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. CR Categories: I.4.10 [Computing Methodologies]: Image Representation—Multidimensional