Image Processing Reference

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P4. Maximality: Both H
R
(A, B) and H
R
(A, B) attain their maximum value of unity

if and only if the roughness values A and B are unity. That is, we have H
R
(A, B)≤

H
R
(1, 1) = 1 and H
R
(A, B)≤H
R
(1, 1) = 1, where A, B∈[0, 1].

P5. Resolution: We have H
R
(A
∗
, B
∗
)≤H
R
(A, B) and H
R
(A
∗
, B
∗
)≤H
R
(A, B),

where A
∗
and B
∗
are respectively the sharpened version of A and B, that is,

A
∗
≤A and B
∗
≤B.

P6. Symmetry: It is evident that H
R
(A, B) = H
R
(B, A) and H
R
(A, B) = H
R
(B, A).

Hence H
R
(A, B) and H
R
(A, B) are symmetric about the line A = B.

P7. Monotonicity: The first-order partial and total derivatives of H
R
(A, B), when

A, B∈(0, 1], are

2
h
log
β
A

+

ln β
i

δH
R

δA

dH
R

dA

=−
1

1

=

β

2
h
log
β
B

+

ln β
i

δH
R

δB

dH
R

dB

=−
1

1

=

(3.20)

β

For the appropriate values of β in H
R
(A, B), where A, B∈(0, 1], we have

δH
R

δA

dH
R

dA
≥0 and

δH
R

δB

dH
R

dB
≥0

=

=

(3.21)

Since we have H
R
(A, B) = 0 when A = B = 0 and H
R
(A, B) > 0 when A, B∈

(0, 1], we may conclude from (3.20) and (3.21) that H
R
(A, B) is a monotonically

non-decreasing function. In a similar manner, for appropriate values of β in

H
R
(A, B), where A, B∈(0, 1], we have

2
h
β
(1−A)
−Aβ
(1−A)
ln β
i
≥0

δH
R

δA

dH
R

dA

1

=

=

2
h
β
(1−B)
−Bβ
(1−A)
ln β
i
≥0

δH
R

δB

dH
R

dB

1

=

=

(3.22)

We also have H
R
(A, B) = 0 when A = B = 0 and H
R
(A, B) > 0 when A, B∈

(0, 1], and hence we may conclude from (3.22) that H
R
(A, B) is a monotonically

non-decreasing function.

P8. Concavity: A two dimensional function f un(A, B) is concave on a two dimen-

sional interval < [a
min
, a
max
], [b
min
, b
max
] >, if for any four points a
1
, a
2

∈

[a
min
, a
max
] and b
1
, b
2

∈[b
min
, b
max
], and any λ
a
, λ
b
∈(0, 1)

f un(λ
a
a
1
+ (1−λ
a
)a
2
, λ
b
b
1
+ (1−λ
b
)b
2
)≥λ
11
f un(a
1
, b
1
) + λ
12
f un(a
1
, b
2
)

+λ
21
f un(a
2
, b
1
) + λ
22
f un(a
2
, b
2
)

(3.23)

where

λ

= λ
a
λ
b
, λ

= λ
a
(1−λ
b
), λ

= (1−λ
a
)λ
b
, λ

= (1−λ
a
)(1−λ
b
)

11

12

21

22

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