Biomedical Engineering Reference
In-Depth Information
This gives a partial differential equation on
φ
:
s
≡
s
(
t
)
d
s
dt
=∇
φ
F
(
s
,
n
,φ,
D
φ,
D
2
∂φ
∂
t
=−∇
φ
·
φ,...
)
,
(8.4a)
d
s
dt
,
F
()
≡
n
·
(8.4b)
where
F
() is a user-created “speed” term that defines the speed of the level set
at point
s
in the direction of the local surface normal
n
at
s
.
F
() may depend
on a variety of local and global measures including the order-
n
derivatives of
φ
,
D
n
φ
, evaluated at
s
, as well as other functions of
s
,
n
,
φ
, and external data.
Because this relationship applies to every level set of
φ
, i.e. all values of
k
, this
equation can be applied to all of
U
, and therefore the movements of
all
the level
set surfaces embedded in
φ
can be calculated from Eq. (8.4).
The level set representation has a number of practical and theoretical advan-
tages over conventional surface models, especially in the context of deformation
and segmentation. First, level set models are topologically flexible, they easily
represent complicated surface shapes that can form holes, split to form multiple
objects, or merge with other objects to form a single structure. These models
can incorporate many (millions) degrees of freedom, and therefore they can ac-
commodate complex shapes such as the dendrite in Fig. 8.7. Indeed, the shapes
formed by the level sets of
φ
are restricted only by the resolution of the sampling.
Thus, there is no need to reparameterize the model as it undergoes significant
changes in shape.
The solutions to the partial differential equations described above are com-
puted using finite differences on a discrete grid. The use of a grid and discrete
time steps raises a number of numerical and computational issues that are impor-
tant to the implementation. However, it is outside of the scope of this chapter to
give a detailed mathematical description of such a numerical implementation.
Rather we shall provide a summary in a later section and refer to the actual
source code which is publicly available
5
.
Equation (8.4) can be solved using finite forward differences if one uses the
up-wind scheme, proposed by Osher
et al.
[30], to compute the spatial deriva-
tives. This up-wind scheme produces the motion of level set models over the
entire range of the embedding, i.e., for all values of
k
in Eq. (8.2). However, this
5
The level set software used to produce the morphing results in this chapter is available
for public use in the VISPACK libraries at http://www.cs.utah.edu/
∼
whitaker/vispack.
