Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors (2004)
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1 Abstract
Diffusion tensor magnetic resonance imaging (DT-MRI) is emerging as an important tool in medical image analysis of the brain. However, relatively little work has been done on producing statistics of diffusion tensors. A main difficulty is that the space of diffusion tensors, i.e., the space of symmetric, positivedefinite matrices, does not form a vector space. Therefore, standard linear statistical techniques do not apply. 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. In our previous work we introduced principal geodesic analysis, a generalization of principal component analysis, to compute the modes of variation of data in Lie groups. In this work we expand the method of principal geodesic analysis to symmetric spaces and apply it to the computation of the variability of diffusion tensor data. 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.
2 NOTES
This paper is very similar to @fletcherPrincipalGeodesicAnalysis2004a, however in here their application is similar to what is done in most approaches nowadays. They apply it direct to matrices, which allows them to visualize some properties of SPD matrices under PCA and PGA. With it they see the swelling effect appearing in PCA, while in PGA it does not appear, as expected. Furthermore, there is not any new information in here, most have already been explained in their other paper, and more information on the topic is on: Principal Geodesic Analysis (PGA).