Biomedical Engineering Reference
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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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