Biomedical Engineering Reference
In-Depth Information
2.2.7
Reconciling Level Set and Distance Function
In a recent paper [34], Gomes and Faugeras introduced a reformulation of the
Hamilton-Jacobi equation of Eq. (2.5) underlying the level set initial formulation
from Osher and Sethian [7] to eliminate problems related to reinitialization of
the distance function and the need to extend the velocity field away from the
level zero.
The fact that the solution to Hamilton-Jacobi equations of the form in Eq. (2.5)
are not distance functions has been demonstrated formally in [35]. In [34] the
authors provide two simple examples illustrating this result. There are both
theoretical and practical reasons pointed out by the authors to motivate the
preservation of the signed distance function during the segmentation process.
Theoretically, the signed distance function gives a unique equivalence to the
implicit description of the moving front. From a practical point of view, the
use of a signed distance function enables to directly extract from the level set
function geometrical properties of the front and guarantees bounded values of
the level set function gradient, ensuring numerical stability of the segmentation
iterative process.
To derive the new dynamic equation, the authors initialize the level set func-
tion
φ
0
=
φ
(
x
,
0) at
t
=
0 as the signed distance function from the initial front.
The goal is to redefine a speed function
F
such that
∂
∂
t
=
F
which (1) preserves
φ
as the signed distance function from the level zero, and (2) ensures that the level
zero of
φ
evolves as in Eq. (2.2). These constraints are expressed mathematically
as:
⎧
⎨
F
/φ
=
0
=
V
∂φ
∂
t
=
F
|∇
φ
| =
1
(2.25)
⎩
where
F
/φ
=
0
denotes the restriction of
F
to the zero-level of
φ
. The authors
derived the following dynamic equation as the solution to this system:
∂φ
∂
t
=
V
(
x
−
φ
∇
φ
)
(2.26)
3
, which is not a Hamilton-Jacobi equation.
Implementation of the equation is proposed with a narrow-band framework,
shock-detecting gradient computation and as described in [14].
for any point
x
∈ R
