Riemannian geometry for the statistical analysis of diffusion tensor data

Abstract

The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positivedefinite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision.1

Adjugate Diffusion Tensors for Geodesic Tractography in White Matter

“…Another advantage with respect to other types of tracking algorithms is that geodesic tractography tends to be more robust to noise. Finally, it has the conceptual advantage that Riemannian geometry is a well understood and powerful theoretical machinery, facilitating mathematical modelling and algorithmics [2][3][4]6,12,[15][16][17][24][25][26]30,31,33,35].…”

Automatic Segmentation of the Optic Radiation Using DTI in Healthy Subjects and Patients with Glaucoma

“…The space of diffusion tensors is a convex subset of the vector space R (3) 2 and does not form a vector space using a Euclidean metric [12,31]. Thus, the decomposition of the diffusion tensors could result in non-physical negative eigenvalues.…”

“…is the matrix trace operator, n is the size of the diffusion tensors DT 1 and DT 2 . Fletcher and Joshi [12] deal with the space of diffusion tensors as a curved manifold called Riemannian symmetric space. They derived a Riemannian metric on the space of diffusion tensors.…”

Learning the Fréchet Mean over the Manifold of Symmetric Positive-Definite Matrices

“…Namely, at each iteration the determinant of the mean value l s equals the geometric average of the determinants of data points y n . Volume conservation is particularly important in applications such as diffusion magnetic resonance imaging [1,7].…”

“…• Statistical analysis of diffusion tensor data in medicine [7]. The tensors yield by diffusion tensor magnetic resonance imaging represent the covariance within a Brownian motion model of water diffusion.…”