Graphics Reference
In-Depth Information
Using the same relaxation technique as above, we solve the first
u
equation
(23) by the iterative scheme:
=
u
i,j
+
ˉ
3
ʾI
i,j
+
ʾg
i,j
+2
μ
[
C
E
u
i,j
+1
+
C
W
u
n
+1
+
C
S
u
i
+1
,j
]
+
C
N
u
n
+1
i−
1
,j
u
n
+1
i,j
i,j−
1
1+
ˉ
3
2
μ
(
C
E
+
C
W
+
C
N
+
C
S
)+
ʾ
,
(24)
where
ˉ
3
>
0 is the relaxation factor and
C
E
=(
v
n
)
i,j
+1
+
o
,C
W
=(
v
n
)
i,j−
1
+
o
,C
N
=(
v
n
)
i−
1
,j
+
o
,C
S
=(
v
n
)
i
+1
,j
+
o
.
Also, we solve
v
by the iterative scheme:
=
v
i,j
+
ˉ
4
4
+
(
v
i
+1
,j
+
v
n
+1
i,j−
1
+
v
i,j
+1
)
i−
1
,j
+
v
n
+1
v
n
+1
i,j
,
(25)
1+
ˉ
4
(
4
+2
μ
u
n
+1
i,j
|∇
|
2
+4
)
where
ˉ
4
>
0 is the relaxation factor.
Now, we present the full algorithm as follows.
AT Algorithm
1. Initialization: set
o
=
p
(
p>
1), and input
u
0
,v
0
,g
0
,μ,ʽ,ʾ, p, , ˉ
3
,ˉ
4
.
2. For
n
=0
,
1
,
,
(i) Update
u
n
+1
by (24);
(ii) Update
v
n
+1
by (25);
(iii) Update
g
by the third equation in (23);
3. Endfor till some stopping rule meets.
···
3Num lExp imen s
In this section, we compare the above two algorithms: the SB algorithm and the
AT algorithm. Here, we set the stopping rule by using the relative error
E
(
u
n
+1
,u
n
):=
u
n
+1
u
n
2
−
≤
ʷ,
(26)
where a prescribed tolerance
ʷ>
0
.
The choice of the parameters are specified
in the captions of the figures.
In the first experiment, we test three real clean images (see Fig. 1), and
generally choose the same parameters in the SB algorithm and the AT algorithm
with
10
3
, ʽ
=1
, ˉ
1
=1
10
−
3
, ʷ
=1
10
−
6
μ
=5
×
×
×
and the parameters
ʾ
= 300
,ˉ
2
=1
,ʻ
= 500 in the SB algorithm and
=2
×
10
−
3
, p
= 2 in the AT algorithm, respectively. For simplicity, we let
ˉ
3
=
ˉ
4
=
ˉ
2
in AT algorithm. These parameters are chosen by experimental experiences to
get good edges in visual. We present the input images and the results detected
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