From Diffusion MRI to Brain Connectomics - Modeling in Computational Biology and Biomedicine - page 213

the computation of the mean and the standard deviation of the set of 3D Gaussian
distributions N constituting Ω 1 and Ω 2 . This mean and standard deviation of

a set of Gaussian distributions would require a metric to be defined on the space of
Gaussian distributions. A number of examples are provided in [ 23 ]—the Euclidean
metric, the Kullback-Leibler divergence, and the Riemannian metric. We reproduce
here only the final example.
Using the affine invariant Riemannian metric on

( x, r )

Sym 3 , which also forms a

Riemannian metric on the space of 3D Gaussian distributions N ( x, r ) , it is possible

to compute the mean distribution N of a set of Gaussian distributions by

a process of Riemannian geodesic descent—a modified gradient de scent process.
Similarly the empirical covariance matrix re lative t o t he mean N is defined to
be Λ Rm =

( X,r )

i =1

N − 1
i

1
n− 1

β i β i , with β i = N ln( N ) which is the gradient of

∇D Rm (

the squared geodesic distance in vector form. Using these it is

possible to define a generalized Gaussian distribution over the space of 3D Gaussian
distributions with a covariance matrix Λ Rm of small variance σ 2 =

N i , N )

tr ( Λ Rm ) :

−

σ 3 )+ β T γβ

2

N,Λ Rm )= 1+

O ( ε ( σ/η )

∈Sym 3 ,
(6.32)

P Rm ( N |

Λ Rm | exp

∀ N

(2 π ) m ( m +1) / 2 |

Λ − 1 wh ere γ =

Rm −R

/ 3+ O ( σ )+ ε ( σ/η ) with R as the Ricci curvature tensor at

lim 0 + x −β ε ∈ R + . N , η as the injection radius at N and ε such that ( x )=0 ∀ β

6.6 Tractography: Inferring the Connectivity

When DTI/DSI/QBI is performed on the brain, the DT/EAP/ODF—hereafter
referred to as the spherical diffusion function (SDF), is a local indicator of coherent
structures or fiber bundles in the cerebral white matter. However, the process
of reconstructing the global structures of fiber bundles by connecting the local
information is known as fiber tracing or tractography . Tractography graphically
reconstructs the connectivity of the cerebral white matter by integrating along the
direction indicated by the local geometry of the SDF. It is a modern tool that is
unique in the sense that it permits an indirect dissected visualization of the brain in
vivo and non-invasively [ 16 ]. The underpinnings of tractography are also based on
the fundamental assumption of dMRI—the diffusion of water molecules is hindered
to a greater extent perpendicular to coherent fiber bundle structures than parallel
to these. Therefore, following the geometry of the local diffusion function and
integrating along reveals the continuous dominant structure of the fiber bundle.
However, in spite of the gain due to its non-invasive nature, tractography can only
infer such structures indirectly. Therefore, tractography is acutely sensitive to the
local geometry and the error is cumulative. The correct estimation of the local
geometry is crucial.