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section, we will use this property to estimate the transmission
and the atmospheric light.
Note that, we neglect the sky regions because the dark
channel of a haze-free image may has high intensity here.
Fortunately, we can gracefully handle the sky regions by
using the haze imaging Equation (1) and our prior together.
It is not necessary to cut out the sky regions explicitly. We
discuss this issue in Section 4.1.
Our dark channel prior is partially inspired by the well
known dark-object subtraction technique widely used in
multi-spectral remote sensing systems. In [1], spatially homogeneous
haze is removed by subtracting a constant value
corresponding to the darkest object in the scene. Here, we
generalize this idea and proposed a novel prior for natural
image dehazing.
4. Haze Removal Using Dark Channel Prior
4.1. Estimating the Transmission
Here, we first assume that the atmospheric light A is
given. We will present an automatic method to estimate
the atmospheric light in Section 4.4. We further assume
that the transmission in a local patch Ω(x) is constant. We
denote the patch’s transmission as ˜t(x). Taking the min operation
in the local patch on the haze imaging Equation (1),
we have:
min
y∈Ω(x)
(Ic(y)) = ˜t(x) min
y∈Ω(x)
(Jc(y))+(1−˜t(x))Ac. (6)
Notice that the min operation is performed on three color
channels independently. This equation is equivalent to:
min
y∈Ω(x)
( Ic(y)
Ac ) = ˜t(x) min
y∈Ω(x)
(Jc(y)
Ac ) + (1 − ˜t(x)). (7)
Then, we take the min operation among three color channels
on the above equation and obtain:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) = ˜t(x)min
c
( min
y∈Ω(x)
(Jc(y)
Ac ))
+(1 − ˜t(x)). (8)
According to the dark channel prior, the dark channel
Jdark of the haze-free radiance J tend to be zero:
Jdark(x) = min
c
( min
y∈Ω(x)
(Jc(y))) = 0. (9)
As Ac is always positive, this leads to:
min
c
( min
y∈Ω(x)
(Jc(y)
Ac )) = 0 (10)
Putting Equation (10) into Equation (8), we can estimate the
transmission ˜t simply by:
˜t(x) = 1 − min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (11)
In fact, minc(miny∈Ω(x)( Ic(y)
Ac )) is the dark channel of the
normalized haze image Ic(y)
Ac . It directly provides the estimation
of the transmission.
As we mentioned before, the dark channel prior is not a
good prior for the sky regions. Fortunately, the color of the
sky is usually very similar to the atmospheric light A in a
haze image and we have:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) → 1, and ˜t(x) → 0,
in the sky regions. Since the sky is at infinite and tends to
has zero transmission, the Equation (11) gracefully handles
both sky regions and non-sky regions. We do not need to
separate the sky regions beforehand.
In practice, even in clear days the atmosphere is not absolutely
free of any particle. So, the haze still exists when
we look at distant objects. Moreover, the presence of haze
is a fundamental cue for human to perceive depth [3, 13].
This phenomenon is called aerial perspective. If we remove
the haze thoroughly, the image may seem unnatural
and the feeling of depth may be lost. So we can optionally
keep a very small amount of haze for the distant objects by
introducing a constant parameter ω (0<ω≤1) into Equation
(11):
˜t(x) = 1 − ω min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (12)
The nice property of this modification is that we adaptively
keep more haze for the distant objects. The value of ω is
application-based. We fix it to 0.95 for all results reported
in this paper.
Figure 5(b) is the estimated transmission map from an
input haze image (Figure 5(a)) using the patch size 15 ×
15. It is reasonably good but contains some block effects
since the transmission is not always constant in a patch. In
the next subsection, we refine this map using a soft matting
method.
4.2. Soft Matting
We notice that the haze imaging Equation (1) has a similar
form with the image matting equation. A transmission
map is exactly an alpha map. Therefore, we apply a soft
matting algorithm [7] to refine the transmission.
Denote the refined transmission map by t(x). Rewriting
t(x) and ˜t(x) in their vector form as t and˜t, we minimize
the following cost function:
E(t) = tT Lt + λ(t −˜t)T (t −˜t). (13)
where L is the Matting Laplacian matrix proposed by Levin
[7], and λ is a regularization parameter. The first term is the
smooth term and the second term is the data term.
1959
section, we will use this property to estimate the transmission
and the atmospheric light.
Note that, we neglect the sky regions because the dark
channel of a haze-free image may has high intensity here.
Fortunately, we can gracefully handle the sky regions by
using the haze imaging Equation (1) and our prior together.
It is not necessary to cut out the sky regions explicitly. We
discuss this issue in Section 4.1.
Our dark channel prior is partially inspired by the well
known dark-object subtraction technique widely used in
multi-spectral remote sensing systems. In [1], spatially homogeneous
haze is removed by subtracting a constant value
corresponding to the darkest object in the scene. Here, we
generalize this idea and proposed a novel prior for natural
image dehazing.
4. Haze Removal Using Dark Channel Prior
4.1. Estimating the Transmission
Here, we first assume that the atmospheric light A is
given. We will present an automatic method to estimate
the atmospheric light in Section 4.4. We further assume
that the transmission in a local patch Ω(x) is constant. We
denote the patch’s transmission as ˜t(x). Taking the min operation
in the local patch on the haze imaging Equation (1),
we have:
min
y∈Ω(x)
(Ic(y)) = ˜t(x) min
y∈Ω(x)
(Jc(y))+(1−˜t(x))Ac. (6)
Notice that the min operation is performed on three color
channels independently. This equation is equivalent to:
min
y∈Ω(x)
( Ic(y)
Ac ) = ˜t(x) min
y∈Ω(x)
(Jc(y)
Ac ) + (1 − ˜t(x)). (7)
Then, we take the min operation among three color channels
on the above equation and obtain:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) = ˜t(x)min
c
( min
y∈Ω(x)
(Jc(y)
Ac ))
+(1 − ˜t(x)). (8)
According to the dark channel prior, the dark channel
Jdark of the haze-free radiance J tend to be zero:
Jdark(x) = min
c
( min
y∈Ω(x)
(Jc(y))) = 0. (9)
As Ac is always positive, this leads to:
min
c
( min
y∈Ω(x)
(Jc(y)
Ac )) = 0 (10)
Putting Equation (10) into Equation (8), we can estimate the
transmission ˜t simply by:
˜t(x) = 1 − min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (11)
In fact, minc(miny∈Ω(x)( Ic(y)
Ac )) is the dark channel of the
normalized haze image Ic(y)
Ac . It directly provides the estimation
of the transmission.
As we mentioned before, the dark channel prior is not a
good prior for the sky regions. Fortunately, the color of the
sky is usually very similar to the atmospheric light A in a
haze image and we have:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) → 1, and ˜t(x) → 0,
in the sky regions. Since the sky is at infinite and tends to
has zero transmission, the Equation (11) gracefully handles
both sky regions and non-sky regions. We do not need to
separate the sky regions beforehand.
In practice, even in clear days the atmosphere is not absolutely
free of any particle. So, the haze still exists when
we look at distant objects. Moreover, the presence of haze
is a fundamental cue for human to perceive depth [3, 13].
This phenomenon is called aerial perspective. If we remove
the haze thoroughly, the image may seem unnatural
and the feeling of depth may be lost. So we can optionally
keep a very small amount of haze for the distant objects by
introducing a constant parameter ω (0<ω≤1) into Equation
(11):
˜t(x) = 1 − ω min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (12)
The nice property of this modification is that we adaptively
keep more haze for the distant objects. The value of ω is
application-based. We fix it to 0.95 for all results reported
in this paper.
Figure 5(b) is the estimated transmission map from an
input haze image (Figure 5(a)) using the patch size 15 ×
15. It is reasonably good but contains some block effects
since the transmission is not always constant in a patch. In
the next subsection, we refine this map using a soft matting
method.
4.2. Soft Matting
We notice that the haze imaging Equation (1) has a similar
form with the image matting equation. A transmission
map is exactly an alpha map. Therefore, we apply a soft
matting algorithm [7] to refine the transmission.
Denote the refined transmission map by t(x). Rewriting
t(x) and ˜t(x) in their vector form as t and˜t, we minimize
the following cost function:
E(t) = tT Lt + λ(t −˜t)T (t −˜t). (13)
where L is the Matting Laplacian matrix proposed by Levin
[7], and λ is a regularization parameter. The first term is the
smooth term and the second term is the data term.
1959
and the atmospheric light.
Note that, we neglect the sky regions because the dark
channel of a haze-free image may has high intensity here.
Fortunately, we can gracefully handle the sky regions by
using the haze imaging Equation (1) and our prior together.
It is not necessary to cut out the sky regions explicitly. We
discuss this issue in Section 4.1.
Our dark channel prior is partially inspired by the well
known dark-object subtraction technique widely used in
multi-spectral remote sensing systems. In [1], spatially homogeneous
haze is removed by subtracting a constant value
corresponding to the darkest object in the scene. Here, we
generalize this idea and proposed a novel prior for natural
image dehazing.
4. Haze Removal Using Dark Channel Prior
4.1. Estimating the Transmission
Here, we first assume that the atmospheric light A is
given. We will present an automatic method to estimate
the atmospheric light in Section 4.4. We further assume
that the transmission in a local patch Ω(x) is constant. We
denote the patch’s transmission as ˜t(x). Taking the min operation
in the local patch on the haze imaging Equation (1),
we have:
min
y∈Ω(x)
(Ic(y)) = ˜t(x) min
y∈Ω(x)
(Jc(y))+(1−˜t(x))Ac. (6)
Notice that the min operation is performed on three color
channels independently. This equation is equivalent to:
min
y∈Ω(x)
( Ic(y)
Ac ) = ˜t(x) min
y∈Ω(x)
(Jc(y)
Ac ) + (1 − ˜t(x)). (7)
Then, we take the min operation among three color channels
on the above equation and obtain:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) = ˜t(x)min
c
( min
y∈Ω(x)
(Jc(y)
Ac ))
+(1 − ˜t(x)). (8)
According to the dark channel prior, the dark channel
Jdark of the haze-free radiance J tend to be zero:
Jdark(x) = min
c
( min
y∈Ω(x)
(Jc(y))) = 0. (9)
As Ac is always positive, this leads to:
min
c
( min
y∈Ω(x)
(Jc(y)
Ac )) = 0 (10)
Putting Equation (10) into Equation (8), we can estimate the
transmission ˜t simply by:
˜t(x) = 1 − min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (11)
In fact, minc(miny∈Ω(x)( Ic(y)
Ac )) is the dark channel of the
normalized haze image Ic(y)
Ac . It directly provides the estimation
of the transmission.
As we mentioned before, the dark channel prior is not a
good prior for the sky regions. Fortunately, the color of the
sky is usually very similar to the atmospheric light A in a
haze image and we have:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) → 1, and ˜t(x) → 0,
in the sky regions. Since the sky is at infinite and tends to
has zero transmission, the Equation (11) gracefully handles
both sky regions and non-sky regions. We do not need to
separate the sky regions beforehand.
In practice, even in clear days the atmosphere is not absolutely
free of any particle. So, the haze still exists when
we look at distant objects. Moreover, the presence of haze
is a fundamental cue for human to perceive depth [3, 13].
This phenomenon is called aerial perspective. If we remove
the haze thoroughly, the image may seem unnatural
and the feeling of depth may be lost. So we can optionally
keep a very small amount of haze for the distant objects by
introducing a constant parameter ω (0<ω≤1) into Equation
(11):
˜t(x) = 1 − ω min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (12)
The nice property of this modification is that we adaptively
keep more haze for the distant objects. The value of ω is
application-based. We fix it to 0.95 for all results reported
in this paper.
Figure 5(b) is the estimated transmission map from an
input haze image (Figure 5(a)) using the patch size 15 ×
15. It is reasonably good but contains some block effects
since the transmission is not always constant in a patch. In
the next subsection, we refine this map using a soft matting
method.
4.2. Soft Matting
We notice that the haze imaging Equation (1) has a similar
form with the image matting equation. A transmission
map is exactly an alpha map. Therefore, we apply a soft
matting algorithm [7] to refine the transmission.
Denote the refined transmission map by t(x). Rewriting
t(x) and ˜t(x) in their vector form as t and˜t, we minimize
the following cost function:
E(t) = tT Lt + λ(t −˜t)T (t −˜t). (13)
where L is the Matting Laplacian matrix proposed by Levin
[7], and λ is a regularization parameter. The first term is the
smooth term and the second term is the data term.
1959
section, we will use this property to estimate the transmission
and the atmospheric light.
Note that, we neglect the sky regions because the dark
channel of a haze-free image may has high intensity here.
Fortunately, we can gracefully handle the sky regions by
using the haze imaging Equation (1) and our prior together.
It is not necessary to cut out the sky regions explicitly. We
discuss this issue in Section 4.1.
Our dark channel prior is partially inspired by the well
known dark-object subtraction technique widely used in
multi-spectral remote sensing systems. In [1], spatially homogeneous
haze is removed by subtracting a constant value
corresponding to the darkest object in the scene. Here, we
generalize this idea and proposed a novel prior for natural
image dehazing.
4. Haze Removal Using Dark Channel Prior
4.1. Estimating the Transmission
Here, we first assume that the atmospheric light A is
given. We will present an automatic method to estimate
the atmospheric light in Section 4.4. We further assume
that the transmission in a local patch Ω(x) is constant. We
denote the patch’s transmission as ˜t(x). Taking the min operation
in the local patch on the haze imaging Equation (1),
we have:
min
y∈Ω(x)
(Ic(y)) = ˜t(x) min
y∈Ω(x)
(Jc(y))+(1−˜t(x))Ac. (6)
Notice that the min operation is performed on three color
channels independently. This equation is equivalent to:
min
y∈Ω(x)
( Ic(y)
Ac ) = ˜t(x) min
y∈Ω(x)
(Jc(y)
Ac ) + (1 − ˜t(x)). (7)
Then, we take the min operation among three color channels
on the above equation and obtain:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) = ˜t(x)min
c
( min
y∈Ω(x)
(Jc(y)
Ac ))
+(1 − ˜t(x)). (8)
According to the dark channel prior, the dark channel
Jdark of the haze-free radiance J tend to be zero:
Jdark(x) = min
c
( min
y∈Ω(x)
(Jc(y))) = 0. (9)
As Ac is always positive, this leads to:
min
c
( min
y∈Ω(x)
(Jc(y)
Ac )) = 0 (10)
Putting Equation (10) into Equation (8), we can estimate the
transmission ˜t simply by:
˜t(x) = 1 − min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (11)
In fact, minc(miny∈Ω(x)( Ic(y)
Ac )) is the dark channel of the
normalized haze image Ic(y)
Ac . It directly provides the estimation
of the transmission.
As we mentioned before, the dark channel prior is not a
good prior for the sky regions. Fortunately, the color of the
sky is usually very similar to the atmospheric light A in a
haze image and we have:
min
c
( min
y∈Ω(x)
( Ic(y)
Ac )) → 1, and ˜t(x) → 0,
in the sky regions. Since the sky is at infinite and tends to
has zero transmission, the Equation (11) gracefully handles
both sky regions and non-sky regions. We do not need to
separate the sky regions beforehand.
In practice, even in clear days the atmosphere is not absolutely
free of any particle. So, the haze still exists when
we look at distant objects. Moreover, the presence of haze
is a fundamental cue for human to perceive depth [3, 13].
This phenomenon is called aerial perspective. If we remove
the haze thoroughly, the image may seem unnatural
and the feeling of depth may be lost. So we can optionally
keep a very small amount of haze for the distant objects by
introducing a constant parameter ω (0<ω≤1) into Equation
(11):
˜t(x) = 1 − ω min
c
( min
y∈Ω(x)
( Ic(y)
Ac )). (12)
The nice property of this modification is that we adaptively
keep more haze for the distant objects. The value of ω is
application-based. We fix it to 0.95 for all results reported
in this paper.
Figure 5(b) is the estimated transmission map from an
input haze image (Figure 5(a)) using the patch size 15 ×
15. It is reasonably good but contains some block effects
since the transmission is not always constant in a patch. In
the next subsection, we refine this map using a soft matting
method.
4.2. Soft Matting
We notice that the haze imaging Equation (1) has a similar
form with the image matting equation. A transmission
map is exactly an alpha map. Therefore, we apply a soft
matting algorithm [7] to refine the transmission.
Denote the refined transmission map by t(x). Rewriting
t(x) and ˜t(x) in their vector form as t and˜t, we minimize
the following cost function:
E(t) = tT Lt + λ(t −˜t)T (t −˜t). (13)
where L is the Matting Laplacian matrix proposed by Levin
[7], and λ is a regularization parameter. The first term is the
smooth term and the second term is the data term.
1959
