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Introduction to statistical pattern recognition (2nd ed.)
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Riemannian Framework for Estimating Symmetric Positive Definite 4th Order Diffusion Tensors
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Incremental learning of bidirectional principal components for face recognition
Pattern Recognition
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MICCAI '09 Proceedings of the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part I
Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI
IPMI'07 Proceedings of the 20th international conference on Information processing in medical imaging
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ISBI'10 Proceedings of the 2010 IEEE international conference on Biomedical imaging: from nano to Macro
Decomposition and visualization of fourth-order elastic-plastic tensors
SPBG'08 Proceedings of the Fifth Eurographics / IEEE VGTC conference on Point-Based Graphics
The multilinear normal distribution: Introduction and some basic properties
Journal of Multivariate Analysis
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IPMI'13 Proceedings of the 23rd international conference on Information Processing in Medical Imaging
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We propose a novel spectral decomposition of a 4th-order covariance tensor, @S. 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, @S. 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 @S 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 @S, 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.