Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI

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.