The MAP-MRI approach [11] uses a functional basis to represent the 3D diffusion signal with as little assumptions as possible. It then analytically reconstructs the 3D diffusion propagator by only assuming the short gradient pulse approximation (1 & 0). In this way, it accurately estimates the diffusion propagator in the presence of both non-Gaussian diffusion and crossing tissue configurations.

MAP-MRI represents the discretely measured signal attenuation E(q) using a set of continuous orthogonal basis functions representing the space E(q; c), where the signal is now represented in terms of basis coefficients c and the q- space wave vector q = |q|g with g the gradient direction is related to the b-value as |q| = л/Ь/(Л — 8/3)/2jt. Without going into the formulation of MAP- MRI’s basis functions, we detail the estimation of basis coefficients c in Eq. (3). In short, we regularize the fitting of c such that E(q; c) smoothly interpolates between the measured q-space points by using Laplacian regularization [14], where regularization weight X is set using voxel-wise generalized cross-validation. We also constrain the estimated diffusion Propagator 1°(R; c) to be positive using quadratic programming [11].

Once c is known, the MAP-MRI basis simultaneously represents the 3D dMRI signal and 3D diffusion propagator. We estimate the q-space indices Return-To- Origin, Return-To-Axis and Return-To-Plane Probability (RTOP, RTAP and RTPP), which in theory are related to the volume, surface and length of a cylindrical pore [11]. We also estimate the non-Gaussianity (NG), which describes the ratio between the Gaussian and non-Gaussian volume of the signal. Finally we estimate the propagator anisotropy (PA), which is a normalized metric that describes the anisotropy of the 3D diffusion propagator. As MAP-MRI is designed to represent the entire 3D diffusion signal, we estimate all metrics using the entire 5 shell data up to a b-value of 12,000 s/mm^{2}, using a radial order of 6, resulting in 50 estimated coefficients. We illustrate these metrics in Fig. 3.