IEEE Trans Med Imaging. 2018 Jan;37(1):185-199. doi: 10.1109/TMI.2017.2756072. Epub 2017 Sep 25.
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.
在扩散磁共振成像（dMRI）中，良好的采样方案对于高效采集和稳健重建至关重要。扩散加权信号通常在 q 空间的单个或多个壳层上采集。信号样本通常在不同的壳层上均匀分布，以使它们不受组织内结构或实验室坐标系方向的影响。静电能最小化（EEM）方法最初是为 dMRI 的单壳层采样方案提出的，最近已推广到多壳层方案，称为广义 EEM（GEEM）。GEEM 已成功应用于人类连接组计划（HCP）。然而，EEM 并没有直接解决最佳采样的目标，即实现采样点之间的大角度分离。在本文中，我们提出了一种更自然的公式，称为球面码（SC），以直接最大化单壳或多壳中不同样本之间的最小角度。我们不仅考虑了连续问题来设计单壳或多壳采样方案，还考虑了离散问题来从现有单壳或多壳方案中均匀提取子采样方案，并对现有方案中的样本进行排序。我们提出了五种算法来解决上述问题，包括增量 SC（ISC）、一种称为迭代最大重叠构造（IMOC）的复杂贪婪算法、一种 1-Opt 贪婪方法、混合整数线性规划（MILP）方法和约束非线性优化（CNLO）方法。据我们所知，这是首次将 SC 公式用于 dMRI 的单壳或多壳采样方案。实验结果表明，SC 方法获得的角度分离更大，旋转不变性更好，优于最新的 EEM 和 GEEM。相关代码和教程已在 DMRITool 中发布。
