Image Processing Reference
In-Depth Information
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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