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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