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where the minimizer
u
is intended to get the piecewise smooth approximation of
a given image
I
on an open bounded domain
ʩ
R
2
,
ʓ
is the edge set, and
μ, ʽ
are positive tuning parameters. Many important properties have been obtained
(see, e.g., [9,10]).
According to comparison with region-based image segmentation, edge detec-
tion also focuses on locating open curves belonging to constituent element of
edges, yet do not have interior and exterior regional separation. In a recent con-
ference report [25], we proposed to embed an open (or a closed) curve into a
narrow region (or band), formed by the curve and its parallel curve (also known
as the offset curve [23]). We then combined the Mumford-Shah (MS) model [15]
with the binary leveling processing by introducing the binary level-set function:
ˈ
=1,if
x
is located in small regions around the edges, while
ˈ
= 0 otherwise
[25]. So the modified Mumford-Shah (MMS) model is to replace the length term
in (1) with:
ↂ
ˈ
div
p
d
x
, S
:=
p
∈
1
,
C
c
(
ʩ
;
2
):
TV
(
ˈ
)=sup
p
∈
R
|
p
|≤
(2)
S
ʩ
where
C
c
(
ʩ
;
2
) is the space of vector-valued functions compactly supported
in
ʩ
with first-order partial derivatives being continuous. However, Wang et al.
[25] introduced a very preliminary algorithm for this modified model. Since the
L
1
-norm has showed advantage in handling impulse noise (see, e.g., [14,11]), we
borrowed the above form (2) and consider the modified MS model with
L
1
-norm
as the fidelity term:
R
μ
ʩ
d
x
+
TV
(
ˈ
)
.
2
d
x
+
ʽ
2
ˈ
)
2
min
ˈ∈{
0
,
1
},u
(1
−
|∇
u
|
ʩ
|
u
−
I
|
(3)
.
It is clear that
ˈ
∈{
0
,
1
}
in (3) is non-convex. So, we relax the set and
constrain
ˈ
∈
[0
,
1] (see, e.g., [5]) as follows:
μ
d
x
+
TV
(
ˈ
)
.
2
d
x
+
ʽ
2
ˈ
)
2
min
ˈ∈
[0
,
1]
,u
(1
−
|∇
u
|
ʩ
|
u
−
I
|
(4)
ʩ
.
An auxiliary
g
can be used to handle the last term and face the constraint
g
=
u
I
by a penalty method. Thus (4) can be approximated by the following
problem with
L
1
-norm:
−
μ
2
d
x
+
TV
(
ˈ
)
.
ʩ
|
2
d
x
+
ʽ
2
d
x
+
ʾ
2
ˈ
)
2
min
u,g
ψ∈
[0
,
1]
(1
−
|∇
u
|
g
|
ʩ
|
u
−
I
−
g
|
ʩ
(5)
The rest of the paper is organized as follows. In Section 2, for solving the mini-
mization problem, we separate it into several subproblems, and apply fixed-point
iterative method and split Bregman method [10] to solve the
u
-subproblem and
ˈ
-subproblem. In Section 3, we make a comparison with the seminar Ambrosio-
Tortorelli model [2] with
L
1
fidelity term and present numerical results to demon-
strate the strengths of the proposed method.
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