Cheng Jian, Shen Dinggang, Yap Pew-Thian
Med Image Comput Comput Assist Interv. 2014;17(Pt 3):281-8. doi: 10.1007/978-3-319-10443-0_36.
In diffusion MRI (dMRI), determining an appropriate sampling scheme is crucial for acquiring the maximal amount of information for data reconstruction and analysis using the minimal amount of time. For single-shell acquisition, uniform sampling without directional preference is usually favored. To achieve this, a commonly used approach is the Electrostatic Energy Minimization (EEM) method introduced in dMRI by Jones et al. However, the electrostatic energy formulation in EEM is not directly related to the goal of optimal sampling-scheme design, i.e., achieving large angular separation between sampling points. A mathematically more natural approach is to consider the Spherical Code (SC) formulation, which aims to achieve uniform sampling by maximizing the minimal angular difference between sampling points on the unit sphere. Although SC is well studied in the mathematical literature, its current formulation is limited to a single shell and is not applicable to multiple shells. Moreover, SC, or more precisely continuous SC (CSC), currently can only be applied on the continuous unit sphere and hence cannot be used in situations where one or several subsets of sampling points need to be determined from an existing sampling scheme. In this case, discrete SC (DSC) is required. In this paper, we propose novel DSC and CSC methods for designing uniform single-/multi-shell sampling schemes. The DSC and CSC formulations are solved respectively by Mixed Integer Linear Programming (MILP) and a gradient descent approach. A fast greedy incremental solution is also provided for both DSC and CSC. To our knowledge, this is the first work to use SC formulation for designing sampling schemes in dMRI. Experimental results indicate that our methods obtain larger angular separation and better rotational invariance than the generalized EEM (gEEM) method currently used in the Human Connectome Project (HCP).
在扩散磁共振成像（dMRI）中，确定合适的采样方案对于使用最少的时间获取用于数据重建和分析的最大信息量至关重要。对于单壳采集，通常倾向于无方向偏好的均匀采样。为了实现这一点，一种常用的方法是琼斯等人在dMRI中引入的静电能量最小化（EEM）方法。然而，EEM中的静电能量公式与最优采样方案设计的目标没有直接关系，即实现采样点之间的大角度分离。一种数学上更自然的方法是考虑球码（SC）公式，其目的是通过最大化单位球面上采样点之间的最小角度差来实现均匀采样。尽管SC在数学文献中得到了充分研究，但其当前公式仅限于单壳，不适用于多壳。此外，SC，或者更精确地说是连续SC（CSC），目前只能应用于连续单位球，因此不能用于需要从现有采样方案中确定一个或几个采样点子集的情况。在这种情况下，需要离散SC（DSC）。在本文中，我们提出了用于设计均匀单壳/多壳采样方案的新型DSC和CSC方法。DSC和CSC公式分别通过混合整数线性规划（MILP）和梯度下降方法求解。还为DSC和CSC提供了一种快速贪心增量解。据我们所知，这是第一项使用SC公式设计dMRI采样方案的工作。实验结果表明，我们的方法比人类连接组计划（HCP）目前使用的广义EEM（gEEM）方法获得更大的角度分离和更好的旋转不变性。
