Image Processing Reference
In-Depth Information
[100,101], and they are often used in the surface-matching methods discussed
earlier. For instance, quaternions (vectors in ) can be natively used to describe
a rotation in 3-D space leading to a straightforward solution of the registration
problem [100]. This method is simple to implement. Its precision rapidly increases
with the number of fiducial points, reaching the performance of surface-matching
algorithms cheaply and efficiently.*
L
4
8.4.1.2
Segmentation and Tessellation
PD or T1/T2 3-D MR images can be used to segment different brain tissues
(white matter, gray matter, cerebrospinal fluid [CSF], skull, scalp) as well as
abnormal formations (tumors) [17,102]. Different kinds of MR contrasts are
optimal for the segmentation of the different kinds of head and brain structures.
For instance, PD-weighted MRI yields superior segmentation of the inner and
outer skull surfaces, because bones have much smaller water content than brain
tissue, making the skull easily distinguishable on PD images. On the other hand,
exploiting T1 and T2 relaxation time differences between various sorts of brain
tissue leads to higher quality segmentation of structures within the brain.
Using triangulation (tessellation) and interpolation it is possible to create fine-
grained smooth mesh representations or tetrahedral assemblies of the segmented
tissues [103-105]. Obtained 3-D meshes of the cortical surface alone brings
valuable information to the analysis of E/MEG signals [106]: the physiology of
the neuronal generators can be considered, allowing one to limit the search space
for activated sources to the gray matter regions and oriented orthogonally or
nearly so to the cortical surface [17,107].
Monte Carlo studies [108] tested the influence of the orientation constraint in
the case of the DECD model and showed that it leads to much better conditioning
* To find the minimum of the error function ε(
R
,
v
), we need merely to calculate a principal eigenvector
tr(
Σ ∆
∆ΣΣ Σ
)
T
3
r
=
max_
eigenvector
(8.15)
+−
i
tr
()
I
where
(
ΣΣ
ΣΣ
−
−
i
i
)
P
P
23
1
∑
1
∑
(
)
(
)
i
x
=
x
Σ
=
x
i
E
−
x
E
x
i
M
−
x
M
∆
=
(
)
31
12
i
P
P
i
i
(
ΣΣ
−
i
)
The eigenvector
r
can be assumed to be normalized (unit length). Regarded as a quaternion,
r
=
[
r
0
,
r
1
,
r
2
,
r
3
]
T
uniquely defines the rotation. This can be converted into a conventional rotation matrix
rrrr
2
+−−
2
2
2
2
(
rrr
−
)
2
(
rrr
+
)
0
1
2
3
12
03
13
0 2
R
=
2
(
r rrr
+
)
rrrr
2
+−−
2
2
2
2
(
rrrr
−
)
.
12
03
0
2
1
3
23
01
2
(
rr
−
rr
)
2
(
rrrr
+
)
rrrr
2
+−−
2
2
2
1
3
0 2
2 3
0 1
0
3
1
2
The translation vector is then simply
vx
=
−
R
x
E
.
M
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