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

Abstract

We study three dimensional volumes of higher order tensors, using Finsler geometry. The application considered here is in medical image analysis, specifically High Angular Resolution Diffusion Imaging (HARDI) [1] of the brain. We want to find robust ways to reveal the architecture of the neural fibers in brain white matter. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled with a symmetric positive definite second order tensor, based on the assumption that there exists one dominant direction of fibers restricting the thermal motion of water molecules, leading naturally to a Riemannian framework. HARDI may potentially overcome the shortcomings of DTI by allowing multiple relevant directions, but invalidates the Riemannian approach. Instead Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide the exact criterion to determine whether a field of spherical functions has a Finsler structure. We also show a fiber tracking method in Finsler setting. Our model also incorporates a scale parameter, which is beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as real HARDI data