Multivariate Tensor-Based Morphometry with a Right-Invariant Riemannian Distance on GL+(n)

Tensor-based morphometry (TBM) studies encode the anatomical information in spatial deformations which are locally characterized by Jacobian matrices. Current methods perform voxel-wise statistical analysis on some features, such as the Jacobian determinant or the Cauchy–Green deformation tensor, which are not complete descriptors of the local deformation. This article introduces a right-invariant Riemannian distance on the GL⁺(n) group of Jacobian matrices making use of the complete geometrical information of the local deformation. A numerical recipe for the computation of the proposed distance is given. Additionally, experiments are performed on both a synthetic deformation study and a cross-sectional brain MRI study.