Scaling-Rotation Framework of Symmetric Positive-Definite Matrices and Applications to Diffusion Tensor Imaging

• Activity Number: 328
• Type: Invited
• Date/Time: Tuesday, August 2, 2016 : 10:30 AM to 12:20 PM
• Sponsor: Korean International Statistical Society
• Author(s): Sungkyu Jung* and Armin Schwartzman and David Groisser
• Companies: University of Pittsburgh and University of California at San Diego and University of Florida
• Keywords: symmetric positive-definite matrices ; eigen-decomposition ; geodesics, ; diffusion tensors ; geodesic normal distribution

Abstract

The statistical analysis of symmetric positive-definite (SPD) matrices arises in diffusion tensor imaging (DTI) and tensor-based morphometry. 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. We introduce 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. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices, and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. In the context of DTI, this results in better behavior of the trace, determinant and fractional anisotropy of interpolated SPD matrices in typical cases. We also discuss some issues in statistical analysis of DTI data in the new framework.