Diffusion MRI invariants: from the group of rotations to a complete neuroimaging fingerprint

Water diffusion gives rise to micrometer-scale sensitivity of diffusion MRI (dMR) to cellular-level tissue structure. The advent of precision medicine and quantitative imaging hinges on revealing the information content of dMR, and providing its parsimonious basis- and hardware-independent ``fingerprint". Here we focus on the geometry of a multi-dimensional dMR signal, derive a complete set of 21 diffusion and covariance tensor invariants in terms of irreducible representations of the group of rotations, and relate them to tissue properties. Conventional dMR metrics are shown to be redundant, while most of the invariants provide novel complementary information. Our complete set of invariants for the kurtosis tensor improves multiple sclerosis classification in a cohort of 1189 subjects. We design acquisitions based on icosahedral vertices guaranteeing minimal number of measurements to determine the most used invariants in only 1--2 minutes for the whole brain. Representing dMR signals via scalar invariant maps with definite symmetries will underpin machine learning classifiers of brain pathology, development, and aging, while fast protocols will enable translation of advanced dMR into clinical practice.