Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging

By Ian L. Dryden, Alexey A. Koloydenko, Diwei Zhou. 2009

433 citations recorded by the source corpus.

The statistical analysis of covariance matrices occurs in m any important applications, e.g. in diffusion tensor imaging or longitudinal data analysis. We consider the situation where it is of interest to estimate an average covariance matrix, describe its anisotropy and to carry out principal geodesic analysis of covariance matrices. In medical image analysis a particular type of covariance matrix arises in diffusion weighted imaging called a diffusion tensor. The diffusion tensor is a 3 × 3 covariance matrix which is estimated at each voxel in the brain, and is obtained by fittin g a physically-motivated model on measurements from the Fourier transform of the molecule displacement density (Basser et al., 1994). A strongly anisotropic diffusion tensor indicates a strong direction of white matter fibre tracts, and plots of measures of anisotropy are very useful t o neurologists. A measure that is very commonly used in diffusion tensor imaging is Fractional Anisotropy

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/0a15503913d0c6d493a812033f98ef04da39b317

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Citations in the corpus

Analyzing Fluctuating Asymmetry with Geometric Morphometrics: Concepts, Methods, and Applications (2015, methodology): “The first such analysis used a distance measure calculated as one minus the matrix correlation between covariance matrices [177], and alternative measures were proposed subsequently [408,409].”
Gromov-Wasserstein Averaging of Kernel and Distance Matrices (2016, background): “(Dryden et al., 2009) considers several simple options for covariance matrices, mostly with fixed alignment; their “Procrustes” mean provides one rudimentary strategy for alignment.”
Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices (2013, methodology): “For all non-Euclidean metrics with (non-kernel) k-means, the Karcher mean [7] was used to compute the centroid.”
A smeary central limit theorem for manifolds with application to high-dimensional spheres (2018, background): “…projective space (cf. Kendall (1984); Mardia and Patrangenaru (2005) for certain shape spaces) or the space of positive definite matrices (cf. Dryden et al. (2009) for diffusion tensors), (b) a non-manifold shape space which is a quotient of a Riemannian manifold under an isometric group…”
Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models (2010, methodology): “This embedding is inspired by recent work of Dryden et al. (2009a) who, in effect, model P+(m) as a subspace of m+1m .”
Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces (2014, background): “Symmetric Positive Definite (SPD) matrices, in particular covariance matrices, have been playing an increasingly important role in many areas of machine learning, statistics, and computer vision, with applications ranging from kernel learning [12], brain imaging [9], to object detection [24, 23].”
Distances and inference for covariance operators (2014, methodology): “Since covariance operators are trace class, we can also generalize the square root matrix distance (Dryden et al., 2009).”