Image Processing of Diffusion Tensor MRI Data - Advance Image Processing in Magnetic Resonance Imaging - page 426

where a k is a spatial mask, for example, 3 × 3 voxels, with sum of coefficients

equal to 1, and T is the anisotropic part of the diffusion tensor D . The anisotropic

part of the tensor has trace zero and can be written as

1
3 trace(

TD =− DI ) (13.21)

1
3

=−〈, 〉 DDI (13.22)

=− 〈,〉
〈,〉

DI
II

D (13.23)

where I is the identity tensor. By rewriting Equation 13.20, it becomes clear that
all the tensor inner products are individually normalized

∑

3
8

〈, 〉
〈, 〉 +

TT
DD

3
4

〈, 〉
〈,〉

TT
DD

TT
D k

〈, 〉

k k k

LI = a (13.24)

k

〈 , D 〉

k k

k

Because the components in the sum are normalized, small and large diffusion
tensor components will have equal weight in determining the lattice index. Unfor-
tunately, the smaller tensors are more influenced by noise and, hence, affect the
index more than is desirable.
An alternative measure to the lattice index, which can be seen as an external
measure as it is based on the tensors in a neighborhood, is to use an internal
voxel-based measure on a filtered version of the diffusion tensor field.
For example, local averaging of the tensor field with a spatial mask a (normalized
so that the coefficients sum to one):

( RA FA c lps , , )

,,

∑

D = a D (13.25)

a kk

k

describes the local average diffusion, where the rank of the average tensor D a
describes the complexity of the macroscopic diffusion structure. If the rank is
close to one, the structure is highly linear, which will be the case in regions of
bundles of fibers having the same direction. If the rank is 2, fibers are crossing
in a plane, or the underlying diffusivity is planar. Applying the geometrical linear
measure c l in Equation 13.11 to this tensor gives a measure that is high in regions
with coherent tensors.

Figure 13.5 compares the three original geometrical measures (top) and the

same measures applied to identical tensor data averaged by a Gaussian mask
(bottom). Major white matter tracts such as the corpus callosum show high
linearity in the averaged data set, indicating high macrostructural organization.