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

© The Author(s) 2010. This article is published with open access at Springerlink.com Abstract We study 3D-multidirectional images, using Finsler geometry. The application considered here is in med-ical 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 ten-sor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of wa-ter is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric frame-work. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restrict-ing the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract mul-tiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to deter-mine whether a spherical function satisfies the strong con-vexity 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