Image Processing Reference
In-Depth Information
FIGURE 13.3
Idealized diagram showing the local coordinate system described by the
eigensystem of the diffusion tensor, in two dimensions. Note that where tracts cross in a
voxel, the major eigenvector is not likely to be parallel to either tract.
corresponding eigenvalues, and surrounded by an ellipse describing the shape of
water diffusion.
The diffusion tensor model is reasonable but is limited when describing neu-
roanatomy. In the case where one voxel contains more complicated geometry than
a single tract, it is misleading to only consider the information in the major eigen-
vector. In addition, the diffusion tensor model cannot describe fiber tract crossings
or complicated patterns of tracts that may occur within a voxel. It is important to
take these limitations into account when developing a data analysis method.
In this chapter, we first present background information on diffusion and
diffusion tensor calculation. We continue with a description of diffusion tensor
shape analysis and visualization methods. Finally, we introduce two techniques
for extracting connectivity information from diffusion tensor data sets.
13.2 DIFFUSION AND DIFFUSION TENSOR
CALCULATION
The process of diffusion is described by Fick's first law, which relates a concen-
tration difference to a flux (a flow across a unit area). It states that the flux,
j
, is
proportional to the gradient of the concentration,
∇
u
. The proportionality constant
d
is the diffusion coefficient.
j
=− ∇
du
(13.1)
For an anisotropic material, the flow field does not follow the concentration
gradient directly, because the material properties also affect diffusion. Conse-
quently, the diffusion tensor,
D
, is introduced to model the material locally.
j
=− ∇
Du
(13.2)
The standard model of diffusion says that over time, the concentration of the
solute will change as the divergence of the flux:
u
=∇⋅ ∇ .
(
Du
)
(13.3)
t
Search WWH ::
Custom Search