Image Processing Reference
In-Depth Information
In Figures 8.8 to 8.10, examples of ordered tolerance matrix is shown for more
images. The images are from the Berkeley segmentation dataset (Martin, Fowlkes,
Tal, and Malik, 2001).
Each image size is 321
×
481 which is divided into 160
×
subimages of size 30
30. The only probe function used is the average gray level
values between 0 and 255. The value of
in equation 8.8 is equal to 20 where the
gray values have not been normalized.
8.2.3
Nearness Relations and Near Sets
Nearness relations were introduced in the context of a perceptual system
O,
F
by James Peters (Peters and Wasilewski, 2009) after the introduction of near set
theory in 2007 (see (Peters, 2007c),(Peters, 2007b) and (Peters and Wasilewski,
2009)). These relations are defined between
sets of perceptual objects
. Therefore, a
nearness relation
R
is a subset of
P
(
O
)
×P
(
O
).
DEFINITION 8.8
Weakly Nearness Relation
Let
O,
F
be a perceptual system and let
X, Y ⊆ O
.
Aset
X
is weakly near to a set
Y
within a perceptual system
O,
F
and is shown
X
F
Y
with
, if and only if the following condition is satisfied:
∃ x ∈ X, y ∈ Y, B⊆
F
x ∼
B
y
Consequently, nearness relation is defined on
such that
P
(
O
) as follows:
F
=
{
(
X, Y
)
∈P
(
O
)
×P
(
O
)
| X
F
Y }
(8.14)
Example 8.6
Figure 8.7 shows two images and their corresponding subimages (25
×
25 pixels
each).
Let
X
and
Y
denotes the set of all subimages in image 1 and image 2
respectively. Let
O,
F
be a perceptual system and let
B⊆
F
and
B
=
{φ
1
(
x
)
}
, where
φ
1
(
x
)=
gray
(
x
)is
the gray scale value of subimage
O
=
X ∪ Y
be the set of all subimages in two images. Let
x
.
X
Y
X
F
Y
Images
and
are then weakly near to each other (
) because they have
elements
x ∈ X
and
y ∈ Y
with matching descriptions (
x ∼
B
y
).
Nearness Relation
DEFINITION 8.9
Let
O,
F
be a perceptual system and let
X, Y ⊆ O
.Aset
X
is near to a set
Y
within a perceptual system
O,
F
and is shown with
X
F
Y
, if and only if the
following condition is satisfied (Peters and Wasilewski, 2009):
∃ x ∈ X, y ∈ Y, A,B⊆
F
,f ∈
F
and also
∃ A ∈ O
/∼
A
,B∈ O
/∼
B
,C∈ O
/∼
C
such that
A, B ⊆ C
Consequently, the nearness relation is defined on
P
(
O
) as follows:
F
=
{
(
X, Y
)
∈P
(
O
)
×P
(
O
)
| X
F
Y }
(8.15)
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