Biomedical Engineering Reference
In-Depth Information
tr
ln
2
S
−
1
/
2
1
.
1
2
S
2
S
−
1
/
2
D
Rm
(
S
1
,
S
2
)=
(6.27)
1
Sym
n
An equally well known metric for
is the Log-Euclidean metric [
3
]. Although
it isn't affine invariant and only similarity invariant, computationally it is more
efficient than the affine invariant Riemannian metric and resembles closely the latter.
The distance between DTs induced by this metric is:
tr
S
2
))
2
.
1
2
D
LE
(
S
1
,
S
2
)=
(ln(
S
1
)
−
ln(
(6.28)
Estimation of DTs in
Sym
3
6.5.2
Using the Riemannian
Metric
Sym
3
Using the appropriate Riemannian metric and geometry for
can constrain
Sym
3
all operations to
. For example using the Riemannian metric for DTI
estimation can ensure that the DTs are positive definite or that no negative
diffusion will be estimated even in the presence of noisy DWIs. This can be
done by using the logarithmically transformed version of the Stejskal-Tanner
equation (Eq.
6.13
) and considering its explicit least square minimization
E
(
D
)=
2
. The gradient of this least square functional using
the affine invariant Riemannian metric can be shown to be [
40
]:
b
ln
S
i
S
0
i
=1
1
2
g
i
+
Dg
i
b
ln
S
i
S
0
N
g
i
Dg
i
)
T
.
∇
E
(
D
)=
+
Dg
i
·
(
Dg
i
)(
(6.29)
i
=1
Furthermore the Euclidean gradient descent algorithm required for optimization has
to be appropriately replaced by a Riemannian geodesic descent algorithm to respect
the Riemannian geometry of
Sym
3
.
6.5.3
Segmentation of a Tensor Field
The goal of segmenting a tensor field or an image of DTs is to compute the optimal
3D surface separating an anatomical structure of interest from the rest of the tensor
image (Fig.
6.9
). To do this we follow the method proposed in [
23
]. The idea will
be to treat the tensor field as a field of Gaussian probability density functions and
to utilize the affine invariant Riemannian metric on
Sym
3
, which also forms a
Riemannian metric in the space of Gaussian density functions, to compute the
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