Simple harmonic oscillator based estimation and reconstruction for one-dimensional q-space MR

INTRODUCTION: The q-space spectroscopy/imaging method provides important insights into tissue microstructure by enabling the calculation of ensemble average diffusion propagators through a Fourier transform. Descriptors of the diffusion propagators such as its moments and return-to-origin probability may be indicators of tissue microstructure which could be sensitive to changes due to aging, development and disease. Moreover, the non-monotonic dependence of the q-space signal on the wave number, q, [1] may provide a direct means to determine cell sizes. Estimations of the derived quantities and reconstruction of the propagators can be significantly improved if the signal decay can be expressed parametrically. For this purpose, biexponential fitting [2] and cumulant expansion techniques have been applied to q-space data. However, biexponential functions are monotonic by design, and as such, they can not model diffraction-like features. The cumulant expansion method is bound to fail as well, because the signal minima are typically beyond the radius of convergence [3] for such expansions. In this work, we propose to express the MR signal in terms of the eigenfunctions of the simple harmonic oscillator Hamiltonian, which form a complete orthogonal basis for the space of square integrable functions.