MRI is a versatile modality with endless options for image formation, cf. Although tuned in on water-bound hydrogen, because of its abundance in the human body, the recorded signals are affected by the physicochemical environment and therefore carry (indirect) anatomical and/or functional information depending on the measurement protocol. Magnetic resonance imaging (MRI) allows one to manipulate the magnetization of a hydrogen spin system in a way that enables non-invasive in-vivo acquisition of informative localized signals, i.e. In a typical macroscopic system, however, the Zeeman effect induces a relatively small yet measurable “classical” magnetization, for which these attributes can be taken literally. The attributes refer to the relative alignment with the external field, but should be taken with a grain of salt for quantum systems of individual nuclei. These states are known in the trade as “spin up”, or “parallel”, and “spin down”, or “anti-parallel”. In this situation the Zeeman splitting effect discloses two quantum eigenstates of its nuclear magnetic moment, distinguishable by a relative energy gap. Taking these effects into account clarifies the inherent uncertainty of geodesics, while simultaneosuly offering a dimensionality reduction of the tractography problem.Ī hydrogen nucleus (proton) behaves like a tiny magnet when interacting with an external magnetic field. 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. 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. 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. 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. 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. Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands.Rick Sengers †, Luc Florack* † and Andrea Fuster
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