Source: Wikipedia, the free encyclopedia.
Thinning is the transformation of a digital image into a simplified, but topologically equivalent image. It is a type of topological skeleton , but computed using mathematical morphology operators.
Example
Let
E
=
Z
2
{\displaystyle E=Z^{2}}
, and consider the eight composite structuring elements, composed by:
C
1
=
{
(
0
,
0
)
,
(
−
1
,
−
1
)
,
(
0
,
−
1
)
,
(
1
,
−
1
)
}
{\displaystyle C_{1}=\{(0,0),(-1,-1),(0,-1),(1,-1)\}}
and
D
1
=
{
(
−
1
,
1
)
,
(
0
,
1
)
,
(
1
,
1
)
}
{\displaystyle D_{1}=\{(-1,1),(0,1),(1,1)\}}
,
C
2
=
{
(
−
1
,
0
)
,
(
0
,
0
)
,
(
−
1
,
−
1
)
,
(
0
,
−
1
)
}
{\displaystyle C_{2}=\{(-1,0),(0,0),(-1,-1),(0,-1)\}}
and
D
2
=
{
(
0
,
1
)
,
(
1
,
1
)
,
(
1
,
0
)
}
{\displaystyle D_{2}=\{(0,1),(1,1),(1,0)\}}
and the three rotations of each by
90
o
{\displaystyle 90^{o}}
,
180
o
{\displaystyle 180^{o}}
, and
270
o
{\displaystyle 270^{o}}
. The corresponding composite structuring elements are denoted
B
1
,
…
,
B
8
{\displaystyle B_{1},\ldots ,B_{8}}
.
For any i between 1 and 8, and any binary image X , define
X
⊗
B
i
=
X
∖
(
X
⊙
B
i
)
{\displaystyle X\otimes B_{i}=X\setminus (X\odot B_{i})}
,
where
∖
{\displaystyle \setminus }
denotes the
set-theoretical difference
and
⊙
{\displaystyle \odot }
denotes the
hit-or-miss transform .
The thinning of an image A is obtained by cyclically iterating until convergence:
A
⊗
B
1
⊗
B
2
⊗
…
⊗
B
8
⊗
B
1
⊗
B
2
⊗
…
{\displaystyle A\otimes B_{1}\otimes B_{2}\otimes \ldots \otimes B_{8}\otimes B_{1}\otimes B_{2}\otimes \ldots }
.
Thickening
Thickening is the dual of thinning that is used to grow selected regions of foreground pixels. In most cases in image processing thickening is performed by thinning the background [1]
thicken
(
X
,
B
i
)
=
X
∪
(
X
⊙
B
i
)
{\displaystyle {\text{thicken}}(X,B_{i})=X\cup (X\odot B_{i})}
where
∪
{\displaystyle \cup }
denotes the
set-theoretical difference
and
⊙
{\displaystyle \odot }
denotes the
hit-or-miss transform , and
B
i
{\displaystyle B_{i}}
is the structural element and
X
{\displaystyle X}
is the image being operated on.
References