Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging.

We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data

Fast, Model-Free , Analytical ODF Reconstruction from the Q-Space Signal

International audienceThe orientation distribution function (ODF) is very important in diffusion MRI. There are two types of ODFs. One is proposed using radial projection in Q-ball imaging [7]. Another one is the marginal pdf proposed in diffusion spectrum imaging (DSI) [8]. Since the marginal pdf is much sharper and mathematically correct, it could be more useful. Recently some reconstruction methods were proposed for this kind of ODF [1, 6]. They are both based on mono-exponential model, globally or locally, which has some intrinsic modeling error [4]. Although the authors in [1] extended the mono-exponential model to multi-exponential model, this multi-exponential model needs to be estimated non-linearly for every voxel and only in some special sampling scheme it has a analytical solution. Here we give a model-free analytical reconstruction method based on the Spherical Polar Fourier expression of the signal. It can estimate the ODF fast and analytically from the signal

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

Two canonical representations for regularized high angular resolution diffusion imaging

Two canonical representations for regularization of unit spherefunctions encountered in the context of high angular resolution diffusionimaging (HARDI) are discussed. One of these is based on spherical harmonicdecomposition, and its one-parameter extension via Tikhonov regularization.This case is well-established, and is mainly reviewed for thesake of completeness. The second one is new, and is based on a higherorder diffusion tensor decomposition. A homogeneous representation ofthis type has been proposed in the literature, but we show that thisis inconvenient for the purpose of regularization. We instead construct aheterogeneous representation that can be regarded as "canonical", to theextent that its behaviour under regularization mimics that of sphericalharmonics

Extrapolating fiber crossings from DTI data : can we gain the same information as HARDI?

High angular resolution diffusion imaging (HARDI) has proven to better characterize complex intra-voxel structures compared to its predecessor diffusion tensor imaging (DTI). However, the benefits from the modest acquisitions and significantly higher signal-to-noise ratios (SNRs) of DTI make it more attractive for use in clinical research. In this work we use contextual information derived from DTI data, to obtain similar crossing information as from HARDI data. We conduct synthetic phantom validation under different angles of crossing and different SNRs. We corroborate our findings from the phantom study to real human data. We show that with extrapolation of the contextual information the obtained crossings are the same as the ones from the HARDI data, and the robustness to noise is significantly better

Insight into the fundamental trade-offs of diffusion MRI from polarization-sensitive optical coherence tomography in ex vivo human brain

In the first study comparing high angular resolution diffusion MRI (dMRI) in the human brain to axonal orientation measurements from polarization-sensitive optical coherence tomography (PSOCT), we compare the accuracy of orientation estimates from various dMRI sampling schemes and reconstruction methods. We find that, if the reconstruction approach is chosen carefully, single-shell dMRI data can yield the same accuracy as multi-shell data, and only moderately lower accuracy than a full Cartesian-grid sampling scheme. Our results suggest that current dMRI reconstruction approaches do not benefit substantially from ultra-high b-values or from very large numbers of diffusion-encoding directions. We also show that accuracy remains stable across dMRI voxel sizes of 1 ​mm or smaller but degrades at 2 ​mm, particularly in areas of complex white-matter architecture. We also show that, as the spatial resolution is reduced, axonal configurations in a dMRI voxel can no longer be modeled as a small set of distinct axon populations, violating an assumption that is sometimes made by dMRI reconstruction techniques. Our findings have implications for in vivo studies and illustrate the value of PSOCT as a source of ground-truth measurements of white-matter organization that does not suffer from the distortions typical of histological techniques.Published versio

Efficient Computation of PDF-Based Characteristics from Diffusion MR Signal

International audienceWe present a general method for the computation of PDF-based characteristics of the tissue micro-architecture in MR imaging. The approach relies on the approximation of the MR signal by a series expansion based on Spherical Harmonics and Laguerre-Gaussian functions, followed by a simple projection step that is efficiently done in a finite dimensional space. The resulting algorithm is generic, flexible and is able to compute a large set of useful characteristics of the local tissues structure. We illustrate the effectiveness of this approach by showing results on synthetic and real MR datasets acquired in a clinical time-frame

Beyond the diffusion tensor model: The package dti

Diffusion weighted imaging is a magnetic resonance based method to investigate tissue micro-structure especially in the human brain via water diffusion. Since the standard diffusion tensor model for the acquired data failes in large portion of the brain voxel more sophisticated models have bee developed. Here, we report on the package dti and how some of these models can be used with the package

Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes

In diffusion MRI (dMRI), a good sampling scheme is important for efficient acquisition and robust reconstruction. Diffusion weighted signal is normally acquired on single or multiple shells in q-space. Signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue, or the laboratory coordinate frame. The Electrostatic Energy Minimization (EEM) method, originally proposed for single shell sampling scheme in dMRI, was recently generalized to multi-shell schemes, called Generalized EEM (GEEM). GEEM has been successfully used in the Human Connectome Project (HCP). However, EEM does not directly address the goal of optimal sampling, i.e., achieving large angular separation between sampling points. In this paper, we propose a more natural formulation, called Spherical Code (SC), to directly maximize the minimal angle between different samples in single or multiple shells. We consider not only continuous problems to design single or multiple shell sampling schemes, but also discrete problems to uniformly extract sub-sampled schemes from an existing single or multiple shell scheme, and to order samples in an existing scheme. We propose five algorithms to solve the above problems, including an incremental SC (ISC), a sophisticated greedy algorithm called Iterative Maximum Overlap Construction (IMOC), an 1-Opt greedy method, a Mixed Integer Linear Programming (MILP) method, and a Constrained Non-Linear Optimization (CNLO) method. To our knowledge, this is the first work to use the SC formulation for single or multiple shell sampling schemes in dMRI. Experimental results indicate that SC methods obtain larger angular separation and better rotational invariance than the state-of-the-art EEM and GEEM. The related codes and a tutorial have been released in DMRITool.Comment: Accepted by IEEE transactions on Medical Imaging. Codes have been released in dmritool https://diffusionmritool.github.io/tutorial_qspacesampling.htm