Image Processing Reference
In-Depth Information
In the previous two subsections we have proposed two classes of entropy measures and
we have shown that the expressions for the proposed entropy measures do not have any
unnecessary terms. However, the base parameters β (see (3.7) and (3.8)) of the two classes
of entropy measures incur certain restrictions, so that the proposed entropies satisfy some
important properties. In this subsection, we shall discuss the restrictions regarding the base
parameters and then provide few properties of the proposed entropies.
Range of Values for the Base
β
The proposed classes of entropy measures H
R
and H
R
respectively given in (3.7) and (3.8)
must be consistent with the fact that maximum information (entropy) is available when the
uncertainty is maximum and the entropy is zero when there is no uncertainty. Note that, in
our case, maximum uncertainty represents maximum possible incompleteness of knowledge
about the universe. Therefore, maximum uncertainty occurs when both the roughness values
used in H
R
and H
R
equal unity and uncertainty is zero when both of them are zero. It can
be easily shown that in order to satisfy the aforesaid condition, the base β in H
R
must take
a finite value greater than or equal to e(≈2.7183) and the base β in H
R
must take a value
in the interval (1, e]. When β≥e in H
R
and 1 < β≤e in H
R
, the values taken by both H
R
and H
R
lie in the range [0, 1]. Note that, when β takes an appropriate value, the proposed
entropy measures attain the minimum value of zero only when ρ
R
(X) = ρ
R
(X
{
) = 0 and
the maximum value of unity only when ρ
R
(X) = ρ
R
(X
{
) = 1.
Properties
Here we present few properties of the proposed logarithmic and exponential classes of en-
tropy measures expressing H
R
and H
R
as functions of two parameters representing rough-
ness measures. We may respectively rewrite the expressions given in (3.7) and (3.8) in
parametric form as follows
2
h
A log
β
A
+ B log
β
B
i
(3.18)
H
R
(A, B) =−
1
β
β
Aβ
1−A
+ Bβ
1−B
1
2
H
R
(A, B) =
(3.19)
where the parameters A (∈[0, 1]) and B (∈[0, 1]) represent the roughness values ρ
R
(X)
and ρ
R
(X
{
), respectively. Considering the convention 0 log
β
0 = 0, let us now discuss the
properties of H
R
(A, B) and H
R
(A, B) along the lines of (Ebanks, 1983).
P1. Nonnegativity: We have H
R
(A, B)≥0 and H
R
(A, B)≥0 with equality in both
the cases if and only if A = 0 and B = 0.
P2. Continuity: As all first-order partial and total derivatives of H
R
(A, B) exists
for A, B∈[0, 1], H
R
(A, B) is a continuous function of A and B. On the other
hand, all first-order partial and total derivatives of H
R
(A, B) exists only for
A, B∈(0, 1]. However, it can be easily shown that lim
A→0, B→0
H
R
(A, B) = 0
(H
R
(A, B) tends to zero when A and B tend to zero) using L'hospitals rule and
we have 0 log
β
0 = 0. Therefore, H
R
(A, B) is a continuous function of A and B,
where A, B∈[0, 1].
P3. Sharpness: It is evident that both H
R
(A, B) and H
R
(A, B) equal zero if and
only if the roughness values A and B equal zero, that is, A and B are 'sharp'.
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