A theoretical signal processing framework for linear diffusion MRI: Implications for parameter estimation and experiment design

Abstract

The data measured in diffusion MRI can be modeled as the Fourier transform of the Ensemble Average Propagator (EAP), a probability distribution that summarizes the molecular diffusion behavior of the spins within each voxel. This Fourier relationship is potentially advantageous because of the extensive theory that has been developed to characterize the sampling requirements, accuracy, and stability of linear Fourier reconstruction methods. However, existing diffusion MRI data sampling and signal estimation methods have largely been developed and tuned without the benefit of such theory, instead relying on approximations, intuition, and extensive empirical evaluation. This paper aims to address this discrepancy by introducing a novel theoretical signal processing framework for diffusion MRI. The new framework can be used to characterize arbitrary linear diffusion estimation methods with arbitrary q-space sampling, and can be used to theoretically evaluate and compare the accuracy, resolution, and noise-resilience of different data acquisition and parameter estimation techniques. The framework is based on the EAP, and makes very limited modeling assumptions. As a result, the approach can even provide new insight into the behavior of model-based linear diffusion estimation methods in contexts where the modeling assumptions are inaccurate. The practical usefulness of the proposed framework is illustrated using both simulated and real diffusion MRI data in applications such as choosing between different parameter estimation methods and choosing between different q-space sampling schemes.

Towards optimal linear estimation of orientation distribution functions with arbitrarily sampled diffusion MRI data

“…At first glance, the connection between our ERFO estimator and the ERF-based theoretical characterization from [8] may not be immediately apparent. This connection is formalized in the following subsection.…”

“…Specifically, assuming that we are given P such training pairs {Op(u),Ep(q)}p=1P, as well as the q-space sampling locations {qm}m=1M and the noise variance σ 2 , we optimize the vector of ERFO estimation coefficients a ( u ) = [ a 1 ( u )···· a M ( u )] T by minimizing (with respect to a ( u )) the following minimum mean-squared error loss function: ∑p=1PE[|Op(u)−∑m=1Mam(u)(Ep(boldqm)+n(boldqm))|2], where double-struckE is used to denote statistical expectation with respect to the noise samples. Under our noise assumptions, it can be shown [8] that minimizing Eq. (6) is equivalent to minimizing ∑p=1P|…”

“…Our present work is based on the hypothesis that existing orientation estimation methods are suboptimal. This hypothesis is inspired by a general theoretical framework we recently introduced [8] that provides unique theory-driven performance characterizations of linear estimators. Specifically, our previous theoretical analysis suggests that the angular resolution and noise sensitivity of existing linear estimation approaches may not actually be close to achieving what is theoretically possible within the class of linear estimators.…”

“…Our previous work [8] derived a direct theoretical relationship between the estimated ODF and the true EAP: trueO^false(boldufalse)=∫PΔfalse(boldrfalse)gfalse(boldu,boldrfalse)dboldr+∑m=1Mamfalse(boldufalse)n(qm), with gfalse(boldu,boldrfalse)≜∑m=1Mamfalse(boldufalse) cos(2πqmTboldr). Equation (5) has two components, one that depends on the true EAP but is independent of the noise, and another that depends on the noise but is independent of the true EAP. The function g ( u , r ) that interacts with the true EAP is termed the EAP response function (ERF), and can be used to infer the accuracy and resolution characteristics of a given linear estimator [8]. The noise term appearing in Eq.…”

“…The noise term appearing in Eq. (5) also allows us to predict the variance of O^(u) assuming that we know the noise variance characteristics of n ( q m ) [8].…”