Image Processing Reference
In-Depth Information
Spherical case (
λλλ
1
): isotropic diffusion,
2
3
(
)
.
T
T
T
DD
λ
=
λ
ˆˆ
+
ˆ ˆ
+
ˆˆ
(13.10)
ee
e e
ee
1
s
1
11
2 2
3 3
will be a combination of these cases.
Expanding the diffusion tensor using these cases as a basis gives
In general, the diffusion tensor
D
T
T
T
D
=
λ
+
λ
+
λ
ˆˆ
ˆ ˆ
ˆˆ
ee
e e
ee
11
22
33
1
2
3
(
)
T
T
T
=−
(
λλ
)
ˆ
+− +
(
λλ
)
ˆˆ
ˆ
ˆ ˆ
e
e
e e
e e
11
11
2 2
1
2
2
3
(
)
T
T
T
+
λ
ˆˆ
+
ˆ
ˆ
+
ˆ ˆ
ee
e
eee
3
11
2 2
3
3
=− +− +
(
λλ λλ λ
)
D
(
)
D
D
1
2
i
2
3
p
3
s
where (
λ
−
λ
), (
λ
−
λ
) and
λ
are the coordinates of
D
in the tensor basis
1
2
2
3
3
{
}. A similar tensor shape analysis has been shown to be useful in a
number of computer vision applications [22,23].
The coordinates of the tensor in our new basis classify the diffusion tensor
and describe how close the tensor is to the generic cases of line, plane, and sphere,
and hence, can be used for classification of the diffusion tensor according to its
geometry. Because the coordinates are based on the eigenvalues of the tensor, they
are rotationally invariant, and the values do not depend on the chosen frame of
reference. To obtain quantitative measures of the anisotropy, the derived coordi-
nates have to be normalized, which, in turn, will lead to geometric shape measures.
As in the case of fractional and relative anisotropy, there are several (rotationally
invariant) choices of normalization. For example, the maximum diffusivity,
D
,
D
,
D
i
p
s
λ
, the
1
trace of the tensor,
λ
+
λ
+
λ
, or the norm of the tensor,
2
2
++
2
, can be
λλλ
1
1
2
3
2
3
used as normalization factors.
By using the largest eigenvalues of the tensor, the following quantitative shape
measures are obtained for the linear, planar, and spherical measures:
λλ
λ
1
2
c
l
=
(13.11)
1
λλ
λ
2
3
c
p
=
(13.12)
1
c
s
=
λ
λ
3
(13.13)
1
where all measures lie in the range from zero to one, and their sum is equal to one,
cc c
l
++=
1
(13.14)
p
s
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