Single- and Multiple-Shell Uniform Sampling Schemes for Diffusion MRI Using Spherical Codes

In diffusion MRI (dMRI), a good sampling scheme is important for efficient acquisition and robust reconstruction. Diffusion weighted signal is normally acquired on single or multiple shells in q-space. Signal samples are typically distributed uniformly on different shells to make them invariant to the orientation of structures within tissue, or the laboratory coordinate frame. The electrostatic energy minimization (EEM) method, originally proposed for single shell sampling scheme in dMRI, was recently generalized to multi-shell schemes, called generalized EEM (GEEM). GEEM has been successfully used in the human connectome project. However, EEM does not directly address the goal of optimal sampling, i.e., achieving large angular separation between sampling points. In this paper, we propose a more natural formulation, called spherical code (SC), to directly maximize the minimal angle between different samples in single or multiple shells. We consider not only continuous problems to design single or multiple shell sampling schemes, but also discrete problems to uniformly extract sub-sampled schemes from an existing single or multiple shell scheme, and to order samples in an existing scheme. We propose five algorithms to solve the above problems, including an incremental SC (ISC), a sophisticated greedy algorithm called iterative maximum overlap construction (IMOC), an 1-Opt greedy method, a mixed integer linear programming method, and a constrained non-linear optimization method. To our knowledge, this is the first work to use the SC formulation for single or multiple shell sampling schemes in dMRI. Experimental results indicate that SC methods obtain larger angular separation and better rotational invariance than the state-of-the-art EEM and GEEM. The related codes and a tutorial have been released in DMRITool.