Simple harmonic oscillator based reconstruction and estimation for one-dimensional Q-space magnetic resonance (1D-SHORE)

Abstract

The movements of endogenous molecules during the magnetic resonance acquisition influence the resulting signal. By exploiting the sensitivity of diffusion on the signal, q-space MR has the ability to transform a set of diffusion-attenuated signal values into a probability density function or propagator that characterizes the diffusion process. Accurate estimation of the signal values and reconstruction of the propagator demand sophisticated tools that are well suited to these estimation and reconstruction problems. In this work, a series representation of one-dimensional q-space signals is presented in terms of a complete set of orthogonal Hermite functions. The basis possesses many interesting properties relevant to q-space MR, such as the ability to represent both the signal and its Fourier transform. Unlike the previously employed cumulant expansion, biexponential fit, and similar methods, this approach is linear and capable of reproducing complicated signal profiles, e.g., those exhibiting diffraction peaks. The estimation of the coefficients is fast and accurate while the representation lends itself to a direct reconstruction of ensemble average propagators as well as calculation of useful descriptors of it, such as the return-to-origin probability and its moments. In axially symmetric and isotropic geometries, respectively, two- and three-dimensional propagators can be reconstructed from one-dimensional q-space data. Useful relationships between the one- and higher-dimensional propagators in such environments are derived.