Digital Signal Processing Reference
In-Depth Information
Fig. 2.
Target pixel and its adjacent pixels
∇
==⋅ ∇=
u
1
1
^
Given
12
, hence the target pixel
O
is
v
D u
u
,
u
=
(
v v
,
)
x
y
∇
u
∇
u
∇
u
discretized by central differencing:
∂
u
∂
v
1
∂
v
2
v
1
−
v
1
v
2
−
v
2
=∇⋅
v
=
+
≈
e
w
+
s
n
(3)
O
∂
t
∂
x
∂
y
h
h
O
O
Where
h
denotes the grid size, which is always taken to be 1 in image processing.
Next, take the midpoint
e
, for example:
1
1
∂
u
1
u
−
u
v
1
=
(
u
)
=
(
)
=
(
)
⋅
E
O
(4)
e
x
e
∇
u
∇
u
∂
x
∇
u
h
e
e
e
∂
u
∂
u
1
12
2
2
(5)
∇=
u
(, ) (
u u
=
,
)
=
(
u
−
u
) (
+
u
+ − −
u
u
u
)4]
e
e
e
E
O
NE
E
SE
S
∂
x
∂
y
h
e
e
wsn
to obtain
,,
1
v
,
2
v
,
2
v
. Therefore, at a
Similar discussion applies to the other
pixel
O
, (1) and (3) is discretized to
1
(6)
(
uu Ouu
−+
)
λ
(
)(
−=
0
)
0
O
p
e
O
O
∇
u
p
∈Λ
p
3
The Characteristics of Region Gradient for TV Model
Considering the computing essence of (1), the
diffusion intensity and diffusion
direction are determined by
∇
u
1
1
^
. When
12
v
=
=
D u
⋅ ∇=
u
,
u
=
(
v v
,
)
x
y
∇
u
∇
u
∇
u
inpainting, the size of diffusion coefficient decides the magnitude of smoothing effect
(that is the size of
diffusion intensity
) in different image feature. The image
inpainting should have a large
diffusion
in the smooth region; on the contrary, the
image inpainting should have a large
diffusion
. However, in the original TV algorithm,
computing the gradient value of the diffusion only considered the change of image
