Image Processing Reference
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as
= B ={y∈O|y = B x 0 ∀x 0 ∈x /
x /
= B }.
(7.2)
Note, Defn. 3 covers O instead of partitioning it because an object can belong to more
than one class. As a result, Eq. 7.2 is called a tolerance class instead of an elementary set.
In addition, each pair of objects x,y in a tolerance class x /
= B must satisfy the condition
kφ(x)−φ(y)k≤ . Next, a quotient set for a given a tolerance relation is the set of all
tolerance classes and is defined as
O /
= B ={x /
= B |x∈O}.
Notice that the tolerance relation is a generalization of the perceptual indiscernibility rela-
tion given in Defn. 1 (obtained by setting = 0). As a result, Defn. 2 can be redefined with
respect to the tolerance relation = B,ε .
The following simple example highlights the need for a tolerance relation as well as demon-
strates the construction of tolerance classes from real data. Consider the 20 objects in
Table 7.1 where|φ(x i )|= 1. Letting = 0.1 gives the following tolerance classes:
X /
= B = {{x 1 ,x 8 ,x 10 ,x 11
},{x 1 ,x 9 ,x 10 ,x 11 ,x 14
},
{x 2 ,x 7 ,x 18 ,x 19
},
{x 3 ,x 12 ,x 17
},
{x 4 ,x 13 ,x 20
},{x 4 ,x 18
},
{x
,x
,x
,x
},{x
,x
,x
,x
},
5
6
15
16
5
6
15
20
{x
,x
,x
}}
6
13
20
Observe that each object in a tolerance class satisfies the conditionkφ B (x)−φ B (y)k≤ ,
and that almost all of the objects appear in more than one class. Moreover, there would be
twenty classes if the indiscernibility relation was used since there are no two objects with
matching descriptions.
TABLE 7.1
Tolerance Class Example
x i
φ(x) x i
φ(x) x i
φ(x) x i
φ(x)
x 1 .4518
x 16 .6079
x 2 .9166 x 7 .9246 x 12 .1910 x 17 .1869
x 3 .1398 x 8 .3537 x 13 .7476 x 18 .8489
x 4 .7972
x 6 .6943
x 11 .4002
x 19 .9170
x 5 .6281 x 10 .4523 x 15 .6289 x 20 .7143
x 9 .4722
x 14 .4990
A l 2 norm-based Nearness Measure (NM) is useful in discerning resemblances between
images (Henry and Peters, 2009a,b; Peters, 2009b,c, 2010; Peters and Ramanna, 2009), and
can be defined between two sets X and Y using Defn. 2 (Hassanien et al., 2009; Henry and
The relations was treated separately in the interest of clarity.

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