Geodesic Uncertainty in Diffusion MRI

Abstract

We study theoretical and operational issues of geodesic tractography, a geometric methodology for retrieving biologically plausible neural fibers in the brain from diffusion weighted magnetic resonance imaging. The premise is that true positives are geodesics in a suitably constructed metric space, but unlike traditional first order methods these are not a priori constrained to connect nongeneric points on subdimensional manifolds, such as the characteristics in traditional streamline methods. By virtue of the Hopf-Rinow theorem geodesic tractography furnishes a huge amount of redundancy, ensuring the a priori existence of at least one tentative fiber between any two points and permitting additional tractometric and data-extrinsic constraints for (fuzzy or crisp) classification of true and false positives. In our feasibility study we consider a hybrid paradigm that unifies existing ideas on tractography, combining deterministic and probabilistic elements in a way naturally supported by metric geometry. Particular attention is paid to an analytical prediction of geodesic deviation on numerically computed geodesics, a ‘tidal’ effect induced by small perturbations resulting from data noise. Taking these effects into account clarifies the inherent uncertainty of geodesics, while simultaneosuly offering a dimensionality reduction of the tractography problem.