Single Image Dehazing with V-transform and Dark Channel Prior

Xiaochun WANG; Xiangdong SUN; Ruixia SONG

doi:10.21078/JSSI-2020-185-10

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Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (2) : 185-194. DOI: 10.21078/JSSI-2020-185-10

Single Image Dehazing with V-transform and Dark Channel Prior

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Abstract

Single image dehazing algorithm based on the dark channel prior may cause block effect and color distortion. To improve these limitations, this paper proposes a single image dehazing algorithm based on the V-transform and the dark channel prior, in which a hazy RGB image is converted into the HSI color space, and each component H, I and S is processed separately. The hue component H remains unchanged, the saturation component S is stretched after being denoised by a median filter. In the procession of intensity component, a quad-tree algorithm is presented to estimate the atmospheric light, the dark channel prior and the V-transform are used to estimate the transmission map. To reduce the computational complexity, the intensity component I is decomposed by the V-transform first, coarse transmission map is then estimated by applying the dark channel prior on the low frequency reconstruction image, and the guided filter is finally employed to refine the coarse transmission map. For images with sky regions, the haze removal effectiveness can be greatly improved by just increasing the minimum value of the transmission map. The proposed algorithm has low time complexity and performs well on a wide variety of images. The recovered images have more nature color and less color distortion compared with some state-of-the-art methods.

Key words

dark channel prior / image haze removal / HSI color space / quad-tree / V-transform

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Xiaochun WANG , Xiangdong SUN , Ruixia SONG. Single Image Dehazing with V-transform and Dark Channel Prior. Journal of Systems Science and Information, 2020, 8(2): 185-194 https://doi.org/10.21078/JSSI-2020-185-10

1 Introduction

Haze is an atmospheric obscurity caused by minute suspended solid particles such as dust, mist and gaseous molecules. Under haze weather conditions, the irradiance from the scenes is scattered or absorbed by the constituents of the atmosphere, and the direct transmission of light is attenuated when photographing. The attenuated direct transmission makes the intensity from the scenes become weaker, and the airlight causes the appearance of the scene to be washed out. Many computer vision system and image processing algorithms suffer from the visibility degradation caused by haze, so image haze removal is necessary before performing some high-level scene understanding tasks^[1].

To solve the issue of image degradation caused by haze, various kinds of haze removal approaches are proposed. These approaches can be broadly classified into two classes: image enhancement and image restoration based methods. The former, including Retinex and histogram equalization algorithm, realizes haze removal by improving the image visibility using the contrast enhancement techniques^{[2, 3]}. These methods can visually improve the quality of degraded images, but tend to produce oversaturation and halo effects at depth discontinuities. While the later realizes image defogging by inversely solving some physical models to recover the degraded images. These methods are physically effective and can produce pleased results with less information loss. Therefore, image restoration based haze removal method has received wide attention^{[4, 5]}. However, the used physical models are usually related to image depth information whose estimation is under constrained. In order to estimate the unknown depth information, methods using multiple images or additional information were proposed in the earlier studies. These methods are usually able to achieve impressive results, but obtaining multiple images of the same scene is impractical in real time applications. Due to this, dehazing methods with only a single reference image have drawn a lot of attention in recent years.

The success of most physical model based single image dehazing methods lies in effective prior information^[6-8]. In 2009, He, et al.^[9] proposed the dark channel prior (DCP) and used it to execute image haze removal. In their method, the transmission map is roughly estimated based on DCP, and refined by soft matting. However, soft matting is time consume and makes this technique unsuitable for real time applications. To solve this problem, Gibson, et al.^[10] proposed a fast single image dehazing method by replacing the soft matting with median filter, and HE et al. use the guided filtering^[11] instead of the soft matting algorithm. Moreover, DCP tends to over-estimate the haze thickness in the sky images due to the failure of DCP in sky or bright regions. To address this problem, many improved methods have been proposed. Huang^[12] presented a DCP based method in the HSI color space, in which the dark channel value is adjusted adaptively according to the difference between maximum and minimum values among R, G and B channels. Some sky region segmentation and DCP based dehazing algorithms have also been proposed^{[13, 14]}, these algorithms estimate the transmission map on sky region and non-sky region separately, based on the segmentation results.

In this paper, a new image dehazing algorithm in HSI color space based on DCP and the V-transform is proposed. The input RGB image is firstly converted into the HSI color space, and then each component H, S and I is processed separately. Haze removal is mainly performed on the intensity component I using the V-transform and DCP. A quad-tree algorithm is presented to estimate the atmospheric light, and the reconstructed intensity image by the low frequency coefficients of V-transform is used to estimate the transmission map. For images with a large area of sky, block effect and color distortion can be greatly improved by just increasing the minimum value of the transmission map in our method.

2 Related Works

In computer vision, the atmospheric scattering model is widely used to describe the formation of a haze image:

\begin{array}{rcl} I (x) = t (x) J (x) + (1 - t (x)) A, \end{array}

(1)

where

x

denotes the pixel location,

I (x)

is the observed hazy image,

J (x)

is the clear scene radiance (i.e., haze-free image),

t (x)

is the medium transmission map and

A

is the global atmospheric light. There are three unknowns in Equation (1):

J (x)

t (x)

and

A

. The goal of haze removal is to recover the real scene

J (x)

by giving

A

and

t (x)

a reasonable estimation.

2.1 Dark Channel Prior

In atmospheric scattering model Equation (1), the color channel

c \in R, G, B

of a haze image can be expressed as:

I^{c} (x) = t (x) J^{c} (x) + (1 - t (x)) A .

Dividing both sides of above equation by

A

, it can be rewritten as:

\begin{array}{rcl} \frac{I^{c} (x)}{A} = t (x) \frac{J^{c} (x)}{A} + 1 - t (x) . \end{array}

(2)

DCP has been devised based on the observation: Most of the non-sky local patches in haze-free outdoor images contain some pixels with very low intensities in at least one color channel. The dark channel

J^{d a r k}

of the local patch

Ω (x)

centred at

x

is defined as:

J^{d a r k} (x) = min_{y \in Ω (x)} (min_{c} J^{c} (y)) .

J

is a haze-free outdoor image, and

Ω (x)

doesn't contain sky region, then

J^{d a r k} (x) = min_{y \in Ω (x)} (min_{c} J^{c} (y)) \approx 0.

Performing minimum operation on both side of Equation (2), and assuming the medium transmission in local patch

Ω (x)

to be constant, which is denoted as

\tilde{t} (x)

, we have

\begin{array}{rcl} min_{y \in Ω (x)} [min_{c} \frac{I^{c} (y)}{A}] = \tilde{t} (x) min_{y \in Ω (x)} [min_{c} \frac{J^{c} (y)}{A}] + 1 - \tilde{t} (x) . \end{array}

(3)

According to DCP, we have

min_{y \in Ω (x)} [min_{c} \frac{J^{c} (y)}{A}] \approx 0

, so the coarse medium transmission can be represented as:

\begin{array}{rcl} \tilde{t} (x) = 1 - min_{y \in Ω (x)} [min_{c} \frac{I^{c} (y)}{A}] . \end{array}

(4)

Note that even in clear days there are still some particles in the atmosphere, and the impact of haze on distant objects can be seen clearly. Therefore, retaining a certain degree of haze when computing the medium transmission makes the dehazing results look more realistic^[9]. For this reason, a constant parameter

ω (0 < ω < 1)

is introduced into Equation (4). The transmission is then estimated as:

\begin{array}{rcl} \tilde{t} (x) = 1 - ω min_{y \in Ω (x)} (min_{c} \frac{I^{c} (y)}{A}) . \end{array}

(5)

The value of

ω

is usually chosen as 0.9.

In most of the single image dehazing methods, the atmospheric light is usually estimated from the most haze-opaque pixels^[7]. In the classical dark channel method, atmospheric light

A

is generally estimated as the highest intensity among the top

0.1 %

brightest pixels in the dark channel. Considering too small transmission map

t (x)

may result in too large clear scene radiance

J

, a lower bound

t_{0}

for the transmission map

t (x)

is used. When

\tilde{t}

is less than the value of

t_{0}

, set

\tilde{t} (x) = t_{0}

(generally

t_{0} = 0.1

). Once the transmission map and the atmospheric light are estimated, the clear scene radiance can be recovered as:

\begin{array}{rcl} J (x) = \frac{I (x) - A}{max (\tilde{t} (x), t_{0})} + A . \end{array}

(6)

2.2 Orthogonal V-transform

The V-system of degree

k

^[15] is a complete orthogonal function system in

L^{2} [0, 1]

, which consists of infinitely many

k

th order piecewise polynomials. It is a class of orthogonal multiwavelets and has advantage in expressing object features^[16]. For simplicity, we only present the base functions of the V-system of degree one here.

The base functions of the V-system of degree one are piecewise linear functions. They are divided into different groups and classes. The first group is composed of the first two orthogonal Legendre polynomials on interval

[0, 1]

. Their expressions are given as

\begin{array}{rcl} V_{1}^{1} (x) = 1, V_{1}^{2} (x) = \sqrt{3} (1 - 2 x) . \end{array}

(7)

Two orthogonal generators forms the second group. They are defined as

\begin{aligned} V_{2}^{1} (x) = {\begin{cases} \sqrt{3} (1 - 4 x), & 0 \leq x < \frac{1}{2}, \\ \sqrt{3} (4 x - 3), & \frac{1}{2} < x \leq 1 . \end{cases} \\ V_{2}^{2} (x) = {\begin{cases} 1 - 6 x, & 0 \leq x < \frac{1}{2}, \\ 5 - 6 x, & \frac{1}{2} < x \leq 1 . \end{cases} \end{aligned}

(8)

Each generator produces the base functions in the

n

th (

n \geq 3

) group in a class.

When

n \geq 3

, the

n

th group is divided into two classes, each class comprises of

2^{n - 2}

functions obtained by squeezing, translating and duplicating one of generators in the second group. The

j

th function of the

i

th class, denoted by

V_{n}^{i, j} (x)

, is defined as

\begin{matrix} V_{n}^{i, j} (x) = {\begin{array}{cc} \sqrt{2^{n - 2}} V_{2}^{i} [2^{n - 2} (x - \frac{j - 1}{2^{n - 2}})], & x \in (\frac{j - 1}{2^{n - 2}}, \frac{j}{2^{n - 2}}), \\ 0, & others, \end{array} \\ i = 1, 2; j = 1, 2, \dots, 2^{n - 1}; n = 3, 4, 5, \dots . \end{matrix}

(9)

Note that many continuous wavelets are directly defined in the frequency domain, they do not have analytical expressions, but the V-wavelet has, which may bring us great convenience in using the V-system. For specific applications, once the V-system is discretized, the orthogonal V-transform matrix

U

can be obtained directly, and the V-transform can be defined accordingly. For a given image

F = (f_{i j})

, its V-transform can be mathematically represented as

Λ = U F U^{T}

, the matrix

Λ

becomes sparser gradually from the upper left corner (low frequency) to the lower right. The corresponding inverse transform is

F = U^{T} Λ U

. The approximate image can be obtained only using the low frequency coefficient in matrix

Λ

3 Proposed Algorithm

In this paper, we propose a new haze removal algorithm based on the V-system, DCP and the HSI color space. This algorithm firstly converts the RGB images into the HSI color space, and then performs image dehazing by processing components H, S and I separately. Usually, images are represented in RGB, but this color model is rarely used for computer vision tasks. The main reason is that R, G and B components in RGB model are all highly correlated to the luminance, which will cause color distortion during image processing. Unlike RGB, the HIS color space separates the image intensity from the color information, making it suitable for many of the grayscale techniques. Use of the HSI color space helps to reduce the complexity and maintain the color fidelity.

3.1 Hue Component Processing

Hue is the most basic attribute of color and its change will result in color artifacts. In addition, the Hue channel is invariant to haze degradation. So, we keep it unchanged in the procession of haze removal.

3.2 Saturation Component Processing

Presence of haze will cause saturation component decrease. To restore hazy image, its saturation component should be increased. To this end, the S component of a hazy image is first performed denoising using the median filtering, then adjusted by a scale factor (3.5 is its experimental value in this paper) which is set manually. The reason why we denoise the saturation component S first is to avoid image oversaturation caused by direct stretching. A large number of experiments show this method can make the result image more real and natural.

3.3 Intensity Component Processing

Intensity is the key component in HSI because human eye is more sensitive to changes in luminance than changes in color. The main step of image dehzing in the HSI color space is alleviating the effect of additional airlight, which is superimposed on the intensity channel I acting as an additional radiation. In this paper, the intensity component is processed using the guided filter, DCP and the V-transform. The procession includes three main steps: Atmospheric light estimation, transmission map estimation and a haze-free image restoration.

1) Atmospheric light estimation

Atmospheric light is the ambient light in atmosphere, which is often taken as the highest intensity value in the input image. However, the pixels with the highest intensity may be white objects in the scene. In order to avoid the influence of white objects and estimate background light reliably, we use the quad-tree algorithm to calculate the atmospheric light

A

. The component

I

is first performed the V-transform and reconstructed with the low frequencies to remove some high frequency noise. The reconstructed component

I

is equally divided into four parts, then select the sub-region with the highest average pixel value and divide it further into four smaller regions, repeat above procession until the size of the selected region is smaller than a pre-specified threshold (size

40 \times 40

is adopted here). The average pixel value of the final rectangular region is taken as the atmospheric light

A

2) Transmission map estimation

Transmission map dependents on the distance from the objects to the camera and is related to the light reaching the camera sensor. Transmission estimation directly affects the quality of restored image. Transmission map with block effect will lead to Halo artifacts and transmission map with loss in edge and texture details will produce residual fog at the depth of the scene. To reduce time complexity and get more reliable transmission map, we first apply the V-transform on the intensity component

I

, then use the approximation image which is reconstructed by its low frequencies to estimate the transmission map. Denote the low frequency reconstructed image as

V (x)

, the coarse transmission map can be formulated as

\begin{array}{rcl} \tilde{t} (x) = 1 - ω \frac{{min}_{x \in Ω (x)} V (x)}{A}, \end{array}

(10)

where

x

denotes the pixel location, and

ω (0 < ω < 1)

is a constant parameter as in Equation (5). The smoothness of the coarse transmission still need to be refined, so guided filter is further employed to optimize the coarse transmission map.

3) Haze-free intensity I restoration

The dehazed intensity component

I

is finally obtained by substituting the atmospheric light

A

and the transmission mapinto Equation (6). Considering that the restored image obtained from the DCP based method has the problem of low brightness, the obtained haze-free intensity is further multiplied by a scale factor

1.3

. A large number of experiments show that the streching makes the dehazed image more pleased. For images containing large area of sky, we increase the value of

t_{0}

in Equation (6) from

0.1

0.5

. Experimental results show that increasing

t_{0}

can effectively remove haze and avoid color distortion. The flow chart of this algorithm is depicted in Figure 1.

Figure 1 Flow chart of the proposed algorithm

Full size|PPT slide

4 Experimental Results and Analysis

Experiments on six typical hazy images are conducted to evaluate our algorithm. They are performed on a Windows 7 PC with a 2.50 GHz Intel Core i5 processor and 12 GB RAM, running MATLABR2016a.

4.1 Qualitative Comparison

To demonstrate the improvements achieved by the proposed method, we compare it with four other state-of-the-art algorithms: Tarel, et al. in [8], He, et al. with soft matting in [9], He, et al. with guided filtering in [11] and Meng, et al. in [17]. For simplicity, the 4 selected algorithms for comparison are labeled as [8], [9], [11] and [17], respectively. The qualitative and quantitative comparison are analyzed.

Figure 2 shows some final dehazed images by our algorithm and the four other methods. It can be seen that the image recovered by our method has high clarity, less color distortion and halo artifacts. The algorithms in [17] and [8] produce Halo effect around the boundary of the scene, for example, the treetops in the fourth row images (d) and (e), ridge line in the fifth row images (d) and (e). The algorithm in [9] can't remove the fog effectively, a lot of haze remains in the restored images, such as images in the first, third row and column (b). Our method can effectively removal haze and also avoid the Halo effect. The hazy images in the last three rows all contains large area of sky. From the corresponding dehazed images, we can see that the restored image of our method has natural transition between the sky area and the non-sky area with less color distortion, while the restored images of the other four methods all contains apparent color distortions in the sky regions. Overall, in all the six experimental results whether it is leave, road, car, sky or house, the visual effect of our approach is obviously superior to the other 4 algorithms compared, the dehazed results by our algorithm have higher visual quality and fewer color distortions.

Figure 2 Dehazed results comparison between the proposed method and some state-of-the-art algorithms: From column (a) to column (f), the images are input haze images, the results of He, et al. with soft matting in [9], He, et al. with guided filtering in [11], Meng, et al. in [17], Tarel, et al. in [8] and the proposed algorithm, respectively

Full size|PPT slide

It is noted that the minimum transmission is set to 0.1 for the images in the first three rows, while images in the last three rows, which contain sky regions, we have increased the minimum transmission from 0.1 to 0.5. Experimental results show that the haze removal effect is obviously improved as the value of

t_{0}

raises to 0.5.

4.2 Quantitative Comparison

We also employ two objective indexes, contrast ratio (CR) and image entropy (ENTR), to quantitatively evaluate our approach. Contrast ratio is the ratio between the luminance of the brightest white and the darkest black. The larger the contrast ratio is, the more levels of detail are shown. Image entropy is used to describe the average information richness and is defined as the average number of bits of the gray level set of the image. The higher the entropy is, the more information the image has. In addition, the time consumption is also compared. The results are presented in Table 1.

Table 1 Contrast ration and image entropy of the recovered image by different algorithms

Haze image	Evaluation Index	Ref.[9]	Ref.[11]	Ref.[17]	Ref.[8]	Algorithm Proposed
	CR	22.4399	22.5200	16.3441	21.1455	35.3921
	ENTR	7.0035	7.0025	6.6910	6.8703	7.5011
	Running time	11.5512	4.0987	4.4219	2.3254	3.3686
	CR	45.5734	42.4798	40.9589	32.7106	56.0500
	ENTR	7.6323	7.4823	7.4499	7.1655	7.6666
	Running time	23.7875	7.7270	6.9482	7.9309	6.5310
	CR	29.0327	25.7037	19.5316	29.3748	39.4963
	ENTR	7.1336	6.9370	6.7423	7.2161	7.2881
	Running time	69.2924	20.1301	19.1072	53.9158	17.1177
	CR	61.4359	50.9561	76.5663	24.0770	89.6270
	ENTR	6.8220	6.8384	6.7672	6.5999	6.0086
	Running time	23.6014	7.2706	7.5602	7.1732	6.5191
	CR	33.5944	26.5096	31.4702	21.2569	55.3513
	ENTR	6.8304	6.3279	6.2855	6.6760	7.1086
	Running time	73.5489	20.4175	19.1318	47.6958	18.1229
	CR	52.7999	44.6190	53.7820	30.5305	76.3263
	ENTR	7.2720	7.4145	7.1969	7.1207	6.8399
	Running time	36.3714	11.2226	10.8327	10.7697	9.5336

The numerical results given in Table 1 show that the proposed method has the best object index values in 15 of 18 lines formed by three indexes of the six experiments. Although [11] outperforms our approach in the index of ENTR in two experiments, and the running time of [8] is shorter than our approach in one experiment, there is only a slight difference compared with our results.

5 Conclusion

We proposed an improved single image dehazing approach based on DCP, the V-transform, and the HIS color space. This approach takes full advantage of the independence of the three components of the HSI space, performs haze removal mainly to the intensity

I

using the V-system and the dark channel prior. For images with large area of sky, we found that increasing the minimum value of transmission map can improve the haze removal effect and reduce color distortion in the sky area greatly. The experimental results show that our algorithm is effective and executable compared with some recent algorithms.

References

Publishing order | Descend order by publishing year | Descend order by cited within

1	Wenqi R, Lin M, Jiawei Z, et al. Gated fusion network for single image dehazing. 31st IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2018, 3253- 3261. Cited in this article [1]

2	Tanghuai F, Changli L, Xiao M, et al. An improved single image defogging method based on retinex. IEEE International Conference on Image, Vision and Computing, 2017, 410- 413. Cited in this article [1]

3	Bojun J, Mingxia Z. Improved histogram equalization algorithm in image enhancement. Laser and Infrared, 2014, 44 (6): 702- 706. Cited in this article [1]

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5	Narasimhan S G, Nayar S K. Interactive (De) weathering of an image using physical models. IEEE International Conference on Computer Vision Workshop on Color and Photometric Methods in Computer Vision, 2013, 1- 8. Cited in this article [1]

6	Qingsong Z, Jiaming M, Ling S. A fast single image haze removal algorithm using color attenuation prior. IEEE Transactions on Image Processing, 2015, 24 (11): 3522- 3533. https://doi.org/10.1109/TIP.2015.2446191 Cited in this article [1]

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Funding

National Natural Science Foundation of China(61571046)

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Abstract
Key words
Cite this article
1 Introduction
2 Related Works
2.1 Dark Channel Prior
2.2 Orthogonal V-transform
3 Proposed Algorithm
3.1 Hue Component Processing
3.2 Saturation Component Processing
3.3 Intensity Component Processing
Figure 1 Flow chart of the proposed algorithm
4 Experimental Results and Analysis
4.1 Qualitative Comparison
Figure 2 Dehazed results comparison between the proposed method and some state-of-the-art algorithms: From column (a) to column (f), the images are input haze images, the results of He, et al. with soft matting in [9], He, et al. with guided filtering in [11], Meng, et al. in [17], Tarel, et al. in [8] and the proposed algorithm, respectively
4.2 Quantitative Comparison
Table 1 Contrast ration and image entropy of the recovered image by different algorithms
5 Conclusion
References
Funding

Received	Accepted	Published
2019-09-28	2019-12-20	2020-04-25
Issue Date
2020-04-25

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Abstract

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Cite this article

1 Introduction

2 Related Works

2.1 Dark Channel Prior

2.2 Orthogonal V-transform

3 Proposed Algorithm

3.1 Hue Component Processing

3.2 Saturation Component Processing

3.3 Intensity Component Processing

Figure 1 Flow chart of the proposed algorithm

4 Experimental Results and Analysis

4.1 Qualitative Comparison

4.2 Quantitative Comparison

Table 1 Contrast ration and image entropy of the recovered image by different algorithms

5 Conclusion

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Abstract

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Cite this article

1 Introduction

2 Related Works

2.1 Dark Channel Prior

2.2 Orthogonal V-transform

3 Proposed Algorithm

3.1 Hue Component Processing

3.2 Saturation Component Processing

3.3 Intensity Component Processing

Figure 1 Flow chart of the proposed algorithm

4 Experimental Results and Analysis

4.1 Qualitative Comparison

4.2 Quantitative Comparison

Table 1 Contrast ration and image entropy of the recovered image by different algorithms

5 Conclusion

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References

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