Jan Slovák (Masaryk University): The Geometry of Diffusion Tensor Imaging
Channel: Centrum Fizyki Teoretycznej PAN
Date: 2021-09-01

Jan Slovák (Masaryk University).
The Geometry of Diffusion Tensor Imaging.
Several important medical imaging methods are based on the diffusion of water molecules, measured by MRI. We shall focus on imaging in brain research. For each small unit of space, the so-called voxels, we interpret the data as speed of the diffusion in all direction. While the grey matter is very isotropic, the white matter is highly anisotropic and this is exploited for segmentation and fibre tracking. The Diffusion Tensor Imaging (DTI) modality assumes that the diffusion process is well approximated by a Gaussian distribution, which means the second order tensors are the relevant approximations for the data. Clearly, diffusion must be approximated by positive definite tensors. However, with the current available resolution, about 30% of the white matter voxels involve crossings, kissing or merging of more fibres. This invokes higher order tensors to be exploited. Along our journey through several practical algorithms, we shall meet the mathematical background involving (conformal) Riemannian geometry, geodesics, some algebra related to quartic and quadratic forms, as well as variational and statistical methods.

GRIEG collaboration project: SCREAM workshop 2021.