![](bg6.png)
For the initial expansion level set function Eq. (6),
F
0
refers to g, (u,v) refers to −β∇g, and K refers to
g ·κ in Eq. (6).
For the free expansion level set function Eq. (7),
F
0
≡ 1 and the other terms are eliminated. So the Eq.
(12) is reduced to:
φ
n+1
i j
= φ
n
i j
−∆t ·∇
+
, (14)
For the surface wrapping level set function Eq. (8), F
0
refers to −g, which is negative, K refers to g ·κ in Eq.
(8) and (u, v) ≡ 0. So the equation can be rewritten
as:
φ
n+1
i j
= φ
n
i j
+∆t[−g·∇
−
+(εK
n
i, j
((D
0x
i j
)
2
+(D
0y
i j
)
2
)
1/2
)].
(15)
Another restriction condition for the equation is that
the front may not invade other cells’ region when
the seed grows. But in the iteration procedure, ev-
ery front is moving independently from other fronts.
To avoid the penetration phenomenon, in every it-
eration step, the outcome is considered as a trial
function. By comparing with other fronts in pre-
vious steps using the following standard: φ
i
m+1
=
max{φ
i
m+1(trial)
,−φ
j
m
}, i ≤ j ≤ n, i 6= j, the final
movement of the front is determined.
For the three level set equations, a reinitialisation
phase is necessary. The purpose of reinitialisation
is to keep the evolving level set function close to a
signed distance function during the evolution. It is a
numerical remedy for maintaining stable curve evolu-
tion (Sussman and Fatemi, 1999). The reinitialisation
step is to solve the following evolution equation:
ψ
τ
= sign(φ(t))(1 −|∇ψ|),
ψ(0,·) = φ(t,·).
(16)
Here, φ(t, ·) is the solution φ at time t. This equation
is solved by an iterative method. In this program 5
iterations are used. The result ψ will be the new φ
used in the program.
3.5 Parameter Setting and Operation
Time
The parameters in the Eq. (6) should be chosen care-
fully, since they will influence the accuracy of final
result. The values of the parameters used in the test
are chosen empirically. By testing with different im-
ages, it was found that the given set of parameters in
Table 3 can be used for images within a broad range
of image characteristics. For the first level set equa-
tion, α = 0.015, β = 0.2, ε = 0.005. For the third
level set equation, α = 0.05, ε = 0.005. Since α de-
termines the sensitivity of the flow to the gradient, for
the initial expansion, the value of α should be small
in order to avoid the influence of sub-structures inside
the cells. However, for the interface wrapping part,
the value of α should be large in order to get accu-
rate position of external boundary of cells. The time
step ∆t is also an important parameter, it determines
the speed of movement. With too large time step ∆t,
the front can not converge to correct solution, with
smaller time step ∆t, the evolution speed is very slow,
as it needs to take more steps to get the right solution.
For the first two level set equations, the time step ∆t
is chosen as 0.1. For the last step, the time step ∆t is
chosen smaller value to increase the accuracy.
There are several stopping criteria for this method.
For instance, it can be checked if the volume increases
after each iteration. Setting a minimum threshold of
volume change can interrupt the flow. Alternatively,
a conservatively high number of iterations can be set.
In this work, the second method is chosen. The pa-
rameter setting and iteration numbers can be found in
Table 3. The iteration number for the last two steps
could be a little different for different images. Usu-
ally, the thicker the membrane of cells is, the larger
the iteration number is. The iteration number of the
last level set equation is generally twice the iteration
number of the second level set equation.
Table 3: Parameter Setting for Level Set Method.
Flow α β ε ∆t ] iter
In. Exp. 0.015 0.2 0.005 0.10 200
Fr. Exp. - - - 0.10 20∼40
S. Wr. 0.05 - 0.005 0.05 40∼100
Table 4 shows the operation time for several test
images. For comparison sake, all the test images
listed in this table use 200 iterations for initial expan-
sion, 20 iterations for free expansion and 40 iterations
for interface wrapping. The operation time depends
on the image size. When the image size grows, more
memory space is required and more operation time is
needed. The number of cells is another important fac-
tor determining the operation time. For each cell, the
program needs to solve one PDE. When the number of
the cells increases, the operation time grows sharply.
For an image of 256 ×256 pixels and containing 68
cells, the program will run around two hours. Appar-
ently, this is the drawback of this method, because the
model needs to set up n PDEs for n cells, and solv-
ing each PDE needs a lot of space and time resource.
However, alternative methods utilizing fewer PDEs
tend to merge level sets and result in wrong results
(Zhou et al., 2007).
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