Biomedical Engineering Reference
In-Depth Information
It is easy to show that for isotropic diffusion, when
λ
1
=
λ
2
=
λ
3
, the coef-
ficient
C
a
=
1. In the anisotropic case, this measure is identical for both linear,
directional diffusion (
λ
1
λ
2
≈
λ
3
) and planar diffusion (
λ
1
≈
λ
2
λ
3
) and is
equal to
1
+
λ
1
1
3
λ
3
+
λ
3
C
limit
a
(8.32)
≈
.
λ
1
Thus
C
a
is always
∼
λ
max
/λ
min
and measures the magnitude of the diffusion
anisotropy. We again want to emphasize that we use the eigenvalue representa-
tion here only to analyze the behavior of the coefficient
C
a
, but we use invariants
(
C
1
,
C
2
,
C
3
) to compute it using Eqs. (8.28) and (8.31).
8.5.2 Geometric Modeling
Two options are usually available for viewing the scalar volume datasets, direct
volume rendering [1, 4] and volume segmentation [67] combined with conven-
tional surface rendering. The first option, direct volume rendering, is only capa-
ble of supplying images of the data. While this method may provide useful views
of the data, it is well known that it is difficult to construct the exact transfer
function that highlights the desired structures in the volume dataset [68]. Our
approach instead focuses on extracting geometric models of the structures em-
bedded in the volume datasets. The extracted models may be used for interactive
viewing, but the segmentation of geometric models from the volume datasets
provides a wealth of additional benefits and possibilities. The models may be
used for quantitative analysis of the segmented structures, for example the cal-
culation of surface area and volume, quantities that are important when studying
how these structures change over time. The models may be used to provide the
shape information necessary for anatomical studies and computational simula-
tion, for example EEG/MEG modeling within the brain [69]. Creating separate
geometric models for each structure allows for the straightforward study of
the relationship between the structures, even though they come from different
datasets. The models may also be used within a surgical planning/simulation/VR
environment [70], providing the shape information needed for collision detection
and force calculations. The geometric models may even be used for manufactur-
ing real physical models of the structures [71]. It is clear that there are numerous
reasons to develop techniques for extracting geometric models from diffusion
tensor volume datasets.
