Decomposition of high angular resolution diffusion images into a sum of self-similar polynomials on the sphere

We propose a tensorial expansion of high resolution diffusion imaging (HARDI) data on the unit sphere into a sum of self-similar polynomials, i.e. polynomials that retain their form up to a scaling under the act of lowering resolution via the diffusion semigroup generated by the Laplace-Beltrami operator on the sphere. In this way we arrive at a hierarchy of HARDI degrees of freedom into contravariant tensors of successive ranks, each characterized by a corresponding level of detail. We provide a closed-form expression for the scaling behaviour of each homogeneous term in the expansion, and show that classical diffusion tensor imaging (DTI) arises as an asymptotic state of almost vanishing resolution.