Abstract or Additional Information
The statistical analysis of symmetric positive-definite (SPD) matrices arises in diffusion tensor imaging (DTI) and tensor-based morphometry (TBM). In particular, interpolation of tensors is important for fiber tracking, registration and spatial normalization of diffusion tensor images. Popular geometric frameworks have been shown to be powerful in generalizing statistics to SPD matrices, but they are not easy to interpret in terms of tensor deformations. A new geometric framework for the set of SPD matrices, aimed to characterize deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices, are introduced. The new scaling-rotation framework provides some unique geometric and statistical challenges. In this talk, I will discuss some advances both in terms of geometry and statistics, including stratification of the set of SPD matrices, partial answers to uniqueness questions, computational procedures for a simple statistics and some asymptotic results. The potential of the geometric framework is demonstrated by data examples in DTI and TBM. Open questions, some of which are related to functional data analysis, will be discussed as well. This talk will be based on a collaboration with David Groisser, Armin Schwartzman and Brian Rooks.