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exponentiation to best match a canonically illuminated image
g
O
(
x
,
y
)
, under normal
lighting conditions. The GIC corrected image
g
(
x
,
y
)
is computed by transforming
the input image pixel by pixel over its position
(
x
,
y
)
with an optimal Gamma
ʳ
∗
,
coefficient
g
(
)
,
ʳ
∗
)
x
,
y
)=
G
(
g
(
x
,
y
(7.26)
1
/
ʳ
∗
=
e
·
g
(
x
,
y
)
(7.27)
ʳ
∗
is computed as:
where
e
is a gray stretch parameter, and
ʳ
∗
=
x
,
y
[
G
(
g
(
x
,
y
)
,
ʳ
)
−
g
O
(
x
,
y
)]
2
arg min
(7.28)
ʳ
∗
is approximated by the golden section search with parabolic interpolation
[
357
]. GIC can enhance the local dynamic range of the face in dark or shadowed
regions, compress values of pixels in bright regions, and compensate for the global
brightness changes of an image.
Intensity gradients such as shading effects are removed through a DoG filter, a
popular method to obtain the resulting bandpass behavior for images. The selected
values of smaller or inner Gaussians are typically quite narrow so the detailed spatial
information in high frequency is kept, while the outer ones might have more contents
for the low frequency range.
The main motivation of the application of LHM after GIC and DoG is to take
into account histogram distribution over local windows and integrate it to global
histogram distribution. To get the LHM transfer function, the histogram distribution
of the input image and its local window are calculated first. The levels of the input
image from previous processing are equalized by
k
j
=
0
n
j
n
,
s
k
=
T
(
r
k
)=
k
=
0
,
1
,...,
L
−
1
(7.29)
where
s
k
is the transformation of the pixel value
r
k
in the original image,
n
is the total
number of pixels,
n
j
is the number of pixels with gray level
r
j
, and
L
is the number of
discrete gray levels. The histogram distribution function
G
(
z
)
from the local window
is obtained by:
z
j
=
0
p
z
(
z
j
)
≈
k
i
=
0
n
i
n
=
G
(
z
j
)=
s
k
(7.30)
where
p
z
(
represents the specified desirable probability density function for the
output image in a local window, and follows the transform
G
z
)
(
z
)=
T
(
r
)
.The
G
−
1
inverse transformation function,
z
is then applied to the levels obtained in
Eq. (
7.29
). The new, processed version of the original image consists of gray levels
characterized by the specified density
p
z
(
=
(
s
)
z
)
which is normalized using Eq. (
7.31
),
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