Low-pass filtering based polarimetric dehazing method for dense haze removal

Jian Liang; Jian Liang; Liyong Ren; Rongguang Liang

doi:10.1364/OE.427629

1. Introduction

To enhance the quality of images captured in scattering media, including haze, turbid water and biological tissues, is always very challenging. Researchers have proposed many methods to increase the imaging depth and enhance the contrast of the scattered images. Some are based on the computer vision [1–3], and the others are based on optical degradation models [4–7]. Recently, the dehazing methods based on conventional neural network are also developed fast [8,9].

Among these dehazing methods, although the polarimetric dehazing method seems a little complicated to realize, it has several unique merits. First of all, the basic conception is very simple, and when three or four linear-polarized images of the same scene are obtained, dehazed images (∼1 million pixels) can be obtained with small computational cost and high efficiency [10–12]. Second, the model is very generalized. It implies that this method can be used to nearly all the homogenerous scattering media [4,13]. Finally, it is very effective in dense scattering media, since the polarization state of the incidence and scattering light is always different. Therefore, the polarimetric dehazing method can separate them effectively and reconstruct a dehazed image with much higher contrast [14–17].

However, the polarimetric dehazing method is still not very stable in processing images from different scattering media or using different kinds of cameras. Researchers have proposed several kinds of optimized methods to enhance its capability and stability. At first, bias coefficients are used to increase its stability [6,15,18], but it is found that this process is not convenient, since these bias coefficients need to be set manually in different cases. In the following, other solutions are demonstrated: one kind is combining digital imaging processing algorithms [6,11,14] to stabilize the polarimetric dehazing method in different environment; another is to avoid errors in reconstructing the objects with high polarization state through analyzing the polarization state of the scene [4,17,19]. Very recently, polarimetric dehazing method using deep learning technique has also been proposed to realize better results [20].

These optimized polarimetric dehazing methods are effective, but the assisted algorithms they combined may make the methods more complicated and time consuming. Besides, most of these methods aim at light haze conditions, which may not be suitable for dealing with dense haze. In this paper, we propose a low-pass filtering based polarimetric dehazing method for dense haze removal. As it is known, polarized parameters calculated by the hazy images, such as degree of polarization (DoP), angle of polarization (AoP) and the intensity of the airlight, are very sensitive to the noise [21], which is the main reason leading to the instability [22]. Therefore, low pass filter is used in the method to effectively suppress the noise and moreover maintain the faint information of the scene. When the noise is removed from the hazy images, it is found that the capability and the stability of the polarimetric dehazing method is enhanced a lot. Experimental results show that the proposed method is very effective in many dense haze conditions. Besides, other assisted algorithms are needed no more, which will greatly simplify the polarimetric dehazing method; also, bias coefficients are not needed, and thus the total automatic dehazing process is realized. The polarimetric dehazing theory will be described in Section 2. Experimental results will be shown in Section 3. The discussions on the proposed method will be demonstrated in Section 4.

2. Theory

In the hazy condition, the captured image contains two kinds of incidence. One is directly reflected by the scene, which is also called the direct light; the other is multi-scattered by hazy particles, which is also called the airlight. Therefore, the total incidence I(x, y) can be expressed as:

(1) $$I({x,y} )= A({x,y} )+ D({x,y} ),$$

where A(x, y) is the airlight, and D(x, y) is the direct light, (x, y) represents the coordination of the pixels. To be simplicity, the direct light can be usually seen as the object light exponentially degraded with the distance, expressed as:

(2)$$D({x,y} )= L({x,y} )\cdot {e^{ - \beta z({x,y} )}},$$

where L(x, y) represents the object light, i.e. the original incidence needed to be retrieved, and z(x, y) the distance. β is the degradation coefficient which is normally assumed to be a constant. Similarly, A(x, y) can be expressed as:

(3)$$A({x,y} )= {A_\infty } \cdot [{1 - {e^{ - \beta z({x,y} )}}} ],$$

where A_∞ is the airlight from the infinity. From Eqs. (1)–(3), one can eliminate e^{-βz(x, y)}, and obtain the object light:

(4)$$L({x,y} )= \frac{{I({x,y} )- A({x,y} )}}{{1 - {{A({x,y} )} / {{A_\infty }}}}}.$$

As can be seen from Eq. (4), the key process in dehazing is to accurately estimate A(x, y) and A_∞. Note that, A_∞ is actually a specific value of A(x, y), so we can focus on how to accurately estimate A(x, y).

In polarimetric dehazing method, four polarized images are captured, represented as I(0; x, y), I(45; x, y), I(90; x, y) and I(135; x, y). Then the Stokes vectors can be calculated as:

(5)$$\begin{aligned} {S_0}({x,y} )&= [{I({0;x,y} )+ I({45;x,y} )+ I({90;x,y} )+ I({135;x,y} )} ]/2\\ {S_1}({x,y} )&= I({0;x,y} )- I({90;x,y} )\\ {S_2}({x,y} )&= I({45;x,y} )- I({135;x,y} ), \end{aligned}$$

where S₀(x, y) equals to I(x, y) in Eq. (1). From Eq. (5), the degree of polarization (DoP) can be calculated as:

(6)$$p({x,y} )= \frac{{\sqrt {{S_1}({x,y} )_{}^2 + {S_2}({x,y} )_{}^2} }}{{I({x,y} )}}.$$

It is assumed that the direct light is non-polarized, while the airlight is partial polarized. Therefore, the DoP calculated by Eq. (6) is supposed to be equal to or smaller than the DoP of the airlight (p_A), according to Eq. (1). Therefore, theoretically, p_A can be regarded as the largest value of p(x, y).

Due to the assumption that the direct light is non-polarized, it means that the polarized part of hazy images are totally the contributions of the airlight. Based on this assumption, one can obtain this part of incidence, represented as A_p(x, y). Then, A(x, y) can be calculated as:

(7)$$A({x,y} )= \frac{{{A_p}({x,y} )}}{{{p_A}}}.$$

In the hazy condition, although the airlight is partial polarized, p_A is still very small, usually less than 0.01, according to Mie’s scattering theory. Therefore, if we directly use Eq. (7) to estimate A(x, y), the noise of the image can be amplified to over 100 times. In dense haze conditions, hazy images are dominated by the airlight, so the noise amplification will extremely decrease the quality of the dehazed image, and submerge the faint information of the scene.

In the proposed method, to accurately estimate the airlight, and meanwhile maintain the faint information of the scene, Eq. (7) cannot be used directly. First, we can estimate the polarized part of the airlight:

(8)$${A_p}({x,y} )= {I_p}({x,y} )= I({x,y} )\cdot p({x,y} ).$$

Then except Eq. (7), the airlight can also be written as:

(9)$$A({x,y} )= {A_p}({x,y} )+ {A_n}({x,y} )= A{}_p({x,y} )+ \frac{{1 - {p_A}}}{{{p_A}}} \cdot {A^{\prime}_p}({x,y} ),$$

where A_n(x, y) is the nonpolarized part of the airlight. Since p_A is very small, A_n(x, y) is much larger than A_p(x, y). It means the noise in the calculated A(x, y) mainly comes from that in A_n(x, y). Therefore, the noise in A_n(x, y) should be firstly suppressed. We use low-pass filter to do the noise suppressing process, and it can be expressed as follows:

A′_p(x, y) can be expressed as:

(10)$${A^{\prime}_p}({x,y} )= {A_p}({x,y} )\ast LP({a,b,\sigma } ).$$

In Eq. (10), ‘*’ represents the convolution, LP represents the low-pass filter, and (a, b) represents the patch of the filter. In this paper, Gaussian low-pass filter is used. Note that, LP filter in the spatial domain or the frequency domain are of the same usage in our dehazing process. The kernel of Gaussian LP filter can be expressed as:

(11)$$LP({a,b,\sigma } )= \frac{1}{{2\pi {\sigma ^2}}}\exp \left[ { - \frac{{{{({a\textrm{ - }{a_0}} )}^2} + {{({b - {b_0}} )}^2}}}{{2{\sigma^2}}}} \right].$$

where (a₀, b₀) is the center coordinate of the kernel, σ is chosen to be one fifth of the patch. In our experiments, we set the patch to be 0.5 percent of the whole size of the hazy image.

Compared with Eq. (7), Eq. (9) is much more reasonable in the practical usage, which can ensure the polarimetric dehazing algorithm robust and reliable. This is because in Eq. (9), A_n(x, y) is calculated by the low-pass filtered A_p(x, y), i.e. $A^{\prime}{_p}(x,y)$, which implies A(x, y) will not be dominated by the noise. Besides, $A{_p}(x,y)$A_p(x, y) still maintains some noise of the original images, so that we can suppress part of noise when we subtract A(x, y) from I(x, y) in Eq. (4) to obtain better dehazed results. The comparison of the calculated airlight A(x, y) is shown in Fig. 1, where the blue line is the result calculated by Eq. (7), while the green one is calculated by Eq. (9). The red line is the ground truth of the airlight, which is directly captured by the camera. It can be seen that the red line has some fluctuations due to the noise introduced by the camera. If directly calculating the airlight from the red line using Eq. (7), the noise is amplified a lot, shown as the blue line. However, the green line is much smoother compared with the other two lines, because the LP filter can effectively suppress the noise.

Fig. 1. Example of the calculated airlight A(x, y) compared with custom method (Eq. (7)) and the proposed method (Eq. (9)). The red line is the directly captured airlight, while the green and blue line are the calculated airlight using the proposed method and the custom method.

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As the definition of A_∞, it implies that if there is the sky area in the original image, A_∞ should be the largest value of A(x, y); otherwise, A_∞ should be slightly larger than A(x, y), according to Eq. (3). To make our dehazing method more robust, we set A_∞ a little larger than A(x, y) to adjust most scattering environment, in case there is no sky area in hazy images. From Eq. (4), it can be seen that A_∞ can be regarded as a coefficient, and just affect the average gray value of the dehazed image, so we can easily modify it in the post-processing. The whole workflow is shown in Fig. 2.

Fig. 2. The workflow of the proposed polarimetric dehazing method.

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3. Experimental results

In this section, the experimental results are shown to demonstrate the effectiveness of the proposed method. The images captured in the extreme dense hazy weather are chosen. Figure 3(a) shows a dense hazy image captured on the roof of our laboratory. There is very little information left from the building in distance. The dehazed images are shown in Figs. 3(b)–3(d), where (b) and (c) are the dehazed images using Schechner’s method and our previous method for comparison, and (d) is the dehazed result of the proposed method. To quantitatively estimate the quality of the dehazed methods, the no-reference contrast expression C(I) used for hazy images is introduced [23]:

(12)$$C(I) = \frac{{\sqrt {\frac{1}{N}\sum\nolimits_{x,y} {{{[{I({x,y} )- \bar{I}} ]}^2}} } }}{{\bar{I}}}.$$

where N is the number of pixels in the image, $\bar{I}$ is the mean gray value of the image, and I(x, y) is the gray value of the pixel (x, y). The contrast of Fig. 3(a) is 0.1673, since most detailed information in the image is submerged into the airlight. The contrasts of Figs. 3(b)–3(d) are 0.3748, 0.5375 and 0.7090, respectively. To clearly show the difference of the detailed information in the dehazed images, the region marked in Fig. 3(a) by a black box is amplified under each figure. From the comparison of the amplified images, it can be seen that the noise in Fig. 3(d) is effectively suppressed, so Fig. 3(d) has a much better quality. For example, there is one bright spot in each amplified image, which has been marked by a yellow circle. It is a light source from a room in the building, which is very faint in the original hazy image. However, it is clearly seen from Fig. 3(d) that the faint information of the hazy image can be greatly recovered.

Fig. 3. Experimental results of Building 1. (a) Original hazy image; (b) dehazed image by Schechner’s method [10]; (c) dehazed image by our previous method [22]; (d) dehazed image by the proposed method. The amplified insets show the detail enhancement of the dehazed images.

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Another example captured in different dense hazy weather is shown in Fig. 4. Figures 4(a)–4(d) are the original hazy image, the dehazed images using Schechner’s method, our previous method and the proposed method, respectively. It can be seen that the quality of Fig. 4(d) is much better than Figs. 4(b) and 4(c), where much detailed information can be recognized. The contrast of Fig. 4(a) calculated by Eq. (12) is 0.0650, which is much less than Fig. 3(a). This is because the scene in Fig. 4 is only one building located in distance, showing less detailed information than Fig. 3(a). The contrasts of Figs. 4(b)–4(d) are 0.1465, 0.1737, and 0.2591, respectively. In such a dense haze condition, the object light is degraded severely and the images are dominated by the airlight. In this case, the airlight is approximately equals to the gray value of the hazy image, and thus the dehazed results are sensitive to the accuracy of the estimated polarized coefficients. As a result, the dehazed results may become a little unstable of the custom polarimetric dehazing methods, as shown in Figs. 4(b) and 4(c). The detailed information is just slightly enhanced in Fig. 4(b), while the noise is also enhanced with the object light in Fig. 4(c). However, it can be seen that the contrast is enhanced a lot in Fig. 4(d), while the noise is well suppressed.

Fig. 4. Experimental results of Building 2. (a) Original hazy image; (b) dehazed image by Schechner’s method [10]; (c) dehazed image by our previous method [22]; (d) dehazed image by the proposed method.

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To specifically show the contrast of the building, the gray value of a column shown in Fig. 4(a) of each figure is plotted, as shown in Fig. 5. It can be seen that the plot of the original image is very flat, which means the detailed information is submerged. The plots of Figs. 4(b) and 4(c) shows that the detailed information is slightly enhanced, but the noise is also enhanced. The plot of Fig. 4(d) shows a very good contrast compared with the other plots. The contrast of the dehazed images using the proposed method can be enhanced by about 4 times than the original hazy images.

Fig. 5. Plot of the gray value versus pixels of the line marked in Fig. 4(a).

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In the following, the dehazed results of hazy images with both near and far objects are shown in Fig. 6. Figure 6(a) is the original hazy image, and it can be seen that the near objects can be recognized well, however, the buildings in distance are faint and very difficult to observe. The dehazed images using Schechner’s method, our previous method and the proposed method are shown in Figs. 6(b)–6(d), respectively. From Figs. 6(b) and 6(c), it can be seen that the haze is not removed thoroughly, which is because the accuracy of the polarized coefficients’ estimation of the airlight is decreased due to the noise. Although the buildings in distance can be seen, but the detailed information on these buildings is still missing. As shown in Fig. 6(d), the dehazed image using the proposed method is much better than Figs. 6(b) and 6(c), and much detailed information is clearly seen from the buildings in distance. However, it can also be seen from the near objects in Fig. 6(d) that they are a little darker than Figs. 6(b) and 6(c), which is mainly because when we calculated the dehazed image using Eq. (4), the denominator is stretched in order to fully enhance the contrast of the objects in distance. It is reasonable since the objects in distance is needed to be enhanced much more than the near objects in dense hazy images. The contrasts of Figs. 6(a)–6(d) are 0.1042, 0.2188, 0.2725 and 0.4351, respectively. These experimental dehazed results show the effectiveness of the proposed method in dense haze condition, and better quality than other custom polarimetric dehazing methods.

Fig. 6. Experimental results of another scene. (a) Original hazy image; (b) dehazed image by Schechner’s method [10]; (c) dehazed image by our previous method [22]; (d) dehazed image by the proposed method.

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The contrast enhancement of these examples is listed in Table 1 for better comparison. From Table 1, it can be seen that the dehazed images of the proposed method are enhanced approximately 3 times of hazy images. These experimental results captured by different haze environment demonstrate the capability of the proposed method, while the contrast enhancement demonstrate its stability.

Table 1. Contrast of the images

View Table

4. Discussions

4.1 Scene has objects with large DoP

In polarimetric dehazing principle, in order to simplify the degradation model, the direct light is assumed as the non-polarized light. This is mostly reasonable since the direct light is severely depolarized by the hazy particles, according to Mie’s scattering theory. However, when the hazy images include specular surface, such as window and water, etc., the assumption is invalid [23]. It is because the high polarization will be directly regarded as the large A_p, according to Eq. (8). As shown in Fig. 7(a) is the A_p of a scene with specular surface in the hazy image. The high polarized region actually contains two causes, which is marked in the figure. One is the specular surface, as marked by a yellow box; while the other is due to the misalignment among four polarized images, as marked by a yellow oval. Besides, the noise may lead to discrete high gray-value points all over the image. This high polarization may lead to the error in reconstructing the dehazed image, as shown in the yellow dash box in Fig. 8(b), the dehazed result using Schechner’s method.

Fig. 7. The map of A_p. (a) Before correction, (b) after correction.

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Fig. 8. Experimental results of the scene with specular surface. (a) Original hazy image; (b) dehazed image by Schechner’s method [10]; (c) dehazed image by the proposed method.

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Therefore, in the proposed method, a simple method is applied to avoid the high polarization in practical applications. Since the high polarization directly leads to the high A_p, we just need to suppress the ultrahigh A_p from the original A_p map. Based on the fact that the number of high A_p is very small compared to the whole pixels in the image, the average value of A_p can still be regarded as the threshold. In the A_p map, the value of each pixel which is higher than the threshold will be set to a fixed value slighter than the threshold (98% of the threshold in our experiments). Then, the corrected A_p map can be obtained, as shown in Fig. 7(b). It can be seen that the ultrahigh A_p is effectively suppressed. And the estimating accuracy and thus the quality of the dehazed image is increased, accordingly, as shown in Fig. 8(c). Compared to the hazy image, as shown in Fig. 8(a), the contrast is enhanced a lot in Fig. 8(c). Moreover, it is much better than Figs. 8(b).

4.2 Estimation of A_∞ in hazy images

There is another limitation in the polarimetric dehazing method, which is the estimation of A_∞. Normally, one needs the sky region in the hazy image to accurate estimate A_∞ [10]. The sky region needs to be identified either manually or using complex identification algorithms. This is very inconvenient in practical usage. What’s worse, for hazy images which don’t have sky region, the dehazing process may fail.

Instead of estimating A_∞ directly from the original hazy image, we propose a method to estimate A_∞ from the map of A, as shown in Fig. 2. The noise is fully suppressed by the low-pass filter in the proposed method, and A_p is cleared from the high polarization based on the method discussed in the Section 3.1, the map of A is much more accurate and less noisy compared to that in the custom polarimetric dehazing methods, and thus A_∞ can be regarded as the largest value of the map of A, according to Eq. (3). Figure 9 compares the dehazed results to a hazy image without any sky region, where Figs. 9(a)–9(d) are the hazy image, the dehazed images using Schechner’s method, our previous method and the proposed method, respectively. As shown in Fig. 9(b), the dehazed image has some invalid pixels, due to the incorrect estimation of A_∞. Figures 9(c) and 9(d) can clearly show the detailed information which is blurred in Fig. 9(a). However, the quality of Fig. 9(d) is much better than Fig. 9(c), and the contrasts of Fig. 9(c) and 9(d) are 0.3937 and 0.5821, respectively.

Fig. 9. Experimental results of the scene without sky region. (a) Original hazy image; (b) dehazed image by the proposed method.

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4.3 Patch of the low-pass filter

In the proposed method, a low pass filter is applied into the algorithm to suppress the noise and increase the accuracy of the estimation of the airlight. Here, the patch of the low-pass filter is discussed. As shown in Fig. 10 are the dehazed results using different patch of the low-pass filter. Figures 10(a1)–10(d1) show the map of A calculated by the patch with (102, 100), (17, 17), (6,6) and (2,2), respectively. Note that, the patches of the row and column can be chosen according to the size of the hazy image, therefore, they may not be equal. As can be seen in Fig. 10, as the patch is decreased, there is less high-frequency information and noise in the map of A. Figures 10(a2)-(d2) are the dehazed results accordingly. To clearly see the difference, the region of the branches (showing more detailed information) marked by the box is amplified, as shown in Figs. 10(a3)–10(d3). It can be seen from Fig. 10(a3) that the noise is still dominated, and the detailed information among branches is blurred. However, Figs. 10(b3)-(d3) are approximately the same, and the contrasts of the three dehazed images are also very similar. These results demonstrate that the patch of the low-pass filter is not sensitive to the dehazed results when the patch is set to be small enough.

Fig. 10. Experimental results with different patch of the low-pass filter. From the left column to the right are the patches with (102, 100), (17, 17), (6, 6) and (2, 2), respectively. (a1)-(d1) are the map of A, (a2)-(d2) are the dehazed results, (a3)-(d3) are the amplified image of the region marked in (a2).

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5. Conclusion

In this paper, we demonstrate a simple but very effective polarimetric dehazing method. By introducing low-pass filtering, this method is suitable in dealing with the images captured in the dense scattering media. With the help of the effective noise suppression, the proposed method is completely automatic and very robust in processing the images captured in dense haze conditions. Experimental results demonstrate the capability of the proposed method, where the faint information in hazy images are preserved very well, and the contrast of the dehazed images are increased a lot. This method has potential usages in many scattering applications.

Funding

Xi’an Scientific and Technological Projects (2020KJRC0013); Natural Science Foundation of Shaanxi Province (2021JM-204); National Natural Science Foundation of China (61505246); China Scholarship Council (201804910323).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.

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Low-pass filtering based polarimetric dehazing method for dense haze removal

Abstract

1. Introduction

2. Theory

3. Experimental results