Image Processing Reference
In-Depth Information
a mathematical description of diffusion tensors and their quantification is necessary
[1]. Several different measures of anisotropy have been proposed in the literature.
Among the most popular are two that are based on the normalized variance of the
eigenvalues: relative anisotropy (RA) and fractional anisotropy (FA) [3]. An advan-
tage of these measures is that they can be calculated without first explicitly calcu-
lating any eigenvalues. Both anisotropy measures can be expressed in terms of the
norm and trace of the diffusion tensor. The norm is calculated as the square root
of the sum of the squared elements of the tensor, which equals the square root of
the sum of the squared eigenvalues; and the trace is calculated as the sum of the
diagonal elements, which equals the sum of the eigenvalues:
(
λλ λλ λλ
λλλ
−+−+−
++
)
2
(
)
2
(
)
2
1
2
3
| −
D
trace
trace
1
3
()
()
D I
|
1
2
2
3
1
3
RA
=
=
(13.5)
(
)
D
2
1
2
3
(
λλ λλ λλ
λλλ
−+−+−
++
)
2
(
)
2
(
)
2
1
2
3
2
| −
D
1
3
trace(
D I
)
|
1
2
2
3
1
3
FA
=
=
(13.6)
|
D
|
2
2
2
1
2
3
is the identity tensor. The constants are inserted to ensure that the measures
range from zero to one. In the next section, we will present alternatives to these
measures based on the geometric properties of the diffusion ellipsoid.
where
I
13.3.1
G
M
D
EOMETRICAL
EASURES
OF
IFFUSION
The diffusion tensor can be visualized using an ellipsoid in which the principal
axes correspond to the directions of the eigenvector system. Using the symmetry
properties of this ellipsoid, the diffusion tensor can be decomposed into basic
geometric measures [25], a concept that we will elaborate in this section.
Let
λ
≥
λ
≥
λ
≥
0 be the eigenvalues of the symmetric diffusion tensor
D
,
1
2
3
and let
e
be the normalized eigenvector corresponding to
λ
. The tensor
D
can
i
i
then be described by
D
=
λ
ˆˆ
T
+
λ
ˆ ˆ
T
+
λ
ˆˆ
T
(13.7)
ee
e e
ee
11
22
33
1
2
3
Diffusion can be divided into three basic cases depending on the rank of the
diffusion tensor:
Linear case (
λλλ
1
>>
): diffusion is mainly in the direction corresponding
2
3
to the largest eigenvalue,
DD
ee
λ
=
λ
ˆˆ
T
(13.8)
11
1
l
1
Planar case ( ): diffusion is restricted to a plane spanned by the
two eigenvectors corresponding to the two largest eigenvalues,
λλ λ
1
>>
2
3
(
)
DD
λ
=
λ
ˆˆ
T
+
ˆ ˆ
T
(13.9)
ee
e e
11
2 2
1
p
1
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