Image Processing Reference
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X∪X {
= U , it can be easily deduced that RX∪RX {
= U and RX∪RX {
= U . Using
these deductions, from (3.3) we get
ρ R (X) = 1− |RX|
|RX|
(3.10)
ρ R (X { ) = 1− |RX { |
|RX { | = 1− n−|RX|
(3.11)
n−|RX|
From (3.9), (3.10) and (3.11), we deduce that
|RX|
n−|RX|
!
ρ R (X { ) = ρ R (X) |RX|
1
C
n−|RX| =
(3.12)
We shall now separately consider three cases of (3.12), where we have 1 < C <∞, C = 1
and C =∞.
When we have 1 < C <∞, we get the relation |RX|
C−1
C
|RX| =
from (3.9).
Using this
relation in (3.12) we obtain
|RX|(
!
C
C−1
)
1
C
ρ R (X { ) =
(3.13)
n−|RX|
After some algebraic manipulations, we deduce
1
n
|RX| −1
!
1
C−1
ρ R (X { ) =
(3.14)
Note that, when 1 < C <∞, ρ R (X) takes value in the interval (0, 1). Therefore, in this case,
the value of|RX|could range from a positive infinitesimal quantity, say , to a maximum
value of n. Hence, we have
C−1
C ≤|RX|≤n C−1
(3.15)
C
Using (3.15) in (3.14), we get
nC− (C−1) ≤ρ R (X { )≤1
(3.16)
As 1 < C <∞, << 1 and usually n >> 1, we may write (3.16) as
0 < ρ R (X { )≤1 (3.17)
Thus, we may conclude that for a given non-zero and non-unity value of ρ R (X), ρ R (X { )
may take any value in the interval (0, 1].
When C = 1 or ρ R (X) takes a unity value,|RX|= 0 and the value of|RX|could range
from to a maximum value of n. Therefore, it is easily evident from (3.12) that ρ R (X { )
may take any value in the interval (0, 1] when ρ R (X) = 1.
Let us now consider the case when C =∞or ρ R (X) = 0. In such a case, the value of
|RX|could range from zero to a maximum value of n and|RX|=|RX|. As evident from
(3.12), when C =∞, irrespective of any other term, we get ρ R (X { ) = 0. This is obvious,
as a exactly definable set X should imply an exactly definable set X { .
Therefore, we find that the relation between ρ R (X) and ρ R (X { ) is such that, if one of
them is considered to take a non-zero value (that is, the underlying set is vaguely definable
or inexact), the value of the other, which would also be a non-zero quantity, can not be
uniquely specified. Therefore, there exist no unnecessary terms in the proposed classes of
entropy measures given in (3.7) and (3.8). However, from (3.10) and (3.11), it is easily
evident that ρ R (X) and ρ R (X { ) are positively correlated.
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