together with a method based on generalized cross validation for the optimal choice of the parameter. Our method results in a noticeable dimension reduction as compared with SPF. Tested on synthetic and real data, the reconstruction with this method is more robust to noise and better preserves fiber directions and crossings.

This work has been published in [13]

A Computational Framework for Experimental Design in Diffusion MRI

Participants : Emmanuel Caruyer, Rachid Deriche.

In this work, we develop a computational framework for optimal design of experiment in parametric signal reconstruction. We apply this to the optimal design of one dimensional Q-space, Q-ball imaging and multiple Q-shell experimental design. We present how to construct sampling scheme leading to minimal condition number, and compare to state-of-the-art sampling methods. We show in particular a better noise performance of these scheme through Monte-Carlo simulations for the reconstruction of synthetic signal. This demonstrates the impact of this computational framework on acquisition in diffusion MRI.

This work has been published in [16]

Parametric Dictionary Learning in Diffusion MRI

Participants : Sylvain Merlet, Emmanuel Caruyer, Aurobrata Ghosh, Rachid Deriche.

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

In this work, we propose an approach to exploit the ability of compressive sensing to recover diffusion MRI signal and its characteristics from a limited number of samples. Our approach is threefold. First, we learn and design a parametric dictionary from a set of training diffusion data. This provides a highly sparse representation of the diffusion signal. The use of a parametric method presents several advantages: we design a continuous representation of the signal, from which we can analytically recover some features such as the ODF; besides, the dictionary we train is acquisition-independent. Next, we use this sparse representation to reconstruct the signal of interest, using cross-validation to assess the optimal regularization parameter for each signal reconstruction. The use of cross-validation is critical in the L1 minimization problem, as the choice of the parameter is sensitive to the noise level, the number of samples, and the data sparsity. Third, we use a polynomial approach to accurately extract ODF maxima. Finally, we motivate and describe the choice of experimental parameters for the HARDI contest.

This work has been published in [26] .

Diffusion and Multiple Orientations from 1.5 MR Systems with Limited Gradient Tables

Participants : Sylvain Merlet, Rachid Deriche, Kevin Whittingstall [Radiology department, Université de Sherbrooke, Québec, Canada] , Maxime Descoteaux [Sherbrooke Connectivity Imaging Laboratory, Computer Science Departement, Université de Sherbrooke, Québec, Canada] .

This work has been performed within the framework of the Brain Connectivities Associate Team.

Diffusion MRI (dMRI) enables the quantification of water diffusion, influenced by the structure of biological tissues, from the acquisition of diffusion weighted magnetic resonance images (DW-MRI). While recent advances enable to recover complex fiber geometries using diffusion measurements along various sampling schemes of high order, some older MR systems work with limited gradient tables (ex: maximum of 6 or 12 directions). These systems are designed for Diffusion Tensor Imaging (DTI). Several hospitals and research institutes in the world are limited by these fixed DTI gradient sets. Therefore, groups that want to perform state-of-the-art tractography using high angular resolution diffusion imaging (HARDI) data are penalized and can only perform DTI tractography on their old system. The Gaussian assumption of the tensor model, in DTI, is an over simplification of the diffusion phenomenon of water molecules in the brain and thus cannot resolve crossing fibers. In this work, we show that new diffusion signal modeling and processing techniques enable to capture complex angular structure of the diffusion process even from a reduced gradient direction set arising from an older MR system.

This work has been published in [27] .

A Robust variational approach for simultaneous smoothing and estimation of DTI

Participants : Rachid Deriche, Meizhu Liu [Department of CISE, University of Florida, Gainesville, USA] , Baba C. Vemuri [Department of CISE, University of Florida, Gainesville, USA] .

Estimating diffusion tensors is an essential step in many applications — such as diffusion tensor image (DTI) registration, segmentation and fiber tractography. Most of the methods proposed in the literature for this task are not simultaneously statistically robust and feature preserving techniques. In this work, we propose a novel and robust variational framework for simultaneous smoothing and estimation of diffusion tensors from diffusion MRI. Our variational principle makes use of a recently introduced total Kullback–Leibler (tKL) divergence for DTI regularization. tKL is a statistically robust dissimilarity measure for diffusion tensors, and regularization by using tKL ensures the symmetric positive definiteness of tensors automatically. Further, the regularization is weighted by a non-local factor adapted from the conventional non-local means filters. Finally, for the data fidelity, we use the nonlinear least-squares term derived from the Stejskal–Tanner model. We present experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples.

This work has been accepted for publication in NeuroImage  [63] .

Modeling in Diffusion MRI

Fast and Analytical EAP Approximation from a 4th Order Tensor

Participants : Aurobrata Ghosh, Rachid Deriche.

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

Generalized Diffusion Tensor Imaging (GDTI) was developed to model complex Apparent Diffusivity Coefficient (ADC) using Higher Order Tensors (HOT) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile doesn't correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the Ensemble Average Propagator (EAP). Though interesting methods for estimating a positive ADC using 4th order diffusion tensors were developed, GDTI in general was overtaken by other approaches, e.g. the Orientation Distribution Function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this work, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th order tensors on synthetic data and in vivo human data.

This work has been accepted for publication in the International Journal of Biomedical Imaging  [54] .

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 po- tential 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 submitted to Medical Image Analysis and has been accepted for a publication to appear early 2013  [57] .

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 is under submission.

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 submitted and is under review.

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

Participants : Rachid Deriche, Jian Cheng [ATHENA and LIAMA, China] , Tianzi Jiang [LIAMA, China] .

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

In 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 S2 and R3. 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 work, 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.

This work has been published in [11] and [18] .

An Intrinsic Diffeomorphism Invariant Riemannian Framework for Probability Density Function Computing in diffusion MRI

Participants : Rachid Deriche, Jian Cheng [ATHENA and LIAMA, China] , Aurobrata Ghosh, Tianzi Jiang [LIAMA, China] .

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

In High Angular Resolution Imaging (HARDI),Ensemble Average Propagator (EAP) and Orientation Distribution Function (ODF) are two important Probability Density Functions (PDFs), which describe the diffusion probability respectively in 3D space and along directions. Fisher information metric has been successfully applied in Diffusion Tensor Imaging (DTI) on tensor estimation, filtering, registration, statistical analysis, etc. However, to our knowledge, existing works in HARDI mainly focus on ODF/EAP estimation, not on ODF and EAP data processing. In this work, we propose a general state-of-the-art Riemannian framework as a mathematical tool to process such PDF data, by representing the square root of the PDF, called wavefunction based on quantum mechanics, as a linear combination of some orthonormal basis functions. The proposed Riemannian framework is showed to be a natural extension of previous Riemannian framework for tensors. We deduced the Riemannian metric for the PDF family via orthonormal basis representation, and proved the statistical manifold to be a convex subset of a high dimensional sphere. In this framework, the exponential map, logarithmic map and geodesic have closed forms, and the weighted Riemannian mean and median uniquely exist. Moreover, we generalized the Log-Euclidean framework and the Geodesic Anisotropy (GA) form tensors to ODFs/EAPs. The theoretical results can be applied to any general PDF data under any orthonormal basis representation. Furthermore we analyzed theoretically the similarities and differences between the Riemannian frameworks for EAPs, ODFs and for tensors, and demonstrated the proposed Riemannian metric is diffeomorphism invariant, which is the natural extension of the previous affine-invariant metric for tensors. Some potential applications were proposed via the Riemannian operations on the ODF/EAP field, such as anisotropy description via GA, nonnegative definite ODF/EAP estimation, interpolation, filtering, Principal Geodesic Analysis (PGA) and atlas estimation. The Riemannian framework and its applications were validated in synthetic, phantom and real data. The experiments demonstrated that the Riemannian framework is very useful for ODF/EAP computing, although the results from Riemannian metric and Euclidean metric are similar for ODFs but much different for EAPs.

This work has been published in [11] . A longer version has been submitted and is under revision for the journal IEEE transaction on Medical Imaging.

Ensemble Average Propagator Reconstruction via Compressed Sensing: Discrete or Continuous Bases ?

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

This work has been partly supported within the framework of the Brain Connectivities Associate Team.

In this work, we propose to compare the sparsity of two classes of representations for the EAP : The discrete representations, via the Haar, Daubechies-Cohen-Fauveau (DCF) 5-3, DCF 9-7 wavelets bases, and the continuous representations, via Spherical Polar Fourier (SPF) and 3D-SHORE bases.

This work has been published in [28] .

Parametric dictionary learning for modeling EAP and ODF in diffusion MRI

Participants : Sylvain Merlet, Emmanuel Caruyer, Rachid Deriche.

In this work, we propose an original and efficient approach to exploit the ability of Compressed Sensing (CS) to recover Diffusion MRI (dMRI) signals 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 sparsely represent the diffusion signal have already been performed. However and contrarly to what has been presented in CS dMRI, in this work we propose and advocate the use of a well adapted learned dictionary and show that it leads to a sparser signal estimation as well as to an efficient reconstruction of very important diffusion features. We first propose to learn and design a sparse and parametric dictionary from a set of training diffusion data. Then, we propose a framework to analytically estimate in closed form two important diffusion features : the EAP and the ODF. Various experiments on synthetic, phantom and human brain data have been carried out and promising results with reduced number of atoms have been obtained on diffusion signal reconstruction, thus illustrating the added value of our method over state-of-the-art SHORE and SPF based approaches.

This work has been published in [25] .

Constrained Diffusion Kurtosis Imaging Using Ternary Quartics and MLE

Participants : Tristan Milne [Queen's University, Kingston, Ontario, Canada] , Aurobrata Ghosh, Rachid Deriche.

This work has been partly supported by the Inria International Internship Program.

We present a ternary quartic based approach with an improved gradient based optimization scheme for diffusion kurtosis imaging to estimate constrained and physically realistic diffusion and kurtosis tensors. We account for the signal noise by considering a maximum likelihood estimation based on the Rician noise model. Diffusion kurtosis imaging (DKI) is a recent important improvement over the diffusion tensor imaging (DTI) model that quantifies the degree of non-Gaussian diffusion in a tissue using a 3D 4th order tensor. However, DKI estimation needs to consider three constraints to be physically relevant. By adopting the implicit ternary quartic parameterization which allows to elegantly impose a positivity constraint on the kurtosis tensor and by employing gradient based optimization schemes, we show dramatically improved performance in terms of estimation time and quality. We derive the mathematical framework and show results on extensive synthetic data experiments.

This work has been published in [30] . A longer version has been submitted and is under revision for the journal Magnetic Resonance in Medicine.

From DW-MRI to Fiber Pathways and Microstructures Recovery

From Diffusion MRI to Brain Connectomics

Participants : Aurobrata Ghosh, Rachid Deriche.

Diffusion MRI (dMRI) is a unique modality of MRI which allows one to indirectly examine the microstructure and integrity of the cerebral white matter in vivo and non-invasively. Its success lies in its capacity to reconstruct the axonal connectivity of the neurons, albeit at a coarser resolution, without having to operate on the patient, which can cause radical alterations to the patient's cognition. Thus dMRI is beginning to assume a central role in studying and diagnosing important pathologies of the cerebral white matter, such as Alzheimer's and Parkinson's diseases, as well as in studying its physical structure in vivo. In this work, we present an overview of the mathematical tools that form the framework of dMRI – from modelling the MRI signal and measuring diffusion properties, to reconstructing the axonal connectivity of the cerebral white matter, i.e., from Diffusion Weighted Images (DWIs) to the human connectome.

This work will be published in  [55] .

Biomarkers for HARDI : 2nd & 4th Order Tensor Invariants

Participants : Rachid Deriche, Aurobrata Ghosh, Théodore Papadopoulo.

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

In this paper, we explore the theory of tensor invariants as a mathematical framework for computing new biomarkers for HARDI. We present and explain the integrity basis, basic invariants and principal invariants of 2nd & 4th order tensors to expand on a recently proposed paper on 4th order tensor invariants. We present the mathematical results and compute the basic and principal invariants on a controlled synthetic dataset and an in vivo human dataset. We show how the integrity bases of these two sets of invariants can form a promising framework for developing new biomarkers for HARDI.

This work has been published in [22] .

Generalized Invariants of a 4th order tensor: Building blocks for new biomarkers in dMRI

Participants : Aurobrata Ghosh, Théodore Papadopoulo, Rachid Deriche.

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

This paper presents a general and complete (up to degree 4) set of invariants of 3D 4th order tensors with respect to SO3. The invariants to SO3 for the 2nd order diffusion tensor are well known and play a crucial role in deriving important biomarkers for DTI, e.g. MD, FA, RA, etc. But DTI is limited in regions with fiber heterogeneity and DTI biomarkers severely lack specificity. 4th order tensors are both a natural extension to DTI and also form an alternate basis to spherical harmonics for spherical functions. This paper presents a systematic method for computing the SO3 invariants of 3D 4th order tensors, derives relationships between the new (generalized) invariants and existing invariants and shows results on synthetic and real data. It also presents, hitherto unknown, new invariants for 4th order tensors. Analogously to DTI, these new invariants can perhaps form building blocks for new biomarkers.

This work has been published in [23] .

Tractography via the Ensemble Average Propagator in diffusion MRI

Participants : Sylvain Merlet, Anne-Charlotte Philippe, Rachid Deriche, Maxime Descoteaux [Sherbrooke Connectivity Imaging Laboratory, Computer Science Departement, Université de Sherbrooke, Québec, Canada] .

This work has been partly supported within the framework of the Brain Connectivities Associate Team.

It's well known that in diffusion MRI (dMRI), fibre crossing is an important problem for most existing diffusion tensor imaging (DTI) based tractography algorithms. To overcome these limitations, High Angular Resolution Diffusion Imaging (HARDI) based tractography has been proposed with a particular emphasis on the the Orientation Distribution Function (ODF). In this work, we advocate the use of the Ensemble Average Propagator (EAP) instead of the ODF for tractography in dMRI and propose an original and efficient EAP-based tractography algorithm that outperforms the classical ODF-based tractography, in particular, in the regions that contain complex fibre crossing configurations. Various experimental results including synthetic, phantom and real data illustrate the potential of the approach and clearly show that our method is especially efficient to handle regions where fiber bundles are crossing, and still well handle other fiber bundle configurations such as U-shape and kissing fibers.

This work has been published in [29] .

Using Radial NMR Profiles to Characterize Pore Size Distributions

Participants : Rachid Deriche, John Treilhard [Queen's University, Ontario, Canada] .

This work has been partly supported by the Inria International Internship Program.

Extracting information about axon diameter distributions in the brain is a challenging task which provides useful information for medical purposes; for example, the ability to characterize and monitor axon diameters would be useful in diagnosing and investigating diseases like amyotrophic lateral sclerosis (ALS) or autism. In [78] , three families of operators are defined, whose action upon an NMR attenuation signal extracts the moments of the pore size distribution of the ensemble under consideration; also a numerical method is proposed to continuously reconstruct a discretely sampled attenuation profile using the eigenfunctions of the simple harmonic oscillator Hamiltonian – the SHORE basis. The work we have performed here extends this method to other bases that can offer a better description of attenuation signal behaviour – in particular, we proposed the use of the radial Spherical Polar Fourier (SPF) basis. Testing is performed to contrast the efficacy of the radial SPF basis and SHORE basis in practical attenuation signal reconstruction. The robustness of the method to additive noise is tested and analyzed. We demonstrated that a low-order attenuation signal reconstruction outperforms a higher-order reconstruction in subsequent moment estimation under noisy conditions. We proposed the simulated annealing algorithm for basis function scale parameter estimation. Finally, analytic expressions are derived and presented for the action of the operators on the radial SPF basis (obviating the need for numerical integration, thus avoiding a spectrum of possible sources of error).

This work has been published [20] .

Elliptic Fourier Features of Brain White Matter Pathways

Participants : Rachid Deriche, Ali Demir [Sabancy University, TU] , Gozde Unal [Sabancy University, TU] .

Magnetic resonance imaging provides diffusion weighted images (DMRI), which non-invasively reconstruct the brain white matter pathways. DMRI is used to study brain white matter diseases as well as aid surgical planning. As localization of different white matter pathways surrounding a pathology is crucial for surgical planning, automatic extraction and classification of different anatomical white matter pathways pre-operatively becomes an important computational tool. In this work, we propose a method for classification of brain white matter pathways based on 3D elliptic Fourier descriptors, which are extended from the 2D elliptic Fourier descriptors. We performed experiments and validation of the proposed method on a white matter atlas space and on real pathological cases.

This work has been published [41] .