Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI
Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI
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DOI：
10.1016/j.sigpro.2006.02.050
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发表时间：
2007-02-01
影响因子：
4.4
通讯作者：
Pajevic, Sinisa
中科院分区：
文献类型：
--
作者：
Basser, Peter J.;Pajevic, Sinisa
我们提出了一个新的光谱分解的四阶协方差张量，Sigma。正如向量（即一阶张量）值随机变量的可变性用协方差矩阵（即二阶张量）S来表征一样，二阶张量值随机变量D的可变性用四阶协方差张量Sigma来表征。因此，正如S的谱分解是其特征值与其对应的（一阶张量）特征向量的外积的线性组合一样，Sigma的谱分解是其特征值与其对应的二阶特征量的外积的线性组合。类似地，这些特征值和二阶特征量可以作为特征来表示和可视化张量值数据中的可变性。在这里，我们提出了一个框架来可视化Sigma的角度结构，然后用它来评估和表征合成扩散张量磁共振成像（DTI）数据的可变性。谱分解提出了一个对称的层次，用它来分类张量数据中固有的统计各向异性。我们还提出了与D相关的样本均值和协方差张量的最大似然估计，并推导出D沿特定方向投影的均值和方差的期望值公式（即表观扩散系数或ADC）。如果我们将二阶张量随机变量视为向量值随机变量，那么这些发现将很难（如果不是不可能的话）收集到，这通常是在多变量统计分析中完成的。(c) 2006 Elsevier B.V.版权所有
We propose a novel spectral decomposition of a 4th-order covariance tensor, Sigma. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance matrix (i.e., a 2nd-order tensor), S, the variability of a 2nd-order tensor-valued random variable, D, is characterized by a 4th-order covariance tensor, Sigma. Accordingly, just as the spectral decomposition of S is a linear combination of its eigenvalues and the outer product of its corresponding (1st-order tensors) eigenvectors, the spectral decomposition of Sigma is a linear combination of its eigenvalues and the outer product of its corresponding 2nd-order eigentensors. Analogously, these eigenvalues and 2nd-order eigentensors can be used as features with which to represent and visualize variability in tensor-valued data. Here we suggest a framework to visualize the angular structure of Sigma, and then use it to assess and characterize the variability of synthetic diffusion tensor magnetic resonance imaging (DTI) data. The spectral decomposition suggests a hierarchy of symmetries with which to classify the statistical anisotropy inherent in tensor data. We also present maximum likelihood estimates of the sample mean and covariance tensors associated with D, and derive formulae for the expected value of the mean and variance of the projection of D along a particular direction (i.e., the apparent diffusion coefficient or ADC). These findings would be difficult, if not impossible, to glean if we treated 2nd-order tensor random variables as vector-valued random variables, which is conventionally done in multi-variate statistical analysis. (c) 2006 Elsevier B.V. All rights reserved.