Image Processing Reference
In-Depth Information
DEFINITION 8.6
Tolerance Preclass
Aset
A ⊂
=
B,ε
is a preclass in
=
B,ε
if, and only if
∀x, y ∈ A, x
=
B,ε
y
.
DEFINITION 8.7
Tolerance Class
Aset
A ⊂
=
B,ε
is a tolerance class in =
B,ε
if, and only if
A
is a maximal preclass.
where
=
B,ε
is defined in (8.8) with respect to a given set of probe functions
B
.Let
x
/
=
B,ε
x
. Tolerance classes can overlap and
hence the set of all tolerance classes is a
covering
of
denote a maximal preclass containing
O
denoted by
O
/
=
B,ε
. Recall
that a cover is a family of subsets of
O
whose union is
O
and their intersection is
not necessarily empty.
O
/
=
B,ε
=
{x
/
=
B,ε
| x ∈ O}
(8.9)
x∈O
x
/
=
B,ε
=
O
(8.10)
Example 8.3
Figure 8.3b shows the image in example 8.2 and its subimages. Let
O,
F
be a
perceptual system where
O
denotes the set of 25
×
25 subimages. The image is
divided into 100 subimages of size 25
of
all the 100 subimages. Let
B
=
{φ
1
(
x
)
}⊆
F
where
φ
1
(
x
)=
gray
(
x
) is the average
gray scale value of pixels in subimage
×
25 and can be shown as a set
X
=
O
1. A
subimage
x
has been selected and the marked subimages in the figure belong to a
tolerance class that is represented by the selected subimage because their gray level
values are close to the gray value of subimage
x
, normalized between 0 and 1. Let
=
.
x
within the tolerance level
.
Example 8.4
The simple image in example 8.1
(Figure 8.1)
is considered here again. For each
given subimage, the corresponding tolerance class has been obtained by finding all
the subimages that have the average gray scale values within the tolerance range
(
1) of the average gray value of the given subimage. Figure 8.4 shows all the
tolerance classes which are calculated for
=
.
∀x ∈ O
after removing the redundant
classes. Note that
x
1
/
=
B,ε
=
x
7
/
=
B,ε
=
x
8
/
=
B,ε
,
x
2
/
=
B,ε
=
x
3
/
=
B,ε
=
x
5
/
=
B,ε
and
x
9
/
=
B,ε
and hence there are 4 tolerance classes in total. The set of all
tolerance classes is a covering of
O
and is shown with
O
/
=
B
=
x
6
/
=
B,ε
=
{x
/
=
B
| x ∈ O}
=
{x
4
/
=
B,
,x
5
/
=
B,
,x
8
/
=
B,
,x
9
/
=
B,
, }
.
Tolerance Matrix
In order to demonstrate a tolerance space and all its tolerance classes, a
tolerance
matrix
is defined here to show the tolerance relation between pairs of perceptual
objects. Each row in a tolerance matrix represents one tolerance class and each
column represents one perceptual object (subimage). Corresponding to each probe
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