Polarimetric image recovery in turbid media employing circularly polarized light

Haofeng Hu; Lin Zhao; Xiaobo Li; Hui Wang; Jingyu Yang; Kun Li; Tiegen Liu

doi:10.1364/OE.26.025047

1. Introduction

The imaging quality could be significantly degraded due to the disturbance of the turbid media, especially in the case of dense turbid media [1,2]. The particles existing in the turbid media can scatter and absorb the target signal out of the optical path. Moreover, the particles also scatter undesired light into the optical path, which could veil the object and thus lead to the reduction of image quality and the decrease of visibility range. It subsequently is a main problem in various applications, such as the object detection [3] and video surveillance [4], etc.

Various methods have been developed for the quality enhancement of hazy images [5–17]. In particular, the method of polarimetric image recovery in scattering media (including haze, fog and turbid water, etc.) was proposed by Schechner [18]. Further studies have been performed to develop this method in subsequent years [19–22], which prove that polarimetric method is effective in improving the image quality of underwater scene [23–27]. Previous polarimetric image recovery methods focus on the information of linear polarization. However, in dense turbid media, the linearly polarized light tends to lose its polarized properties, and thus the quality of image recovery could be decreased.

Previous studies have shown that circularly polarized light tends to maintain its original polarization property better than the linearly polarized light, which is called the “circular polarization memory” effect [28,29]. Therefore, we believe if the circularly polarized light is employed, we can overcome the limitation in dense turbid media to further enhance the quality of image recovery.

In this paper, we focus on the conditions of imaging through dense turbid media. We show the limitation of the traditional polarimetric image recovery method based on linearly polarized light, and we try to overcome the limitations by circularly polarized light. We employ the circular polarization beam to illuminate the objects in turbid water, and we investigate the polarization properties of the scattered light received by the detector. It is found that the light received by the detector contains both the circularly polarized light and linearly polarized light, in which the linearly polarized light is generated by the scattering medium. Based on this property, we propose an image recovery method combining the circular polarization information and linear polarization information, which is an improvement of the previous works only using linearly polarized light. In addition, several groups of real-world experiment in the dense turbid water are performed, in order to demonstrate the effectiveness of the proposed method.

2. Traditional method of polarimetric image recovery, based on linearly polarized light

The traditional method of polarimetric image recovery proposed by Schechner is based on the basic physical model of image formation through the homogeneous turbid medium described in [13]. The irradiance received by detector can be expressed as:

I (x, y) = D (x, y) + B (x, y) = L (x, y) t (x, y) + A_{\infty} (1 - t (x, y)),

where

D (x, y)

originates from the radiance of the object in the fields of view, which is attenuated due to the absorption and scattering by the turbid medium.

B (x, y)

is called the veiling light or backscatter, which comes from the light scattered towards the detector by the scattering particles, and

L (x, y)

is the radiance of the object without being attenuated by the particles in water, and

A_{\infty}

corresponds to the value of backscatter at an infinite distance in the water.

t (x, y)

is the medium transmittance.

The traditional method of polarimetric image recovery proposed by Schechner is based on two images captured at two orthogonal polarization states, which are the co-linear image $I^{| |} (x, y)$ and the cross-linear image $I^{⊥} (x, y)$ [18]:

\begin{matrix} I^{| |} (x, y) = \frac{D (x, y)}{2} + B^{| |} (x, y), \\ I^{⊥} (x, y) = \frac{D (x, y)}{2} + B^{⊥} (x, y) . \end{matrix}

The polarization orientation of the backscatter is close to that of the incident linearly polarized light [13,17], then the degree of polarization (DOP) of the backscatter can be described as $p_{s c a t} = (A_{\infty}^{| |} - A_{\infty}^{⊥}) / (A_{\infty}^{| |} + A_{\infty}^{⊥})$ . The veiling light can be described as $B (x, y) = (I^{| |} - I^{⊥}) / p_{s c a t}$ [13], and then the transmission and scene radiance are obtained:

\begin{matrix} t (x, y) = 1 - \frac{B (x, y)}{A_{\infty}} = 1 - \frac{I^{| |} (x, y) - I^{⊥} (x, y)}{p_{s c a t} A_{\infty}}, \\ L (x, y) = \frac{I (x, y) - B (x, y)}{t (x, y)} = \frac{I^{| |} (x, y) + I^{⊥} (x, y) - B (x, y)}{t (x, y)}, \end{matrix}

The traditional method of polarimetric image recovery proposed by Schechner described above is only based on the linearly polarized light. However, this traditional method could lead to poor performance under dense turbid media due to strong scattering. Previous studies have shown that circularly polarized light tends to maintain its degree of polarization in scattering environment better than linearly polarized light, which is called the polarization memory effect [28]. Therefore, it is interesting to realize the polarimetric image recovery with the circularly polarized light. In the next section, we will present a new method of polarimetric image recovery to remove the dense haze in the image in dense turbid media combining the circularly polarized light and the linearly polarized light.

3. Polarimetric dehazing method combining circularly polarized light and linearly polarized light

We employ the Stokes vector S, which consists of four parameters $(s_{0}, s_{1}, s_{2}, s_{3})$ , to describe the polarization state of light. Here, the first element $s_{0}$ defines the total intensity of light. According to Eq. (1), the light received by the detector can be classified into two categories: object radiance and backscatter. We denote the Stokes vectors of the object radiance and the backscatter as $S^{o}$ and $S^{b}$ respectively. Therefore, the intensity of the object radiance can be expressed as:

s_{0}^{o} = L (x, y) t (x, y),

and the intensity of the backscatter can be expressed as:

s_{0}^{b} = A_{\infty} [1 - t (x, y)] .

The Stokes vector of the light received by detector

S

can be expressed as:

S = S^{o} + S^{b} = [\begin{matrix} s_{0}^{o} \\ s_{1}^{o} \\ s_{2}^{o} \\ s_{3}^{o} \end{matrix}] + [\begin{matrix} s_{0}^{b} \\ s_{1}^{b} \\ s_{2}^{b} \\ s_{3}^{b} \end{matrix}],

where

s_{0} (x, y)

indicates the total intensity of light detected by the detector,

s_{0}^{o} (x, y)

and

s_{0}^{b} (x, y)

denote the intensity of the object light and the veiling light, and the total intensity of light can be described as

s_{0} (x, y) = s_{0}^{o} (x, y) + s_{0}^{b} (x, y) .

In this method, the circularly polarized light is employed to illuminate the objects in the turbid media. In this case, there are linearly polarized light and circularly polarized light contained in the radiance acquired by the detector [30]. Thus, the Stokes vector of the light received by detector can be decomposed into linear polarization, circular polarization and un-polarization parts, which can be expressed as:

\begin{array}{l} S = S_{l - p o l a r i z e d} + S_{c - p o l a r i z e d} + S_{u n p o l a r i z e d} \\ = [\begin{matrix} \sqrt{s_{1}^{2} + s_{2}^{2}} \\ s_{1} \\ s_{2} \\ 0 \end{matrix}] + [\begin{matrix} s_{3} \\ 0 \\ 0 \\ s_{3} \end{matrix}] + [\begin{matrix} s_{0} - \sqrt{s_{1}^{2} + s_{2}^{2}} - s_{3} \\ 0 \\ 0 \\ 0 \end{matrix}], \end{array}

where the subscript of l and c indicate the linearly and circularly polarized light respectively.

The degree of linear polarization $p_{l}$ and the degree of circular polarization $p_{c}$ can be calculated based on the Stokes vector as:

\begin{array}{l} p_{l} = \frac{\sqrt{s_{1}^{2} + s_{2}^{2}}}{s_{0}}, \\ p_{c} = \frac{s_{3}}{s_{0}} . \end{array}

We can use different polarization states to recover the image by the traditional polarimetric dehazing model proposed by Schechner, which can dehaze the underwater image sufficiently.

3.1 Image recovery with linearly polarized light

Firstly, we can process the linearly polarized light, which is similar to the traditional polarimetric image recovery method mentioned in Section 2. It can be seen from Eq. (3) that only $A_{\infty}$ and $p_{l_s c a t}$ , which are the value of backscatter at an infinite distance and the degree of linear polarization respectively, are required to recover the image based on the linearly polarized light. $A_{\infty}$ and $p_{l_s c a t}$ can be estimated according to the background region in the image as [31]:

A_{\infty} = \frac{1}{| Ω |} \sum_{(x, y) \in Ω} s_{0}^{} (x, y),

p_{l_s c a t} = \frac{1}{| Ω |} \sum_{(x, y) \in Ω} [\frac{\sqrt{{[s_{1}^{} (x, y)]}^{2} + {[s_{2}^{} (x, y)]}^{2}}}{s_{0}^{} (x, y)}],

where

| Ω |

specifically represents the number of pixels in the background region of the image.

Strictly speaking, the value of $p_{l_s c a t} (x, y)$ could vary slightly if one chooses different regions in the background [13], and thus there could be an error estimation of $p_{l_s c a t}$ , which affect the result of recovery image. Therefore, we employ a parameter $ε_{l}$ slightly above 1 to modify $p_{l_s c a t}$ as $ε_{l} p_{l_s c a t}$ . According the equations above, the intensity of backscatter of partial linearly polarized light $s_{0_l}^{b} (x, y)$ can be expressed as:

s_{0_l}^{b} (x, y) = \frac{\sqrt{{[s_{1}^{b} (x, y)]}^{2} + {[s_{2}^{b} (x, y)]}^{2}}}{ε_{l} p_{l_s c a t}} = \frac{\sqrt{s_{1}^{2} (x, y) + s_{2}^{2} (x, y)}}{ε_{l} p_{l_s c a t}} .

In practice, the objects are often considered to be highly depolarized, and the DOP of the object radiance can be neglected either in the case of passive and active imaging [13,21,23,27], which means

s_{1}^{o} = s_{2}^{o} = s_{3}^{o} = 0

. In other words,

s_{1} = s_{1}^{b}, s_{2} = s_{2}^{b}, s_{3} = s_{3}^{b}

, according to Eq. (6).

Similar to Eq. (3), the medium transmittance and thus the object radiance recovered based on linear polarization $L_{l} (x, y)$ can be deduced as:

\begin{array}{l} t_{l} (x, y) = 1 - \frac{\sqrt{s_{1}^{2} (x, y) + s_{2}^{2} (x, y)}}{ε_{l} p_{l_s c a t} A_{\infty}}, \\ L_{l} (x, y) = \frac{s_{0} (x, y) - A_{\infty} [1 - t_{l} (x, y)]}{t_{l} (x, y)} . \end{array}

3.2 Image recovery with circularly polarized light

According to the recovered image based on linearly polarized light, we can consequently perform the image recovery based on circularly polarized light. We consider $L_{l} (x, y)$ , which is the dehazed object radiance based on linearly polarized light given by Eq. (12), to be the radiance for image recovery with circularly polarized light. Then we need to estimate the global parameters $A_{\infty}$ and $p_{c_s c a t}$ . For estimating $A_{\infty}$ , we measure the gray levels of the pixels corresponding to a region of background in the image of $L_{l} (x, y)$ , while for estimating $p_{c_s c a t}$ , we measure the gray levels of the pixels corresponding to a region of background in the image of $L_{l} (x, y)$ and $s_{3}$ , which can be considered to be the values of $A_{\infty}$ and the difference between right and left circular polarizations. Consequently, $A_{\infty}$ and $p_{c_s c a t}$ can be estimated as follows:

A_{\infty} = \frac{1}{| Ω |} \sum_{(x, y) \in Ω} L_{l} (x, y),

p_{c_s c a t} = \frac{1}{| Ω |} \sum_{(x, y) \in Ω} [\frac{s_{3} (x, y)}{L_{l} (x, y)}],

where

| Ω |

specifically represents the number of pixels in the background region of the image.

For circularly polarized light, we also employ a parameter $ε_{c}$ above 1 to modify $p_{c_s c a t}$ as $ε_{c} p_{c_s c a t}$ . Then, the veiling light of circularly polarized light can be derived as:

s_{0_c}^{b} (x, y) = \frac{s_{3}^{b} (x, y)}{ε_{c} p_{c_s c a t}} = \frac{s_{3}^{} (x, y)}{ε_{c} p_{c_s c a t}} .

Based on the equations above, the medium transmittance and thus the object radiance for $L_{c} (x, y)$ can be deduced as:

\begin{matrix} t_{c} (x, y) = 1 - \frac{s_{3}}{ε_{c} p_{c_s c a t} A_{\infty}}, \\ L_{c} (x, y) = \frac{L_{l} (x, y) - A_{\infty} [1 - t_{c} (x, y)]}{t_{c} (x, y)} . \end{matrix}

According to the Eqs. (12) and (16), the method of image recovery for obtaining the radiance of the scene $L_{c} (x, y)$ can be recovered by the method combining both circularly polarized light and linearly polarized light. The flowchart of the procedure of the proposed method in this paper is shown in Fig. 1.

Fig. 1 The flowchart of the polarimetric image recovery method in turbid media combining circularly polarized light and linearly polarized light.

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4. Experiments of underwater imaging

We perform the real-world experiments of underwater imaging to verify the method of image recovery illustrated in Section 3. The experimental setup is shown in Fig. 2(a). The light-emitting diode (LED) together with an optical filter are employed to generate the active illumination light with the central wavelength of 632.8 nm. The illumination light is modulated by polarization state generator (PSG) composed with a linear polarizer and a quarter-wave plate, which can generate the circularly polarized light as the active illumination light. We used a transparent PMMA tank filled with water and make it turbid by blending the clear water with milk. The images are taken by a monochrome CCD camera (AVT Stingray F-033B).

Fig. 2 (a) Experimental setup for underwater imaging. (b) Intensity image in clear water.

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In our experiment, the target region consists of a plastic cube, which has some words and patterns on its surface. The target is immersed in the turbid water, and the intensity image of this scene in clean water is shown in Fig. 2(b). The depolarization of the object depends on the surface feature and the material of the object. The surface of the plastic cube in our experiment is dull polished instead of specular, which leads to the diffuse reflection of the light, and in addition, the plastic usually leads to high depolarization [32]. Therefore, the object shown in Fig. 2(b) is highly depolarized.

In order to obtain the four Stokes parameters s₀, s₁, s₂, s₃, we need to modulate the orientations of the linear polarizer and the quarter-wave plate in polarization state analyzer (PSA) four times and then capture the images. Figure 3 shows the four images taken with the specific orientations of the linear polarizer and the quarter-wave retarder. Based on the images in Fig. 3, the four parameters of the Stokes vectors can be calculated [33].

Fig. 3 (a)-(d) are the images taken with the specific orientations of the linear polarizer and the quarter-wave retarder.

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The intensity image, which is s₀ of the Stokes vector, in the turbid water is shown in Fig. 4(a). It can be seen that the visibility of the image is poor. The edge of the plastic cube and the details of the scene are severely degraded. The degree of linear polarization $p_{l_s c a t} (x, y)$ and the degree of circular polarization $p_{c_s c a t} (x, y)$ can be calculated based on the Stokes vector, which is shown in Figs. 4(b) and 4(c). The blue square is the region belong to the background, which is used to estimate the intensity of the backscatter at infinity $A_{\infty}$ , as well as the degree of linear polarization and degree of circular polarization of the backscatter ( $p_{l_s c a t}$ and $p_{c_s c a t}$ ).

Fig. 4 (a) Intensity image of the scene, and (b) the degree of linear polarization $p_{l_s c a t} (x, y)$ , (c) the degree of circular polarization $p_{c_s c a t} (x, y)$ .

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It should be noted that the values of $ε_{l}$ and $ε_{c}$ are key parameters in optimizing the quality of the recovered image. Here, we perform the search of $ε_{l}$ and $ε_{c}$ (with a resolution step of 0.01) to maximize the value of EME [34], which are set to 1.0 and 2.0. The optimal values of $ε_{l}$ and $ε_{c}$ are found to be $ε_{l} = 1.28$ , $ε_{c} = 1.50$ . Although the optimal values of $ε_{l}$ and $ε_{c}$ are different, they correspond to the optimal quality of image recovery for linear polarization and circular polarization respectively. This optimal set of $ε_{l}$ and $ε_{c}$ leads to the optimal image recovered by our method. Figure 5(b) shows the recovered object radiance $L_{c} (x, y)$ at optimal values of $ε_{l}$ and $ε_{c}$ by the proposed method. For our Matlab program running on the computer with Intel(R) Core (TM) i5-7300HQ CPU@2.50GHz processor, the iterative algorithm takes about 4 s for finding the optimal values of $ε_{l}$ and $ε_{c}$ in our case. In order to compare with the traditional polarimetric dehazing method proposed by Schechner, we perform the experiment with linearly polarized illuminating, and the recovered image by Schechner’s method [13] is also presented in Fig. 5(c). However, since the DOP of the object radiance is neglected according to Eq. (11), our method is not suitable for the highly polarized object recovery. In particular, our method can lead to negative values of $L_{c} (x, y)$ at the pixels corresponding to the highly polarized objects.

Fig. 5 (a) The intensity image of the scene in the turbid water. (b) Recovered image by our method. (c) Recovered image by the traditional polarimetric dehazing method in [13].

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To further verify the effectiveness of our method in the condition of denser turbid water, we added different amount of milk in water to generate the turbid media with medium density and high density of milk. The intensity images of the scene in such turbid media are shown in Fig. 6(a), and it can be seen that the details of the images are severely degraded. The images recovered by the Schechner’s method are also shown in Fig. 6(b), while the recovered images by our method are shown in Fig. 6(c). It can be seen that our method can recover the details of the scene clearly even though the media is quite turbid, which has a better performance than Schechner’s method.

Fig. 6 (a) The intensity images of the scene in turbid water with different densities of milk. (b) Recovered images by the traditional polarimetric dehazing method in [13]. (c) Recovered images by our method.

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In order to show the details of the image better, we display in Fig. 7 the enlarged views of two regions (A and B in red and blue rectangle region respectively in Fig. 5(a)) for the images recovered by our method and Schechner’s method in different densities of turbid water shown in Fig. 5 and Fig. 6. It can be seen that the performance of different regions recovered by our method are better than those by Schechner’s method in different densities of the scattering particles.

Fig. 7 Enlarged views of details with different densities of milk marked in Fig. 5(a). The images of low density correspond to Fig. 5.

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In addition, in order to quantitatively verify the effectiveness of our method in different densities of turbid media, we use two criterions to evaluate the quality of image, including the value of measure of enhancement (EME) [34] and Michelson’s contrast [35]. The higher value of EME and Michelson’s contrast indicate the higher image quality. The results for different methods with different densities of milk are listed in Table 1. It can be seen in Table 1 that both the values of EME and Michelson’s contrast for our method are greater than those of Schechner’s method, which indicates that the quality of image recovery for our method is relatively better.

Table 1. The contrast of targets in Fig. 7.

View Table

In addition, in order to verify the universality of our method, we perform the experiment for the scenes with different polarization properties, including the scene of rough wooden board with some patterns and words on it and the non-flat plastic toy. Furthermore, we additionally perform the image recovery based on other two representative polarimetric methods, including polarization difference imaging (PDI) method [36] and the method based on distribution analysis of angle of polarization proposed by J. Liang [21]. The results are show in Fig. 8. It can be seen that the visibility of the original intensity images shown in Figs. 8(b) and 8(h) are poor. The recovered images by our method are shown in Figs. 8(c) and 8(i), and the recovered images by Schechner’s method, PDI method and Liang’s method are also presented in Fig. 8 for comparison. It can be seen that the visibility of the recovered image in Figs. 8(c) and 8(i) are greatly enhanced, and the details are considerably clearer than those of other methods.

Fig. 8 (a) The intensity image of the wooden board in clear water. (b) The intensity image of the wooden boards in the turbid water. (c) Recovered image by our method. (d) Recovered image by Schechner’s method. (e) Recovered image by PDI method. (f) Recovered image by Liang’s method. (g) The intensity image of the non-flat plastic toy in clear water. (h) The intensity image of the non-flat plastic toy in the turbid water. (i) Recovered image by our method. (j) Recovered image by Schechner’s method. (k) Recovered image by PDI method. (l) Recovered image by Liang’s method.

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5. Conclusion

In conclusion, based on the phenomenon of circular polarization memory, we propose the method of polarimetric image recovery in turbid media, in which the circularly polarized light instead of the linearly polarized light is employed as the illumination light. Based on the circular and linear polarization information derived from the measured Stokes vectors of the scene, the image recovery using linear polarization occurs is performed first and it is followed by the image recovery based on circular polarization. This new proposed method combines the circular polarization and linear polarization information by means of fully understanding and combing all components of the Stokes vector to recover the image more effectively compared with previously methods. The results of real-world experiments show that our method could considerably enhance the quality of recovered image in turbid water even though the turbid media is quite dense, and the quality of the image recovered by our method is better than that of previous methods, including traditional polarimetric image recovery method proposed by Schechner. In particular, the method proposed in this paper can overcome the attenuation of the polarization information, and thus can have a good performance in the condition of strong scattering for dense turbid media.

Funding

National Natural Science Foundation of China (61775163); National Instrumentation Program (2013YQ030915); Young Elite Scientists Sponsorship Program by CAST (2017QNRC001); China Postdoctoral Science Foundation (2016M601260); Director Fund of Qingdao National Laboratory for Marine Science and Technology (QNLM201717).

Acknowledgments

Haofeng Hu acknowledges the Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA) and the China Scholarship Council (CSC).

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		Low density		Medium density		High density
Regions	Method	EME	Michelson’s contrast (%)	EME	Michelson’s contrast (%)	EME	Michelson’s contrast (%)
Mark A	Intensity image	0.46	6.9	0.42	6.5	0.29	6.1
	Schechner’s method	1.52	14.5	1.27	14.1	0.92	11.2
	Our method	8.59	72.9	6.56	63.9	4.95	52.3
Mark B	Intensity image	0.45	10.7	0.40	9.9	0.31	9.1
	Schechner’s method	0.81	12.9	0.66	12.4	0.58	10.6
	Our method	3.64	44.1	3.41	40.7	3.09	34.1

Polarimetric image recovery in turbid media employing circularly polarized light

Abstract

1. Introduction

2. Traditional method of polarimetric image recovery, based on linearly polarized light

3. Polarimetric dehazing method combining circularly polarized light and linearly polarized light

3.1 Image recovery with linearly polarized light

3.2 Image recovery with circularly polarized light

4. Experiments of underwater imaging

5. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (1)

Equations (16)

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