Geometry of the cumulant series in diffusion MRI

Water diffusion gives rise to micron-scale sensitivity of diffusion MRI (dMRI) to cellular-level tissue structure. Precision medicine and quantitative imaging depend on uncovering the information content of dMRI and establishing its parsimonious hardware-independent fingerprint. Based on the rotational SO(3) symmetry, we study the geometry of the dMRI signal and the topology of its acquisition, identify irreducible components and a full set of invariants for the cumulant tensors, and relate them to tissue properties. Including all kurtosis invariants improves multiple sclerosis classification in a cohort of 1189 subjects. We design the shortest acquisitions based on icosahedral vertices to determine the most used invariants in only 1–2 minutes for whole brain. Representing dMRI via scalar invariant maps with definite symmetries will underpin machine learning classifiers of pathology, development, and aging, while fast protocols will enable translation of advanced dMRI into clinic. Rotational symmetry defines the geometry and topology of the diffusion MRI signal, its irreducible components and their invariants. These metrics are applied to human brain MRI, improve multiple sclerosis classification, and enable faster scans.