Biomedical Engineering Reference
In-Depth Information
S
n
L
n
L
=
(3.16)
k
=
0
L
−
1
n
k
U
n
L
=
(3.17)
k
=
S
+
1
P
L
(
X
k
) =
n
L
n
L
where k
=
1,2
...
S
(3.18)
P
U
(
X
k
) =
n
k
U
n
U
where k
=
S
+
1, S
+
2,
...
,L
−
1
(3.19)
3.3.1.4 Execution of GHE on Each Sub-Histogram
After decomposing the input histogram into two sub-histograms, each sub-histogram
undergoes independent GHE. Let n depicts the total number of the input image,
defined as Eq. (
3.20
). The Cumulative Density Function (CDF) for
X
L
and
X
U
are
represented as Eqs. (
3.21
) and (
3.22
), respectively.
(3.20)
n
=
n
L
+
n
U
S
c
L
(
X
k
) =
P
L
(
X
k
)
(3.21)
k
=
0
L
−
1
c
U
(
X
k
) =
P
U
(
X
k
)
(3.22)
k
=
S
+
1
By definition, it is expected that
c
L
(
X
L
−
1
) =
1
. Once the histogram partition
has completed, the conventional HE is implemented to each sub-histogram by uti-
lizing the defined CDF as a transform function, and subsequently it is combined to
form the output,
Y
, which can be expressed mathematically as follows:
Y
L
(
X
k
) =
X
0
+ (
X
S
+
X
0
)
C
L
(
X
k
)
where k
=
0,1,
...
,S
(3.23)
Y
U
(
X
k
) =
X
S
+
1
+ (
X
L
−
1
+
X
S
+
1
)
C
U
(
X
k
)
where k
=
S
+
1, S
+
2,
...
,L
−
1
(3.24)
(3.25)
Y
=
Y
L
+
Y
U
The histogram equalization operation has remapped the gray level of the sub-
image
X
L
over the range of
(
X
0
,
X
L
)
whereas
Y
U
has remapped the gray level of the
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