Image Processing Reference
In-Depth Information
O /∼ B =
{x /∼ B | x ∈ O},
(8.4)
x∈O x /∼ B =
O,
(8.5)
∀x, y ∈ O
(
x /∼ B )
(
y /∼ B )=
∅.
(8.6)
Example 8.2
Figure 8.3a shows an image of size 256
×
256 pixels and their subimages. Let
O, F
be a perceptual system where
O
denotes the set of 25
×
25 subimages.
A) The original image
B) A subimage in the covering
A) The original image
B) A subimage in the covering
(b)
(a)
(a)
(b)
C) The equivalence class on a white background
D) The equivalence class marked on image
C) The equivalence class on a white background
D) The equivalence class marked on image
(c)
(d)
(c)
(d)
(8.3a) An equivalent class
(8.3b) A tolerance class
FIGURE 8.3: An image and one of its equivalence (Left) and tolerance (Right)
classes
Let B = 1 ( x ) }⊆ F is the set of only one probe function
φ 1 where
φ 1 ( x )=
gray
. The two marked subimages
are perceptually indiscernible with respect to their gray level values and hence they
are named an equivalence class . The equivalence class is shown both individually
(c) and on top of the image where the rest of image is blurred (d).
(
x
) is the average gray scale value of subimage
x
Perceptual indiscernibility relation can also be defined in a weak sense as follows
Perceptual Weak Indiscernibility Relation
(Peters and
DEFINITION 8.3
Wasilewski, 2009; Peters, 2009)
Let
O, F
be a perceptual system. Let
B
=
1 2 , ..., φ l }
and
x, y ∈ O
. A percep-
tual weak indiscernibility relation
B is defined relative to
B
as follows

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