Shape anisotropy: tensor distance to anisotropy measure

Fractional anisotropy (FA), defined as the distance of a diffusion tensor from its closest isotropic tensor, has been extensively studied as quantitative anisotropy measure for diffusion tensor magnetic resonance images (DT-MRI). It has been used to reveal the white matter profile of brain images, as guiding feature for seeding and stopping in fiber tractography and for the diagnosis and assessment of degenerative brain diseases. Despite its extensive use in DT-MRI community, however, not much attention has been given to the mathematical correctness of its derivation from diffusion tensors which is achieved using Euclidean dot product in 9D space. But, recent progress in DT-MRI have shown that the space of diffusion tensors does not form a Euclidean vector space and thus Euclidean dot product is not appropriate for tensors. In this paper, we propose a novel and robust rotationally invariant diffusion anisotropy measure derived using the recently proposed Log-Euclidean and Jdivergence tensor distance measures. An interesting finding of our work is that given a diffusion tensor, its closest isotropic tensor is different for different tensor distance used. We demonstrate qualitatively that our new anisotropy measure reveals superior white matter profile of DT-MR brain images and analytically show that it has a higher signal to noise ratio than FA.