Image Processing Reference
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where the second-order term on the right-hand side represents simple Laplacian
smoothing in the tensor-warped space, i.e., isotropic diffusion associated with
the heat equation.
13.7.1
M
EASURING
D
ISTANCES
IN
THE
T
ENSOR
-W
ARPED
S
PACE
Once we have the metric tensor
G
, we are able to apply results from Riemannian
geometry to describe geometric objects such as geodesic paths and distances
between points in the brain. Unlike tractographic methods based on following
the flow of principal eigenvectors of
D
, these geodesic paths are well defined
even in regions where the tensor diffusion is isotropic.
We have approached the measurement of distances in this space in two ways.
First, we have implemented an Eikonal-type equation using level-set methods to
produce a distance transform that respects the metric
G.
This required the deri-
vation of a formula for the speed of an evolving front in the direction of its
Euclidean normal. Second, we have implemented Dijkstra's algorithm using
G
to determine
distances between neighboring voxels, employing the for-
mula , in which
w
is the vector from a voxel to its neighbor. Though it
can suffer from discretization problems, Dijkstra's algorithm is fast and allows
interactive display of return paths.
For our level-set [19] implementation, we seek a speed function
F
for use in
the evolution equation
1
2
T
(
wGw
)
φ
=|∇| .
F
φ
(13.31)
t
This can be done using the following algorithm, which amounts to finding
the length of the projection of the unit normal in the tensor-warped space onto
the Euclidean normal:
∇
|∇ |
φ
φ
1.
Set
n
=
, the Euclidean normal to the level set.
2.
Find any two linearly independent vectors
t
1
and
t
2
perpendicular to
n
.
These are tangents that span the tangent space to the level set.
3.
Set
w
=
()( )
1
t
×
t
.
2
n
4.
Set
=
w
wGw
.
This is the unit normal with respect to
G
.
1
2
T
(
)
T
n
5.
Set
F
=|
n
n
|.
This is the length of the projection of
onto
n.
13.7.2
E
XPERIMENTS
The data acquisition and preprocessing experiments were the same as described
in Subsection 13.6.1. Other experiments are described in the following text:
Tensor-warped distances:
Figure 13.14
shows a slice through a 3-D dis-
tance transform with respect to the metric derived from the DT-MRI
tensor field. The contours are isodistance contours.
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