Biomedical Engineering Reference
In-Depth Information
Figure 1.
Configuration of the DDC. See the text for symbols. Reprinted with permission
from the AAPM.
to the closest features while keeping the DDC smooth, and is fully described in
Section 2.1.4. This net force causes the vertex to accelerate according to a well-
known formula:
m
i
f
i
(
t
)
,
a
i
(
t
)=
(1a)
where
m
i
is the mass of the vertex. For simplicity, the mass of each vertex is taken
to be unity.
Next, the velocity,
v
i
, and position,
p
i
T
, of each vertex
i
are
computed and updated for the next iteration at time
t
+∆
t
using explicit Euler
integration:
=(
x
i
,y
i
)
v
i
(
t
+∆
t
)=
v
i
(
t
)+
a
i
(
t
)∆
t,
(1b)
and
p
i
(
t
+∆
t
)=
p
i
(
t
)+
v
i
(
t
)∆
t,
(1c)
where ∆
t
is the time step. The initial velocity of each vertex, i.e., the velocity at
t
=0, is taken to be zero. The deformation process continues until all vertices
come to rest, which occurs when the velocity and acceleration of each vertex
become approximately zero. A time step of unity is used for the iterations.
During deformation, the distance between neighboring vertices can become
larger, and the DDC may no longer accurately represent the local geometry of the
boundary. Therefore, after each iteration, the DDC is “re-sampled” so that the
distance between vertices is maintained at a uniform value. Linear interpolation
of the vertices as a function of curve length was found to work well. A sampling
distance of 20 pixels was found to provide a good representation of the prostate
boundary.
