Simple harmonic oscillator based reconstruction and estimation for three-dimensional q-space MRI

lm l j q u l l jlm Y q u L e q u i u π π π π + − − − = Φ . This is a three-dimensional extension of a scheme developed to represent one-dimensional q-space data (3). Here Lj-1 l+1/2 (.) and Ylm(.) are the associated Laguerre polynomials and spherical harmonics, respectively, and u is a constant estimated from the data at each voxel location. The inverse Fourier transform of the E(q) data yields the average propagator, P(R), whose orientational dependence provides the desired connectivity information. Since the Fourier transform is linear, the propagator has an expansion with the same coefficients Ajlm. Moreover, the orientation-dependent radial moments defined by () ( ) R R R