Riemannian geometry for the statistical analysis of diffusion tensor data
Riemannian geometry for the statistical analysis of diffusion tensor data
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DOI：
10.1016/j.sigpro.2005.12.018
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发表时间：
2007-02-01
影响因子：
4.4
通讯作者：
Joshi, Sarang
中科院分区：
文献类型：
--
作者：
Fletcher, P. Thomas;Joshi, Sarang
扩散张量磁共振成像(DT-MRI)产生的张量表示水扩散的布朗运动模型中的协方差。在这种物理解释下，扩散张量必须是对称的、正定的。然而，目前的扩散张量数据统计分析方法将扩散张量视为线性实体，不考虑这种正定约束。这一困难是由于扩散张量空间不形成向量空间的事实。本文证明了扩散张量空间是一种称为黎曼对称空间的曲流形。然后，我们开发了产生统计数据的方法，即在这个空间中的平均值和变化模式。我们证明了这些统计量保持了张量的自然几何性质，包括其本征值为正的约束。对称空间公式还导致了扩散张量内插的自然定义和新的各向异性度量。我们期望这些方法将用于扩散张量图像的配准，从扩散张量数据产生统计图谱，以及量化由疾病引起的解剖变异。本文提出的框架在出现对称正定张量的其他应用中也是有用的，例如力学和计算机视觉。(C)2006爱思唯尔B.V.保留所有权利。
The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positive-definite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision. (c) 2006 Elsevier B.V. All rights reserved.