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

Abstract

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 (HCP). 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 (MILP) method, and a Constrained Non-Linear Optimization (CNLO) 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.

Fig. 1. Sketch map of IMOC

(a) demonstrates MOC algorithm 2 for P-C-S in 𝕊¹, where u₁, u₂ and u₃ are selected one by one with a given θ, and the yellow arc denotes the total coverage set. Figures (b) and (c) are sketch maps for the multi-shell case P-C-M. (b) shows the compromise between the covering radius θ₀ in the combined shell and the covering radius θ_s in the s-th shell, and demonstrates the binary search path (the dashed blue line segment) and the obtained covering radii in IMOC. The grey area contains all feasible covering radii, and the red curve denotes all Pareto optimal covering radii. (c) demonstrates MOC for 3 shell case. Three colors denote three shells. The numbers near the points are the selection orders in MOC for the points. Point 1 (i.e., u_1,1) is first selected for shell 1, then point 2 (u_2,1) for shell 2 and point 3 (u_3,1) for shell 3. The red circle around u_1,1 denotes its coverage set C(u_1,1, θ₁) which is added to CS₁, and the dashed black circle around u_1,1 denotes C(u_1,1, θ₀) which is added to CS₀. Point 4 (u_1,2) is selected based on CS_s, ∀s = 0, 1, 2, 3.

Fig. 2. Effect of discretization

The left figure shows the covering radii of single shell schemes with K samples obtained by methods using two uniform sets with 81 samples and 20481 samples. The right figure shows the covering radii of multi-shell schemes with K × 3 samples, where θ_s is the mean of covering radii in three shells, and θ₀ is the covering radius of the combined shell.

Fig. 3

Multi-shell sampling schemes with 90 × 3 samples generated by three methods, and the covering radii of 3 shells and the combined shell in schemes shown in Table III. The colors differentiate the sampling points from the three shells. Note that there is a hole area in the scheme by IMOC, which can be fixed by 1-Opt and CNLO.

Fig. 4

First row: full HCP scheme, sub-sampled schemes by MILP and random selection, respectively. Second row: fODFs fields estimated from the data sets corresponding to schemes, where fODF glyphs are colored by directions, the background is the GFA map, and the yellow tubes denote detected peaks.