Biomedical Engineering Reference
In-Depth Information
model tissue deformations in image-guided surgery. They propose a three-
component model to simulate the properties of rigid, elastic, and fluid struc-
tures. For this purpose the image is divided into a triangular mesh with
n
connected nodes
i
. Each node is labeled according to the physical properties
of the underlying anatomical structures: for example, bone is labeled as rigid,
soft tissues as elastic, and CSF as fluid. While nodes labeled as rigid are kept
fixed, nodes labeled as elastic or fluid are deformed by minimizing an energy
function. Edwards et al.
38
proposed a number of different energy terms to
constrain deformations: for example, nodes labeled as elastic can be con-
strained by a tension energy
2
0
E
tension
i
,
(
j
)
j
i
i
,
j
0
where corresponds to the relaxed distance between two nodes. An alter-
native choice for nodes labeled as elastic is a stiffness energy term:
i
,
j
2
E
stiffness
i
,
(
j
,
k
)
j
k
2
i
Nodes labeled as fluid do not have any associated tension or stiffness energy.
Instead they have an associated folding energy
2
A
2
A
2
if
A
A
0
A
2
----------
----------
------
E
fold
i
,
(
j
,
k
)
2
A
2
2
otherwise
where
A
0
is the area of the undeformed triangle,
A
is the area of the deformed
triangle, and
is a threshold for the triangular area above which the energy
contribution is constant. This energy term prevents the development of sin-
gularities in the transformation, i.e., the collapsing or folding over of triangles.
In the implementation proposed by Edwards et al.
38
the registration is
driven by a similarity measure which minimizes the distance between cor-
responding landmarks, but other similarity measures can be easily integrated
into the energy function.
13.2.6
Registration Using Optical Flow
A well known registration technique which is equivalent to the equation
of motion for incompressible flow in physics is the so-called optical flow.
39
The concept of optical flow was originally introduced in computer vision
in order to recover the relative motion of an object and the viewer in
between two successive frames of a temporal image sequence. Its funda-
mental assumption is that the image brightness of a particular point stays
constant, i.e.,
I
(
x
,
y
,
z
,
t
)
I
(
x
x
,
y
y
,
z
z
,
t
t
).
(13.13)
