Biomedical Engineering Reference
In-Depth Information
Figure 6.
Level set formulation of equations of motion: (a) arrows show the propagating
direction of the front
γ
(
t
=
t
1
)
; (b) the corresponding surface
ψ
(
x, y
)
at time
t
2
.
above equation is also called an Eulerian Hamilton-Jacobi equation (see [16,32]).
Equation (7) for the 2D and 3D cases can be generalized as
∂ψ
∂t
=
F
κ
(
x, y
)
|∇
,
∂ψ
|
∂t
=
F
κ
(
x, y, z
)
|∇
|
ψ
ψ
,
(8)
where
F
κ
(
x, y
) and
F
κ
(
x, y, z
) are curvature-dependent speed functions in 2D
and 3D, respectively.
The topological change of the level set segmentation method is illustrated in
Figure 6. The arrows in Figure 6a show the propagating direction of the front
γ
(
t
=
t
1
). During a period of time
t
2
, the front
γ
(
t
=
t
2
) and the zero level set
ψ
(
x, y, t
=
t
2
) is shown in Figure 6b. In this way, the actual object contour can
be obtained using the level set method.
4.2. Analogies of Front Propagation
There are three analogies of the front propagation equation (see Suri et al. [16]).
The first is that these equations can be compared with the Euclidean geometric
heat equation (see Grayson [36]), given as
∂C
∂t
=
κN,
(9)
where
κ
is the curvature,
N
is the inward unit normal, and
C
are the curve co-
ordinates. The second analogy is that Eq. (7) is also called the curvature motion
