Image Processing Reference
In-Depth Information
Notice (O,
∼
=
B
) is an example of a tolerance space. The relation
∼
=
B
in Defn. 3 defines a
covering of O instead of partitioning O because an object can belong to more than one class.
Let A⊂
∼
=
B
. A is a preclass in
∼
=
B
, if∀x, y∈A, x
∼
=
B
y, i.e. kφ
B
(x)−φ
B
(y) k
2
≤
ε (Schroeder and Wright, 1992b). Of course, the idea of a preclass can be generalized to any
metric space. A maximal preclass of tolerance
∼
=
B,ε
is called a class. The particular form
of distance-based tolerance relation
∼
=
B,ε
used in this chapter works well for image analysis,
applied mathematics, engineering and biomedicine, where
∼
=
B,ε
is used to express inaccuracy
of measurement or what Poincare calls an indistinguishable difference in the intensity of a
sensation (Poincare, 1913). In other words, a set A⊂
∼
=
B
is a tolerance class if, and only if
A is a maximal preclass.
The following simple example highlights the need for a tolerance relation as well as demon-
strates the construction of tolerance classes from sample data. Let X
/
∼
=
B,ε
denote a covering
of X defined by
∼
=
B,ε
. Consider the 20 objects in Table 12.2 where|φ(x
i
)|= 1. Put ε = 0.1
and obtain the following tolerance classes:
X
/
∼
=
B
= {{x
, x
, x
, x
},{x
, x
, x
, x
, x
},
1
8
10
11
1
9
10
11
14
{x
, x
, x
, x
},
2
7
18
19
{x
, x
, x
},
3
12
17
{x
},
{x
5
, x
6
, x
15
, x
16
},{x
5
, x
6
, x
15
, x
20
},
{x
6
, x
13
, x
20
}}.
, x
, x
},{x
, x
4
13
20
4
18
Observe that each pair of objects x, y in a tolerance class satisfies the conditionkφ(x)−
φ(y)k
2
≤ε, and many of the objects appear in more than one class.
TABLE 12.2
Tolerance Class Example
x
i
φ(x)
x
i
φ(x)
x
i
φ(x)
x
i
φ(x)
x
1
.4518
x
6
.6943
x
11
.4002
x
16
.6079
x
.9166
x
.9246
x
.1910
x
.1869
2
7
12
17
x
3
.1398
x
8
.3537
x
13
.7476
x
18
.8489
x
.7972
x
.4722
x
.4990
x
.9170
4
9
14
19
x
5
.6281
x
10
.4523
x
15
.6289
x
20
.7143
Definition 4 Tolerance Near Sets (Peters, 2009)
LethO,
be
a set of probe functions representing image features. LetBcontains probe functions used
to measure features of subimages in X, Y⊂O. A set X is perceptually near a Y within
the perceptual systemhO,
F
ibe a perceptual system, where O is a set of images (sets of points) andB⊆
F
F
i(i.e., (X ./
F,ε
Y )) iff there are x∈X and y∈Y and there is
such that x
∼
=
B,ε
y.
B⊆
F
12.3
Resemblance Measures
Tolerance classes can be viewed as structural elements in representing an image. The moti-
vation for using tolerance classes in perceptual image analysis is the conjecture that visual
perception in the human perception is performed in the class level rather than pixel level.
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