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A polynomial approach for extracting the extrema of a spherical function and its application in diffusion MRI

机译：提取球形函数极值的多项式方法及其在核磁共振成像中的应用

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Antipodally symmetric spherical functions play a pivotal role in diffusion MRI in representing sub-voxel-resolution microstructural information of the underlying tissue. This information is described by the geometry of the spherical function. In this paper we propose a method to automatically compute all the extrema of a spherical function. We then classify the extrema as maxima, minima and saddle-points to identify the maxima. We take advantage of the fact that a spherical function can be described equivalently in the spherical harmonic (SH) basis, in the symmetric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. We extract the extrema of the spherical function by computing the stationary points of its constrained HP representation. Instead of using traditional optimization approaches, which are inherently local and require exhaustive search or re-initializations to locate multiple extrema, we use a novel polynomial system solver which analytically brackets all the extrema and refines them numerically, thus missing none and achieving high precision.To illustrate our approach we consider the Orientation Distribution Function (ODF). In diffusion MRI the ODF is a spherical function which represents a state-of-the-art reconstruction algorithm whose maxima are aligned with the dominant fiber bundles. It is, therefore, vital to correctly compute these maxima to detect the fiber bundle directions. To demonstrate the potential of the proposed polynomial approach we compute the extrema of the ODF to extract all its maxima. This polynomial approach is, however, not dependent on the ODF and the framework presented in this paper can be applied to any spherical function described in either the SH basis, ST basis or the HP basis.
机译：对立对称球面功能在弥散MRI中代表基础组织的亚体素分辨率微结构信息起着关键作用。该信息由球面函数的几何形状描述。在本文中，我们提出了一种自动计算球面函数所有极值的方法。然后，我们将极值分为最大值，最小值和鞍点，以识别最大值。我们利用这样一个事实，即可以以球谐函数（SH）为基础，以球为约束的对称张量（ST）为基，以球为均质多项式（HP）来等效地描述球函数。我们通过计算约束HP表示的固定点来提取球面函数的极值。与其使用传统的优化方法（这些方法固有地是本地的并且需要穷举搜索或重新初始化以定位多个极值），我们使用一种新颖的多项式系统求解器来分析所有极值并对其进行数值精炼，从而不遗漏任何结果并实现高精度。为了说明我们的方法，我们考虑了方向分布函数（ODF）。在扩散MRI中，ODF是一个球形函数，代表了一种最新的重建算法，其最大值与主导纤维束对齐。因此，正确计算这些最大值以检测光纤束方向至关重要。为了证明所提出的多项式方法的潜力，我们计算了ODF的极值以提取其所有最大值。但是，该多项式方法不依赖于ODF，并且本文介绍的框架可以应用于以SH为基础，以ST为基础或以HP为基础描述的任何球面函数。

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