Mixed Integer Linear Programming for Discrete Sampling Scheme Design in Diffusion MRI

In diffusion MRI (dMRI), a uniform single or multiple shell sampling scheme is typically required for data acquisition in \(\textbf{q}\) -space, because uniform spherical sampling offers the advantage of capturing more information using fewer samples, leading to superior reconstruction results. Uniform sampling problems can be categorized into continuous and discrete types. While most existing sampling methods focus on the continuous problem that is to design spherical samples continuously from single or multiple shells, this paper primarily investigates two discrete optimization problems, i.e., 1) optimizing the polarity of an existing scheme (P-P), and 2) optimizing the ordering of an existing scheme (P-O). Existing approaches for these two problems mainly rely on greedy algorithms, simulated annealing, and exhaustive search, which fail to obtain global optima within a reasonable timeframe. We propose several Mixed Integer Linear Programming (MILP) based methods to address these problems. To the best of our knowledge, this is the first work that solves these two discrete problems using MILP to obtain global optimal or sufficiently good solutions in 10 min. Experiments performed on single and multiple shells demonstrate that our MILP methods can achieve larger separation angles and lower electrostatic energy, resulting better reconstruction results, compared with existing approaches in commonly used software (i.e., CAMINO and MRtrix).