4th order tensors for multi-fiber resolution and segmentation in white matter

Abstract

Since its inception, DTI modality has become an essential tool in the clinical scenario. In principle, it is rooted in the emergence of symmetric positive definite (SPD) second-order tensors modelling the difusion. The inability of DTI to model regions of white matter with fibers crossing/merging leads to the emergence of higher order tensors. In this work, we compare various approaches how to use 4th order tensors to model such regions. There are three different projections of these 3D 4th order tensors to the 2nd order tensors of dimensions either three or six. Two of these projections are consistent in terms of preserving mean diffusivity and isometry. The images of all three projections are SPD, so they belong to a Riemannian symmetric space. Following previous work of the authors, we use the standard k-means segmentation method after dimension reduction with affinity matrix based on reasonable similarity measures, with the goal to compare the various projections to 2nd order tensors. We are using the natural affine and log-Euclidean (LogE) metrics. The segmentation of curved structures and fiber crossing regions is performed under the presence of several levels of Rician noise. The experiments provide evidence that 3D 2nd order reduction works much better than the 6D one, while diagonal components (DC) projections are able to reveal the maximum diffusion direction.