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2 The Minimization Algorithms
In this section, we introduce the minimization algorithms for solving (5) and
use an iterative method for computing
u
, and the split Bregman method for
resolving the binary level-set function
ˈ
.
Similarly as [6,26], we use the alternating optimization technique to split (5)
into three subproblems:
-
u
-subproblem: for fixed
ˈ
and
g
,wesolve
μ
ʩ
ʩ
|
2
d
x
.
2
d
x
+
ʾ
2
ˈ
)
2
min
u
(1
−
|∇
u
|
u
−
I
−
g
|
(6)
-
ˈ
-subproblem: for fixed
u
,wesolve
μ
2
d
x
+
TV
(
ˈ
)
.
(1
− ˈ
)
2
min
ˈ∈
[0
,
1]
|∇u|
(7)
ʩ
-
g
-subproblem: for fixed
u
,wesolve
ʽ
2
g
)
2
d
x
.
d
x
+
ʾ
2
min
g
ʩ
|
g
|
(
u
−
I
−
(8)
ʩ
The solution of (8) can be easily expressed as:
max
0
,
1
.
ʽ
g
=
|
u
−
I
|
−
2
ʾ
|
u
−
I
|
The detailed process can be referred to [21,20]. Next, we present the algorithms
for (6) and (7).
2.1 Fixed-Point Iterative Method for Solving
u
We first consider (6). Notice that the functional in (6) is convex for fixed
ˈ
,
so it admits a minimizer. The corresponding Euler-Lagrange equation takes the
form:
⊧
⊨
−
2
μ
div
(1
− ˈ
)
2
∇u
+
ʾ
(
u − I − g
)=0
,
in
ʩ,
∂ʩ
=0
,
(9)
∂u
∂
n
⊩
on
∂ʩ.
where
n
is the unit outer normal vector to
∂ʩ
as before. We expect
ˈ
take value
0 at the homogeneous region, i.e., 1
1. So, we propose to use a fixed-point
iterative scheme based on relaxation method to solve this elliptic problem with
variable coecient (see, e.g., [17] for similar ideas). We start with the difference
equation:
−
ˈ
≈
ˈ
)
i,j
+1
(
u
i,j
+1
−
ˈ
)
i,j−
1
(
u
i,j−
1
−
ʾu
i,j
=2
μ
[(1
−
u
i,j
)+(1
−
u
i,j
)
(10)
ˈ
)
i−
1
,j
(
u
i−
1
,j
−
ˈ
)
i
+1
,j
(
u
i
+1
,j
−
+(1
−
u
i,j
)+(1
−
u
i,j
)] +
ʾI
i,j
+
ʾg
i,j
,
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