From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 210

more complex and generic function images such as in DSI or QBI, where each voxel
contains a diffusion function such as the EAP in DSI or a spherical function such
as the ODF in QBI, represented as coefficients in a particular basis of choice—such
as the SH basis. Therefore, processing such higher dimensional images requires
sophisticated mathematical and computational tools.
Processing diffusion images also forms an important part of the dMRI pipeline
from acquisition to extraction of meaningful physical and medical information from
the data. Operations such as regularization are important for denoising diffusion
images as they render the tensor field in DTI or ODF field in QBI more coherent
and therefore greatly improve the results of post-processing algorithms such as
tractography. A rich body of literature for regularizing tensor fields in DTI can be
found in [ 3 , 15 , 46 , 50 , 51 ]. The capacity to segment tensor images or ODF images
makes it possible to identify and reconstruct white matter structures in the brain
such as the corpus callosum, which is not possible from simple scalar MR images.
However, the extension of such operations from scalar images to tensor fields or
ODF fields requires the correct mathematical definitions for spaces of tensors (or
EAPs or ODFs) with the appropriate metric. In this section we will present the tools
required to process tensor images (DTI), which has seen extensive mathematical
development recently. In particular we will present appropriate metrics for the
space of symmetric positive definite matrices (or diffusion tensors) Sym n

,an

Sym n

estimation algorithm for DTI that ensures that the DT is estimated in

Sym n using the Riemannian metric of and a segmentation algorithm that uses the

Sym n Riemannian metric of to segment regions in a tensor field.

6.5.1 The Affine Invariant Riemannian Metric for Diffusion

Tensors

Diffusion tensors are symmetric matrices. However, since negative diffusion is

non-physical these matrices are also required to be positive definite. In other words
DTs belong to the space

3 × 3

Sym 3 symmetric positive definite matrices, which

is a non-Euclidean space. Therefore an appropriate metric needs to be defined on this
space which would render it into a Riemannian manifold and which would permit to
constrain all operations naturally to

of 3 × 3

Sym 3
naturally inherits the Euclidean and the Frobenius metrics from the space of all
matrices, however,

Sym 3 by using Riemannian geometry.

Sym 3 is neither complete nor closed under these metrics.

A number of works have recently proposed the affine invariant Riemannian
metric for Sym n which has been used extensively to compute on DTs [ 40 , 45 , 51 ].

In [ 51 ] the Riemannian metric is derived to be g ij =

g ( X,Y )= X,Y S =

tr S − 2 X S − 1 Y S − 2 , ∈Sym 3 T S Sym 3 = Sym 3

the
tangent space at S . The geodesic distance between DTs induced by this metric can
be computed to be:

∀ S and with X,Y ∈