Challenges in Reconstructing the Propagator via a Cumulant Expansion of the One-Dimensional q-space NMR signal

Generalized Diffusion Tensor Imaging (GDTI) [1] is one of the few methods that estimate the ensemble average diffusion propagator from the diffusion weighted signal. It has a statistical approach and views the signal, which under the q-space formalism is the Fourier transform of the propagator, as the characteristic function of the propagator. Instead of taking the inverse Fourier transform of the signal, GDTI estimates the cumulants of the propagator from the signal (characteristic function) and then approximates the propagator using the Gram-Charlier Type-A series, which is a series approximation of a probability density function based on its cumulants. However, it is well known that the Gram-Charlier series has a poor convergence, especially since only a truncated series is considered (order-4 in [1]). The Edgeworth series, which is a reordering of the terms from the Gram-Charlier series, is known to perform better since it is a true asymptotic expansion [2]. GDTI has never been validated numerically. We propose, here, to compare the Gram-Charlier and the Edgeworth series in 1D on known diffusion propagators, where the propagator, the signal and the cumulants have analytical forms. We also compare with cumulants estimated from the signal. Our experiments strongly suggest that for analytical cumulants the Edgeworth series improves on the Gram-Charlier series, and estimating the cumulants from the signal is numerically a sensitive and important problem.