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 )
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 b , λ
= (1−λ a )(1−λ b )
11
12
21
22
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