In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas and is publicly available.

Configurations of dense locally parallel 3D curves occur in medical imaging, computer vision and graphics. Examples include white matter fibre tracts, textures, fur and hair. We develop a differential geometric characterization of such structures by considering the local behaviour of the associated 3D frame field, leading to the associated tangential, normal and bi-normal curvature functions. Using results from the theory of generalized minimal surfaces we adopt a generalized helicoid model as an osculating object and develop the connection between its parameters and these curvature functions. These developments allow for the construction of parametrized 3D vector fields (sampled osculating objects) to locally approximate these patterns. We apply these results to the analysis of diffusion MRI data via a type of 3D streamline flow. Experimental results on data from a human brain demonstrate the advantages of incorporating the full differential geometry.