Christophe Lenglet

DTI segmentation by statistical surface evolution

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). A DTI produces, from a set of diffusion-weighted MR images, tensor-valued images where each voxel is assigned with a 3 times 3 symmetric, positive-definite matrix. This second order tensor is simply the covariance matrix of a local Gaussian process, with zero-mean, modeling the average motion of water molecules. As we will show in this paper, the definition of a dissimilarity measure and statistics between such quantities is a nontrivial task which must be tackled carefully. We claim and demonstrate that, by using the theoretically well-founded differential geometrical properties of the manifold of multivariate normal distributions, it is possible to improve the quality of the segmentation results obtained with other dissimilarity measures such as the Euclidean distance or the Kullback-Leibler divergence. The main goal of this paper is to prove that the choice of the probability metric, i.e., the dissimilarity measure, has a deep impact on the tensor statistics and, hence, on the achieved results. We introduce a variational formulation, in the level-set framework, to estimate the optimal segmentation of a DTI according to the following hypothesis: Diffusion tensors exhibit a Gaussian distribution in the different partitions. We must also respect the geometric constraints imposed by the interfaces existing among the cerebral structures and detected by the gradient of the DTI. We show how to express all the statistical quantities for the different probability metrics. We validate and compare the results obtained on various synthetic data-sets, a biological rat spinal cord phantom and human brain DTIs

Inferring White Matter Geometry from Diffusion Tensor MRI: Application to Connectivity Mapping

We introduce a novel approach to the cerebral white matter connectivity mapping from diffusion tensor MRI. DT-MRI is the unique non-invasive technique capable of probing and quantifying the anisotropic diffusion of water molecules in biological tissues. We address the problem of consistent neural fibers reconstruction in areas of complex diffusion profiles with potentially multiple fibers orientations. Our method relies on a global modelization of the acquired MRI volume as a Riemannian manifold M and proceeds in 4 majors steps: First, we establish the link between Brownian motion and diffusion MRI by using the Laplace-Beltrami operator on M. We then expose how the sole knowledge of the diffusion properties of water molecules on M is sufficient to infer its geometry. There exists a direct mapping between the diffusion tensor and the metric of M. Next, having access to that metric, we propose a novel level set formulation scheme to approximate the distance function related to a radial Brownian motion on M. Finally, a rigorous numerical scheme using the exponential map is derived to estimate the geodesics of M, seen as the diffusion paths of water molecules. Numerical experimentations conducted on synthetic and real diffusion MRI datasets illustrate the potentialities of this global approach.

Control Theory and Fast Marching Techniques for Brain Connectivity Mapping

We propose a novel, fast and robust technique for the computation of anatomical connectivity in the brain. Our approach exploits the information provided by Diffusion Tensor Magnetic Resonance Imaging (or DTI) and models the white matter by using Riemannian geometry and control theory. We show that it is possible, from a region of interest, to compute the geodesic distance to any other point and the associated optimal vector field. The latter can be used to trace shortest paths coinciding with neural fiber bundles. We also demonstrate that no explicit computation of those 3D curves is necessary to assess the degree of connectivity of the region of interest with the rest of the brain. We finally introduce a general local connectivity measure whose statistics along the optimal paths may be used to evaluate the degree of connectivity of any pair of voxels. All those quantities can be computed simultaneously in a Fast Marching framework, directly yielding the connectivity maps. Apart from being extremely fast, this method has other advantages such as the strict respect of the convoluted geometry of white matter, the fact that it is parameter-free, and its robustness to noise. We illustrate our technique by showing results on real and synthetic datasets. OurGCM(Geodesic Connectivity Mapping) algorithm is implemented in C++ and will be soon available on the web.

Variational, Geometric and Statistical Methods for Modeling Brain Anatomy and Function

We survey the recent activities of the Odyssée Laboratory in the area of the application of mathematics to the design of models for studying brain anatomy and function. We start with the problem of reconstructing sources in MEG and EEG, and discuss the variational approach we have developed for solving these inverse problems. This motivates the need for geometric models of the head. We present a method for automatically and accurately extracting surface meshes of several tissues of the head from anatomical magnetic resonance (MR) images. Anatomical connectivity can be extracted from diffusion tensor magnetic resonance images but, in the current state of the technology, it must be preceded by a robust estimation and regularization stage. We discuss our work based on variational principles and show how the results can be used to track fibers in the white matter (WM) as geodesics in some Riemannian space. We then go to the statistical modeling of functional magnetic resonance imaging (fMRI) signals from the viewpoint of their decomposition in a pseudo-deterministic and stochastic part that we then use to perform clustering of voxels in a way that is inspired by the theory of support vector machines and in a way that is grounded in information theory. Multimodal image matching is discussed next in the framework of image statistics and partial differential equations (PDEs) with an eye on registering fMRI to the anatomy. The paper ends with a discussion of a new theory of random shapes that may prove useful in building anatomical and functional atlases.

Variational Approaches to the Estimation, Regularizatinn and Segmentation of Diffusion Tensor Images

Diffusion magnetic resonance imaging probes and quantifies the anisotropic diffusion of water molecules in biological tissues, making it possible to non-invasively infer the architecture of the underlying structure. In this chapter, we present a set of new techniques for the robust estimation and regularization of diffusion tensor images (DTI) as well as a novel statistical framework for the segmentation of cerebral white matter structures from this type of dataset. Numerical experiments conducted on real diffusion weighted MRI illustrate the techniques and exhibit promising results.

DT-MRI estimation, regularization and fiber tractography

Diffusion tensor MRI probes and quantifies the anisotropic diffusion of water molecules in biological tissues, making it possible to non-invasively infer the architecture of the underlying structures. In this article, we present a set of new techniques for the estimation and regularization of diffusion tensors MRI datasets as well as a novel approach to the cerebral white matter connectivity mapping. Numerical experimentations conducted on real diffusion weighted MRI will exhibit promising results.

A statistical framework for DTI segmentation

We address the problem of the segmentation of cerebral white matter structures from diffusion tensor images (DTI). DTI can be estimated from a set of diffusion weighted images and provides tensor-valued images where each voxel is assigned with a 3 times 3 symmetric, positive-definite matrix. As we will show in this paper, the definition of a dissimilarity measure and statistics between tensors is a non trivial task which must be carefully tackled. We claim that, by using the differential geometrical properties of the manifold of multivariate normal distributions, it is possible to improve the quality of the segmentation obtained with other dissimilarity measures such as the Euclidean distance or the Kullback-Leibler divergence. Our goal is to prove that the choice of this probability metric has a deep impact on the tensor statistics and, hence, on the achieved results. We introduce a variational formulation to estimate the optimal segmentation of a diffusion tensor image. We show how to estimate diffusion tensors statistics for three different probability metrics and evaluate their respective performances. We validate and compare the results obtained on synthetic and real datasets

A fast and rigorous anisotropic smoothing method for DT-MRI

Tensors are nowadays an increasing research domain in different areas, especially in image processing, motivated for example by DT-MRI (diffusion tensor magnetic resonance imaging). In this paper, we exploit the theoretically well-founded differential geometrical properties of the space of multivariate normal distributions, where it is possible to define a Riemannian metric and express statistics on the manifold of symmetric positive definite matrices. We focus on the contributions of these tools to the anisotropic filtering and regularization of tensor fields. We present promising results on synthetic and real DT-MRI data