Underwater biomimetic orientation method using imaging polarization sensor based on direct sunlight compensation

Guangmin Li; Ya Zhang; Shiwei Fan; Fei Yu

doi:10.1364/OE.520710

1. Introduction

The underwater unmanned platforms such as underwater unmanned vehicles (UUV), autonomous underwater robots, torpedoes, and frogman carriers play a pivotal role as essential tools for humanity in exploring and exploiting the ocean [1–3]. These platforms are capable of discreetly executing various military or civilian tasks, contributing to a wide range of applications in both maritime defense and civilian sectors [4]. The robustness of navigation systems is a crucial assurance for the smooth execution of tasks by underwater unmanned platforms. The traditional navigation methods have corresponding shortcomings [5–7]. However, biomimetic polarization navigation can utilize imaging polarization sensors to project polarized light scattered from the atmosphere through wide-angle lenses and polarizers onto complementary-metal-oxide semiconductor (CMOS) sensors, and convert the light intensity information into the heading of underwater unmanned platforms. This approach offers advantages such as full autonomy, resistance to interference, and absence of accumulated errors [8–10], which is an emerging and up-and-coming underwater navigation method.

In underwater scenarios, Waterman [11] conducted extensive field measurements in various marine environments. In clear weather conditions, the E-vector patterns correlated with the solar vector can still be detected at a depth of 200 meters, which provides experimental evidence for underwater biomimetic heading measurements of IPS. Powell [12] constructed a refraction transmission model for underwater polarized light and utilized the K-nearest neighbors regression method to solve for the solar vector. Heading measurement experiments were conducted worldwide, resulting in a root mean square error (RMSE) for heading measurements of 6$^{\circ }$. Dupeyroux [13] improved the optical path structure of a point-source polarization sensor using ultraviolet filters, and the mean heading error reached 1$^{\circ }$ with a standard deviation of 4$^{\circ }$. Although the underwater imaging polarization sensor built by Hu [14] considers air-water refraction, the influence of direct sunlight interference on underwater polarization orientation is still not considered. The measured orientation error in the static experiment is 1.3$^{\circ }$. However, these methods have not been tested in actual underwater dynamic scenarios. Cheng [15,16] employed an UUV for polarization orientation tests at different water depths. The mean square error (MSE) of heading at a depth of 5 meters was 16.57$^{\circ }$. It can be seen that due to the lack of consideration of polarization sensor detection model errors and environmental stray light interference in the above methods, significant errors still persist in underwater polarization orientation results.

In the study of polarization detection models, Fan [17] considered the consistency error of IPS light intensity and the installation angle error of polarizers. Ren [18] improved the IPS model by introducing the extinction ratio error of polarizers, thereby further enhancing the accuracy of the angle of polarization (AOP) detection. Li [19] proposed a field calibration method based on the Berry model, improving the directional robustness in scenarios with severe multiple scattering. Wan [20] further considered the distortion effect of the optical system in imaging PS and constructed the Muller matrix of the optical system to improve the accuracy of AOP. However, the above methods did not account for the impact of direct sunlight interference on the polarization channels. Liu [21] introduced a point-source polarization sensor detection model that overcomes direct sunlight interference, enhancing the sensor’s adaptability to different solar elevation angles. However, this model did not consider the influence of air-water interface refraction on the polarization detection model. Due to lens distortion, the optical structure of IPS is different from the point-source polarization sensor, resulting in the inapplicability of the detection model. Moreover, the least squares method for solving polarization state makes it difficult to meet the real-time calculation requirements of IPS with millions of pixel resolutions. Additionally, some robust polarization orientation methods considering adverse scenarios also neglect direct sunlight interference [10,22–25], significantly affecting the accuracy of heading measurements.

In summary, the traditional underwater polarization orientation methods do not consider the influence of direct sunlight interference. Moreover, the existing point-source direct sunlight compensation method has the problems of poor model applicability and low computational efficiency for underwater imaging polarization sensors. Therefore, this paper proposes an underwater biomimetic orientation method using an imaging polarization sensor based on direct sunlight compensation. The main contributions are as follows:

1) The underwater IPS detection model based on direct sunlight compensation is proposed. On the basis of the underwater polarization transmission model, the weight coefficient of polarized light intensity direct sunlight is used to compensate for the influence of underwater direct sunlight intensity on polarization channel detection, thereby improving the accuracy of the underwater polarization detection model;
2) An analytical solution method for the polarization state information of underwater IPS is derived, and a linear equation for solving the weight coefficients of direct sunlight is constructed using augmented Stokes vectors to improve the computational efficiency;
3) The underwater polarization orientation experimental platform is established and the algorithm’s effectiveness is validated through both simulation experiments and actual underwater experiments.

The remaining structure of this paper is organized as follows: Section 2 introduces the traditional underwater polarization detection models and analyzes the AOP inaccuracy caused by direct sunlight interference. Section 3 presents the proposed underwater polarization orientation method. Section 4 details the simulations and experiments of underwater polarization orientation, and discusses the results. Section 5 summarizes the work of this article.

2. Traditional underwater polarization detection model

The accuracy of IPS detection directly impacts the precision of underwater polarization orientation, necessitating the establishment of an accurate underwater polarization detection model.

First, the coordinate systems are defined. The $e$-frame represents the "East-North-Up" navigation coordinate system; the $b$-frame denotes the body coordinate system of the carrier; and the $c$-frame signifies the sensor coordinate system, typically assumed to coincide with the $b$-frame; the $l$-frame represents the horizontal reference coordinate system, obtained by compensating the $b$-frame with pitch and roll angles. In traditional underwater polarization navigation, light refracts when passing through the interface of air and water, resulting in a Snell’s window. As a result, the atmospheric scattered light in the celestial hemisphere converges within an angle of 97.5$^{\circ }$ and incident on the polarization sensor. The full-sky polarization pattern will also be compressed within Snell’s window. The normal direction $\mathbf {n}$ of the refraction interface is perpendicular to the horizontal plane. The incident light after scattering through the atmosphere is defined as $\mathbf {i}$, and the transmitted light after refraction at the air-water interface is $\mathbf {t}$, as illustrated in Fig. 1(a).

Fig. 1. Illustration of the underwater polarization detection process. a) Refraction process of atmospheric scattered light at the water-air interface; b) Mechanism of direct sunlight influence.

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The Stokes vectors are used to describe the polarization states of incident light and transmitted light, which are represented as ${{\mathbf {S}}{i}}={{\left ( {{I}_{i}},{{Q}_{i}},{{U}_{i}},{{V}_{i}} \right )}^{T}}$ and ${{\mathbf {S}}{t}}={{\left ( {{I}_{t}},{{Q}_{t}},{{U}_{t}},{{V}_{t}} \right )}^{T}}$ respectively. Here, $I$ represents the total light intensity, $Q$ denotes the intensity difference of horizontal polarization light, $U$ represents the intensity difference of vertical polarization light, and $V$ signifies the intensity difference of circular polarization light. In natural light, $V$ is usually negligible. Defining ${{\theta }_{i}}$ and ${{\theta }_{t}}$ as the angles of incidence and refraction with respect to the normal of the refraction interface, the polarization transfer Mueller matrix for a calm water surface is expressed as a function of ${{\theta }_{i}}$ and ${{\theta }_{t}}$:

(1)$${\bf{M}}({\theta _i},{\theta _t}) = \frac{1}{2}\frac{{{n_t}\cos {\theta _t}}}{{{n_i}\cos {\theta _i}}}\left( {\begin{array}{cccc} {t_s^2 + t_p^2} & {t_s^2 - t_p^2} & 0 & 0\\ {t_s^2 - t_p^2} & {t_s^2 + t_p^2} & 0 & 0\\ 0 & 0 & {2{t_s}{t_p}} & 0\\ 0 & 0 & 0 & {2{t_s}{t_p}} \end{array}} \right)$$

where ${{n}_{i}}$ and ${{n}_{t}}$ represent the refractive indices of the incident medium and refractive medium, respectively. ${{t}_{s}}$ and ${{t}_{p}}$ denote the transmittance in the vertical and parallel directions, and their calculation formulas are given by:

(2)$${t_s} = \frac{{2{n_i}\cos {\theta _i}}}{{{n_i}\cos {\theta _i} + {n_t}\cos {\theta _t}}},{t_p} = \frac{{2{n_i}\cos {\theta _i}}}{{{n_t}\cos {\theta _i} + {n_i}\cos {\theta _t}}}$$

Therefore, defining the observation vector of a single pixel of the polarization sensor in the $n$-frame as $\mathbf {OP}$. ${{\theta }_{2}}$ represents the angle of light exiting the air-water interface, which is equal to the angle of incidence (observed vector zenith angle ${{\theta }_{t}}$) of underwater light onto the IPS. Therefore, the zenith angle can be used as the exit angle ${{\theta }_{t}}$. Subsequently, based on the Fresnel formula, the incident angle can be calculated as ${{\theta }_{i}}=\arcsin ({{n}_{t}}\sin {{\theta }_{t}}/{{n}_{i}})$, and the Mueller matrix $\mathbf {M}({{\theta }_{i}},{{\theta }_{t}})$ can be determined. Thus, the polarization state of scattered light before refraction can be represented as:

(3)$${{\bf{S}}_i} = {\bf{M}}{({\theta _i},{\theta _t})^{ - 1}} \cdot {{\bf{S}}_t}$$

${{\mathbf {S}}_{t}}$ represents the measured Stokes vector detected by the polarization sensor through the four polarization channels. The mapping function between the intensity values $I_{out}^{k}$ of the polarization channels and ${{\mathbf {S}}_{t}}$ is the polarization detection model. The traditional imaging polarization sensor model considers the impact of errors such as intensity consistency, polarizer installation angle, and extinction ratio on the AOP measurement, which is expressed as:

(4)$$I_{out}^k = {\beta _k}{I_{in}}\left[ {1 + {\eta _k}d\cos \left( {2\xi - 2{{\tilde \alpha }_k}} \right)} \right],k = 1,2,3,4$$

${{\beta }_{k}}$ is the intensity response parameter for the $k$-th polarization channel, ${{\tilde {\alpha }}_{k}}={{\alpha }_{k}}+\delta {{\alpha }_{k}}$ represents the polarizer installation angle with errors, ${{\eta }_{k}}$ is the extinction ratio consistency error parameter. $\xi$ is the polarization angle, and $d$ is the degree of polarization. $I_{out}^{k}$ is the output intensity after passing through the polarizer for the input intensity ${{I}_{in}}$. Thus, the Stokes vector of refracted light that ignores the influence of circularly polarized light can be obtained:

(5)$${{\bf{S}}_t} = {\left[ {I_{out}^1 + I_{out}^3,I_{out}^1 - I_{out}^3,I_{out}^2 - I_{out}^4} \right]^T}$$

From ${{\mathbf {S}}{t}}$, the polarization angle and degree of polarization information of the refracted light can be inversely deduced. Although the traditional model improves the detection accuracy of polarized light formed by natural light scattering, it does not consider the coupling effect between direct sunlight and polarized light on the detection of the polarization channels. As shown in Fig. 1(b), when sunlight enters the atmosphere, part of the light is scattered by atmospheric scattering, forming the full-sky polarization pattern, while another part directly enters the sensor as direct sunlight without polarization effects. The detection process of direct sunlight does not conform to the traditional model. The electric vector will decompose into ${{E}_{1}}$ in the sensor plane and ${{E}_{2}}$ perpendicular to ${{E}_{1}}$. As a result, model errors will be introduced in the calculation of polarization state, seriously affecting the accuracy of polarization orientation. Hence, an underwater polarization orientation method based on direct sunlight compensation is proposed.

3. Underwater polarization orientation based on direct sunlight compensation

3.1 Improved detection model of underwater IPS

Figure 1(a) shows that there is still a depth-related vertical distance between the center of the sensor $O$ and the point of atmospheric scattering light refraction $O'$. The working environment of the polarization sensor is within 200 meters below the water surface [11] with a Snell’s window radius of approximately 200 meters. Therefore, the influence of the refraction position deviation on the heading solution can be neglected. Hence, the observation directions before refraction at different times can still be considered at the same origin $O$, as shown in Fig. 2.

Fig. 2. Illustration of underwater polarization sensor detection process under direct sunlight interference.

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Let $\mathbf {OS}'$ be the direction of the refracted solar vector, and $\mathbf {OP}$ be the direction of the observed vector below the water surface. The electric field vector of the direct sunlight can be decomposed into components ${{\mathbf {E}}_{1}}$ and ${{\mathbf {E}}_{2}}$. ${{\mathbf {E}}_{1}}$ lies on the surface of the polarizer, while ${{\mathbf {E}}_{2}}$ is perpendicular to the plane formed by ${{\mathbf {E}}_{1}}$ and $\mathbf {OS}'$. When ${{\mathbf {E}}_{1}}$ passes through the polarizer, the transmitted light intensity can be expressed as:

(6)$$I_{out}^{{E_1}} = 0.5I_{in}^{{\rm{sun}}}{\cos ^2}\psi$$

$I_{in}^{\text {sun}}$ represents the incident light intensity of direct sunlight, $I_{out}^{{{E}_{1}}}$ represents the transmitted light intensity of ${{\mathbf {E}}_{1}}$, and $\psi$ represents the angle between ${{\mathbf {E}}_{1}}$ and the polarization direction of the polarizer. According to the basic transmission mechanism of polarized light, the transmitted light intensity of the electric field vector ${{\mathbf {E}}_{2}}$ through the polarizer can be expressed as:

(7)$$I_{out}^{{E_2}} = 0.5I_{in}^{{\rm{sun}}}{\left( {\cos \alpha - \frac{{\sin \alpha }}{{\tan \beta }}} \right)^2} = 0.5I_{in}^{{\rm{sun}}} \cdot {P^{{E_2}}}\left( {\theta _s^t,\psi } \right)$$

where

(8)$$\left\{ \begin{array}{l} \cos \alpha ={-} {{\sin \psi } \mathord{\left/ {\vphantom {{\sin \psi } {\sin {\gamma _y}}}} \right.} {\sin {\gamma _y}}}\\ \cos \beta = {{ - \sin 2\psi {{\sin }^2}{\theta _s}} \mathord{\left/ {\vphantom {{ - \sin 2\psi {{\sin }^2}{\theta _s^t}} {(2\sin {\gamma _y}\sin {\gamma _x})}}} \right.} {(2\sin {\gamma _y}\sin {\gamma _x})}} \end{array} \right.$$

$\theta _s^t$ denotes the solar vector zenith angle after refraction, and $\psi$ is the angle between ${{\mathbf {E}}_{\mathbf {1}}}$ and polarization direction. ${\gamma }_{x}$ represents the angle between the incident light and the x-axis, and ${\gamma }_{y}$ represents the angle between the incident light and the y-axis. Therefore, the square term in Eq. (7) can be expressed as a function of $\psi$ and $\theta _{s}^{t}$, characterized by ${P}^{{{E}_{2}}}(\theta _{s}^{t},\psi )$. As $\theta _s^t$ tends towards 0, ${\gamma }_{x}$ and ${\gamma }_{y}$ are approaching 90${}^\circ$. $\theta _s^t$ satisfies Fresnel formula with the solar vector zenith angle $\theta _{s}^{i}$ before refraction:

(9)$$\theta _s^t = \arcsin ({n_i}\sin \theta _s^i/{n_t})$$

$\theta _{s}^{i}$ can be obtained from the carrier’s position and solar ephemeris. Point-source polarized sensors directly detect the polarization information in the direction of the CMOS chip’s normal. In contrast, for an imaging polarization sensor, as shown in Fig. 3. When direct sunlight passes through the wide-angle lens, the light transmission path changes. Subsequently, after passing through the lens hood, the direct sunlight incidents into the polarizing plate array and the CMOS chip in a nearly vertical direction. At this point, the expression for output light intensity of ${{\mathbf {E}}_{\mathbf {2}}}$ is $I_{out}^{{{E}_{2}}}\approx 0.5I_{in}^{sun}{{\sin }^{2}}\psi$, enabling the derivation of the polarization response intensity expression for direct sunlight:

(10)$$\mathop {I_{out}^{sun}}_{{\gamma _s} \to 0} {\mkern 1mu} = I_{out}^{{E_1}} + I_{out}^{{E_2}} \approx 0.5I_{in}^{{\rm{sun}}}({\cos ^2}\psi + {\sin ^2}\psi ) = 0.5I_{in}^{{\rm{sun}}}$$

However, in the actual measurement process, $I_{in}^{sun}$ is usually a part of the total light intensity and varies with the azimuth angle $\psi$. Therefore, by analogy with Eq. (6), $I_{out}^{sun}$ is redefined:

(11)$$I_{out}^{sun} = 0.5{I_{in}}{k_m}{\cos ^2}\psi = {I_{in}}{k_m}P$$

where $P$ represents the polarization response coefficient of direct sunlight, which is related to $\psi$.

Fig. 3. Principle diagram of direct sunlight detection for underwater IPS.

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Combining with Eq. (4), the single-channel light intensity response model of the underwater four-channel polarization unit can be established:

(12)$$I_{out}^k = \frac{1}{2}{\beta _k}[1 + {I_{in}}{\eta _k}d\cos (2\xi - 2{\tilde \alpha _k})] + 0.5{\beta _k}{I_{in}}(1 - {\eta _k}d){k_m}({P_k} - 0.5)$$

The corresponding error-free installation angles for the polarizers corresponding to ${{\tilde {\alpha }}_{k}}$ are 0$^{\circ }$, 45$^{\circ }$, 90$^{\circ }$, and 135$^{\circ }$; ${{k}_{m}}$ is the weight of direct sunlight under non-polarized light; ${{P}_{k}}$ represents the polarization response coefficient of direct sunlight in channel $k$, which can be calculated by:

(13)$${P_k} = 0.5{\cos ^2}{\psi _k}$$

${{\psi }_{k}}$ is the angle between the azimuthal angle of ${{\mathbf {E}}_{1}}$ and the direction of the polarizer ${{\tilde {\alpha }}_{k}}$. The azimuthal angle of ${{\mathbf {E}}_{1}}$ differs from the azimuthal angle $\alpha _{s}^{b}$ of the solar vector in the carrier system by $\pi /2$. Therefore, defining counterclockwise as positive, the expression for ${{\psi }_{k}}$ is given by:

(14)$${\psi _k} = (\alpha _s^b - \pi /2) - {\tilde \alpha _k}$$

After rearranging Eq. (12), the final underwater IPS detection model under direct sunlight interference can be obtained as follows:

(15)$$\left\{ \begin{array}{c} I_{out}^k = \frac{1}{2}{\beta _k}{I_{in}}[1 + {\eta _k}d\cos (2\xi - 2{{\tilde \alpha }_k})] + I_{ou{t_k}}^{sun}\\ I_{ou{t_k}}^{sun} = 0.5{\beta _k}{I_{in}}(1 - {\eta _k}d){k_m}({P_k} - 0.5) \end{array} \right.$$

Compared to the traditional model, the improved imaging PS detection model additionally considers the impact of direct sunlight. $I_{out_{k}}^{sun}$ represents the intensity response value of direct sunlight transmitted through the polarizing plate to the CMOS chip. Therefore, the environmental adaptability of the improved model can be enhanced. Moreover, when the quality of direct sunlight ${k_m}$ is zero, the improved IPS model degenerates to the traditional model. It is worth noting that the solar azimuth angle $\alpha _{s}^{b}$ in the carrier system in Eq. (14) can be obtained by the solar azimuth angle $\alpha _{s}^{e}$ in $e$-frame and the approximate heading of the carrier.

(16)$$\alpha _s^b = \alpha _s^e - headin{g_n}$$

$headin{{g}_{n}}$ is the rough heading value output by other navigation systems.

3.2 Analytical solution for polarization state based on the improved detection model

After establishing the detection model of the imaging polarization sensor, it is necessary to further combine the response light intensity value of the measured polarization channel to solve the polarization state. Therefore, it is necessary to establish the mapping model between detection value $I_{m}^{k}$ and the Stokes vector ${{\mathbf {S}}_{t}}$ of underwater refracted light. The traditional polarization state calculation method based on Model 1 is:

(17)$$\begin{array}{l} {{\bf{S}}_t} = {[{I_{in}},{I_{in}}d\cos 2\xi ,{I_{in}}d\sin 2\xi ]^T}\\ = {[I_m^1 + I_m^3,I_m^3 - I_m^1,I_m^4 - I_m^2]^T} \end{array}$$

${\{ I_m^k\} _{k = 1,2,3,4}}$ represent the measured values of the single-channel polarization intensity. The initial polarization angle $\xi$ and the degree of polarization $d$ can be calculated as:

(18)$$\xi = \arctan ({{\bf{S}}_t}(2),{{\bf{S}}_t}(1))/2,d = \sqrt {{{\bf{S}}_t}{{(1)}^2} + {{\bf{S}}_t}{{(2)}^2}} /{{\bf{S}}_t}(0)$$

Although this model does not consider the impact of direct sunlight, we can use the calculated degree of polarization and incident light intensity as prior information for this algorithm. For the underwater IPS model under direct sunlight interference, Eq. (15) is expanded as:

(19)$$\begin{array}{l} \hat I_{out}^k = {I_{in}} + {\eta _k}\cos 2{{\tilde \alpha }_k} \cdot {I_{in}}d\cos 2\xi + \\ {\rm{ }}{\eta _k}\sin 2{{\tilde \alpha }_k} \cdot {I_{in}}d\sin 2\xi + {{\hat P}_k} \cdot {I_{in}}{k_m} \end{array}$$

where $\hat I_{out}^k = I_{out}^k/{\beta _k}$ and ${\hat P_k} = 0.5(1 - {\eta _k}d)({P_k} - 0.5)$. The solution process for the polarization state involves determining the parameters $\{ {\beta _k},{\eta _k},{\tilde \alpha _k},{k_m}|k = 1,2,3,4\}$, and ensuring these parameters satisfy the target function for the polarization channel response:

(20)$$\mathop {\min }_{\{ {\beta _k},{\eta _k},{{\tilde \alpha }_k},{k_m}\} } \sum_{k = 1}^4 {{{\left\| {I_m^k - I_{out}^k} \right\|}^2}}$$

The parameters $\{ {\beta _k},{\eta _k},{\tilde \alpha _k}\}$ can be obtained through pre-calibration, while ${{k}_{m}}$ is primarily determined by the interference of direct sunlight, closely related to the carrier environment, and requires real-time solution. Therefore, a detailed derivation is performed for the solution process of ${{k}_{m}}$ and the polarization state. The Stokes vector is augmented and Eq. (19) is rewritten into linear matrix form.

(21)$${{\bf{\hat S}}_t} = {\left[ {\begin{array}{cccc} {{I_{in}}} & {{I_{in}}d\cos 2\xi } & {{I_{in}}d\sin 2\xi } & {{I_{in}}{k_m}} \end{array}} \right]^T}$$

(22)$$F = \left[ {\begin{array}{c} {\hat I_{out}^1}\\ {\hat I_{out}^2}\\ {\hat I_{out}^3}\\ {\hat I_{out}^4} \end{array}} \right],D = \left[ {\begin{array}{cccc} 1 & {{\eta _1}\cos 2{{\tilde \alpha }_1}} & {{\eta _1}\sin 2{{\tilde \alpha }_1}} & {{{\hat P}_1}}\\ 1 & {{\eta _2}\cos 2{{\tilde \alpha }_2}} & {{\eta _2}\sin 2{{\tilde \alpha }_2}} & {{{\hat P}_2}}\\ 1 & {{\eta _3}\cos 2{{\tilde \alpha }_3}} & {{\eta _3}\sin 2{{\tilde \alpha }_3}} & {{{\hat P}_3}}\\ 1 & {{\eta _4}\cos 2{{\tilde \alpha }_4}} & {{\eta _4}\sin 2{{\tilde \alpha }_4}} & {{{\hat P}_4}} \end{array}} \right]$$

Equations (21) and (22) satisfy $D \cdot {{\bf {\hat S}}_t} = F$. Therefore, Eq. (20) can be solved by the least squares method to obtain the optimal Stokes vector:

(23)$${{\bf{\hat S}}_t} = {({D^T}D)^{ - 1}} \cdot {D^T}F$$

By once again utilizing Eq. (18), the Stokes vector of a single polarization channel can be solved by considering the interference of direct sunlight. However, for IPS with millions of polarization channels, the matrix inversion operation in Eq. (23) can be significantly time-consuming. Therefore, it is necessary to derive an analytical solution of ${{\bf {\hat S}}_t}$. Neglecting the influence of installation angle error $\delta {{\alpha }_{k}}$ and extinction ratio error ${{\eta }_{k}}$ of polarizers, the installation angle ${{\tilde {\alpha }}_{k}}=\{0{}^\circ,45{}^\circ,90{}^\circ,135{}^\circ \}$ is substituted into Eq. (19) to obtain the equation system:

(24)$$\left\{ \begin{array}{l} \hat I_{out}^1 = {I_{in}} + {I_{in}}d\cos 2\xi + {{\hat P}_1}{I_{in}}{k_m}\\ \hat I_{out}^2 = {I_{in}} + {I_{in}}d\sin 2\xi + {{\hat P}_2}{I_{in}}{k_m}\\ \hat I_{out}^3 = {I_{in}} - {I_{in}}d\cos 2\xi + {{\hat P}_3}{I_{in}}{k_m}\\ \hat I_{out}^4 = {I_{in}} - {I_{in}}d\sin 2\xi + {{\hat P}_4}{I_{in}}{k_m} \end{array} \right.$$

The weight of non polarized direct sunlight ${{k}_{m}}$ can be calculated:

(25)$${k_m} = {k_c}\frac{{(\hat I_{out}^1 + \hat I_{out}^2 + \hat I_{out}^3 + \hat I_{out}^4) - 4{I_{in}}}}{{{I_{in}} \cdot ({{\hat P}_1} + {{\hat P}_2} + {{\hat P}_3} + {{\hat P}_4})}}$$

where ${{k}_{c}}$ is a constant, representing the theoretical error compensation coefficient for the analytical method relative to the least squares method. This paper takes an empirical value of 20. Then the light intensity of the polarization channel after direct sunlight compensation can be obtained:

(26)$$\hat I_m^k = I_m^k - {\hat P_k}{I_{in}}{k_m},k = 1,2,3,4$$

The Stokes vector after direct sunlight compensation can be obtained:

(27)$${{\bf{\hat S}}_t} = {\left[ {\begin{array}{ccc} {\hat I_m^1 + \hat I_m^3} & {\hat I_m^3 - \hat I_m^1} & {\hat I_m^4 - \hat I_m^2} \end{array}} \right]^T}$$

Additionally, in underwater polarized light navigation, it is necessary to convert the detected polarization information of the refracted light $\mathbf {t}$ into the polarization information of the incident light $\mathbf {i}$.

(28)$${{\bf{\hat S}}_i} = {\bf{M}}{({\theta _i},{\theta _t})^{ - 1}} \cdot {{\bf{\hat S}}_t}$$

The final AOP measurement value after underwater direct sunlight compensation can be obtained as $\hat {\xi }=0.5\arctan ({{\mathbf {\hat {S}}}_{i}}(2),{{\mathbf {\hat {S}}}_{i}}(1))$. The direct sunlight compensation flow for AOP images is shown in Fig. 4. The polarization channel simulation data under error interference is used for direct sunlight compensation, and the initial AOP and DOP are used to calculate the error parameters and get the compensated output light intensity value. Finally, the underwater AOP image with error suppression is solved by the Stokes vector.

Fig. 4. The direct sunlight compensation flow for AOP images.

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3.3 Underwater polarization orientation

Once the AOP image compensated for underwater direct sunlight is obtained, the underwater polarization heading can be determined, as shown in Fig. 5. On the basis of obtaining the underwater IPS polarization channel and prior polarization state, an improved polarization detection model and analytical solution method are used for direct sunlight compensation, achieving a significant improvement in polarization orientation accuracy at different solar altitude angles. The specific steps of underwater polarization orientation are as follows.

Fig. 5. Flow chart of underwater polarization orientation based on direct sunlight compensation.

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Assuming the number of effective pixel points for the polarized sensor is $J$, the observed vectors in the sensor coordinate system after camera intrinsic distortion calibration are denoted as $\{ {\bf {OP}}_j^c|j = 1,\ldots,J\}$. Then, The pre-refraction observation vectors in $l$-frame after real-time attitude compensation are represented as $\{ {\bf {OP}}_j^l|j = 1,\ldots,J\}$ [26], with observation azimuth angles $\{ \varphi _i^{l,j}|j = 1,\ldots,J\}$. The corresponding zenith angles before refraction are $\{ \theta _i^{l,j}|j = 1,\ldots,J\}$, and the refracted zenith angles below the water surface are $\{ \theta _t^{l,j}|j = 1,\ldots,J\}$. The Mueller matrix corresponding to the $j$-th observation vector is denoted as ${\bf {M}}(\theta _i^{l,j},\theta _t^{l,j})$. The compensated AOP values after direct sunlight compensation can be obtained using Eq. (28) as $\{ {\hat \xi _j}|j = 1,\ldots,J\}$. To calculate the solar vector ${{\mathbf {s}}^{l}}$ by the Rayleigh scattering model, the expression of the polarization vector ${{\mathbf {e}}^{l}}$ in $l$-frame needs to be calculated:

(29)$${{\bf{e}}^l} = {\bf{P}}{{\bf{E}}^l} = \cos {\hat \xi _j}{\bf{P}}{{\bf{X}}^l} + \sin {\hat \xi _j}{\bf{P}}{{\bf{Y}}^l}$$

$\mathbf {P}{{\mathbf {X}}^{l}}$ and $\mathbf {P}{{\mathbf {Y}}^{l}}$ are the coordinate basis vectors and can be expressed as:

(30)$$\left\{ \begin{array}{l} {\bf{P}}{{\bf{X}}^l} = {(\begin{array}{ccc} {\cos \varphi _i^{l,j}\sin \theta _i^{l,j}} & {\sin \varphi _i^{l,j}\sin \theta _i^{l,j}} & { - \cos \theta _i^{l,j}} \end{array})^T}\\ {\bf{P}}{{\bf{Y}}^l} = {(\begin{array}{ccc} { - \sin \varphi _i^{l,j}} & {\cos \varphi _i^{l,j}} & 0 \end{array})^T} \end{array} \right.$$

Substituting Eq. (30) into Eq. (29) allows for the calculation of the polarized vector ${{\mathbf {e}}^{l}}$. Furthermore, based on the single Rayleigh scattering model, the polarization vector of the scattered light is perpendicular to the scattering plane. Similarly, the polarization vector is perpendicular to the solar vector ${{\mathbf {s}}^{l}}$, denoted as ${{({{\mathbf {e}}^{l}})}^{T}}{{\mathbf {s}}^{l}}={{\mathbf {e}}^{l}}{{({{\mathbf {s}}^{l}})}^{T}}$. Therefore, defining $\mathbf {E}=[\mathbf {e}{1}^{l}\cdots \mathbf {e}{J}^{l}]$, and ${\mathbf {E}}^{T}{{\mathbf {s}}^{l}}=0$. The solution for the solar vector in the $l$-frame can be formulated as the following optimization problem:

(31)$${\min _s}\left( {{{\left( {{{\bf{s}}^l}} \right)}^T}{\bf{E}}{{\bf{E}}^T}{{\bf{s}}^l}} \right),{\rm{ s}}{\rm{.t}}{\rm{. }}{\left( {{{\bf{s}}^l}} \right)^T}{{\bf{s}}^l} = 1$$

The optimal estimate for the solar vector is given by the eigenvector corresponding to the minimum eigenvalue of the matrix $\mathbf {E}{{\mathbf {E}}^{T}}$. This can be obtained through singular value decomposition (SVD). The solar meridian direction in the $l$-frame is calculated as $\alpha _{sun}^{l}=\arctan (\mathbf {\lambda }\text { }(2)/\mathbf {\lambda }\text { }(1))$, where $\mathbf {\lambda }(1)$ and $\mathbf {\lambda }(2)$ are the first and second elements of the vector $\mathbf {\lambda }$, respectively. The solar zenith angle ${{\theta }{s}}$ and azimuth angle ${{\varphi }_{s}}$ in $n$-frame can be determined based on the local time and location. The heading is then calculated by computing the difference between the theoretical azimuth angle in $e$-frame $\alpha _{s}^{e}$ and the measured value in $l$-frame $\alpha _{sun}^{l}$, denoted as:

(32)$$heading = \alpha _s^e - \alpha _{sun}^l{\rm{\quad or \quad}}heading = \alpha _s^e - \alpha _{sun}^l + \pi$$

The solution for the heading angle $heading$ has a 180$^{\circ }$ ambiguity, which can be resolved by the rough heading values $headin{{g}_{n}}$ output by other navigation sensors. The final underwater polarized orientation algorithm based on direct sunlight compensation is outlined in Table 1.

Table 1. Process of underwater polarization orientation algorithm based on direct sunlight compensation.

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4. Experimental results and discussion

This section verifies the effectiveness of the proposed underwater polarization orientation method based on direct sunlight compensation through simulation and underwater experiments. Use the final heading measurement accuracy as the evaluation indicator.

4.1 Simulation

Firstly, simulate the underwater IPS output under the interference of direct sunlight, and perform polarization orientation based on different sensor detection models to verify the effectiveness of the proposed method. The specific simulation parameter settings are as shown in Table 2. According to the factory calibration of the polarized camera, set the consistency parameters of light intensity response ${{\beta }_{k}}$, extinction ratio consistency parameter ${{\eta }_{k}}$, and installation deviation of the polarizer ${{\tilde {\alpha }}_{k}}$. The camera’s equivalent focal length $[{{f}_{c,x}},{{f}_{c,y}}]$, principal point position $[{{k}_{c,x}},{{k}_{c,y}}]$, and fisheye lens distortion parameters $[{{D}_{0}},{{D}_{1}},{{D}_{2}},{{D}_{3}}]$ are also set based on prior calibration values. The underwater direct sunlight weight parameter ${{k}{m}}$ is set to 0.3. Finally, set the solar zenith angle $\theta _{s}^{n}$, azimuth angle $\alpha _{s}^{n}$, and the geographical latitude and longitude of the carrier.

Table 2. Parameters for simulation settings.

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By setting the rotation angle of the underwater IPS to 720 $^{\circ }$ within 100 seconds, multiple polarization channels can output light intensity under direct sunlight interference. The experimental setting of the solar azimuth angles is $[0^{\circ } ,60^{\circ } ,120^{\circ } ,180^{\circ } ,240^{\circ } ,300^{\circ } ]$, and that of the zenith angles are $[45^{\circ } ,55^{\circ } ,60^{\circ } ,65^{\circ } ,70^{\circ } ,85^{\circ } ]$. The output results of the central region within the Snell’s window are shown in Fig. 6

Fig. 6. Simulation results of underwater polarization channels under direct sunlight interference.

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Under different polarization directions, the amplitude and phase of underwater IPS imaging are significantly disturbed by direct sunlight interference. Different underwater orientation methods are then employed for polarization heading calculation. The traditional method uses the detection model that does not consider the impact of direct sunlight interference [14,19,20,27], in which [14] realized underwater polarization orientation. The proposed method realizes underwater polarization orientation based on direct sunlight compensation. In the proposed method, a normal distribution random noise with a mean of 0$^{\circ }$ and a variance of 0.5$^{\circ }$ is added to the heading reference value as the prior heading ($headin{{g}_{n}}$). In the simulation experiments, the heading measurement results of different methods are shown in Fig. 7. It can be seen that the PS model with direct sunlight compensation effectively suppresses the impact of direct sunlight on the detection intensity of the polarization channel. Compared to the traditional model, the heading measurement results of the proposed model are closer to the reference heading.

Fig. 7. Heading results of simulation experiment under different solar zenith angles.

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The heading errors of different underwater polarization detection models are shown in Fig. 8. Under direct sunlight interference, the traditional method exhibits significant fluctuation errors in heading measurement due to the lack of corresponding error suppression mechanisms. However, the proposed method based on underwater direct sunlight compensation suppresses direct sunlight interference by improving the underwater sensor detection model and deriving an error compensation algorithm of polarization channel intensity. This enhances the accuracy of AOP measurements, thereby reducing the heading errors of underwater IPS. The heading RMSE is shown in Table 3. Compared to the traditional method, the proposed polarization orientation method can significantly improve heading measurement accuracy through direct sunlight compensation. Subsequent underwater experiments will further validate the progressiveness of the proposed algorithm through practical underwater experiments.

Fig. 8. Heading errors of simulation experiment under different solar zenith angles.

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Table 3. Heading RMSE ($^{\circ }$) of simulation experiment under different solar zenith angles.

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4.2 Underwater polarization orientation experiment

This section validates the performance of the proposed model under actual underwater direct sunlight interference. The equipments used in the underwater experiment are shown in Fig. 9. The IPS consists of a polarized camera and a fisheye lens. The polarization sensor is rigidly attached to a small turntable along with the fiber optic inertial navigation system (FINS) and is supported by an aluminum alloy bracket. A polyethylene container is placed above the bracket, filled with water to a depth of 30cm, allowing sunlight scattering to be detected by the polarization sensor through the transparent bottom of the container. For the effect of water-glass-air interface refraction in front of the polarization sensor, we used the single refraction imaging model of the underwater camera [28] for underwater image correction. The relative rotation of FINS is used as the reference for the PS relative heading. The orientation accuracy of FINS can reach 0.02$^{\circ }/secL$, and this heading accuracy can serve as the reference for polarization orientation. The specific parameters of the equipment are shown in Table 4. The experiment was conducted on September 12, 2023, near Harbin (longitude 126.7264$^{\circ }$, latitude 45.6234$^{\circ }$, altitude 148.74m).

Fig. 9. Experimental platform for underwater polarization orientation .

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Table 4. Equipment models and parameters.

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During the experimental process, the FINS was initially aligned, and then the turntable was controlled to rotate. Underwater full-sky polarization images were captured at different azimuths and time intervals throughout the day. The AOP detection results of the underwater polarization sensor are shown in Fig. 10. The quality of the detected AOP images is significantly poorer when the time is near noon, as shown in Fig. 10( a), (b), and (c). This is because during the noon period, the solar zenith angle is lower, and direct sunlight is stronger, resulting in significant noise interference in the solar meridian detected by the sensor. Therefore, the solar zenith angle of (c) is the smallest, and the AOP image interference is the most obvious. In this study, different polarization orientation methods were used to solve the heading from actual underwater data.

Fig. 10. AOP detection results of underwater IPS.

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The heading results of different models in underwater experiments are shown in Fig. 11, and the heading errors are shown in Fig. 12. Both the traditional method and the proposed method perform well in measuring polarization heading when the solar zenith angle is large (as shown in the second row of Fig. 11). However, as the sun rises and the zenith angle decreases (as shown in the first row of Fig. 11), the traditional method is affected by direct sunlight interference, resulting in a significant deviation of the heading measurement from the reference value. The variation trend of heading deviation is also related to the orientation of the carrier, consistent with the simulation results. However, proposed method considers the influence of direct sunlight and is closer to the reference heading in actual underwater experiments, with smaller heading errors. The heading errors of the detection model based on direct sunlight compensation are significantly smaller than those of the traditional model under different solar zenith angles.

Fig. 11. Heading results of underwater polarization orientation experiments.

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Fig. 12. Heading errors of underwater polarization orientation experiments.

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The heading RMSE for different methods is shown in Table 5. At different solar zenith angles, compared with the traditional method, the proposed method has an average reduction of 92.53${\% }$ in heading error. This validates the progressiveness of the proposed orientation method for underwater IPS based on direct sunlight compensation and suggests broad application prospects.

Table 5. The heading RMSE ($^{\circ }$) of underwater polarization orientation experiments.

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Furthermore, from Fig. 11 and Fig. 12, the heading errors in the first row are significantly larger than those in the second row for both of the traditional method and the proposed method. This is because the solar zenith angles in the first row are smaller, resulting in sufficient sunlight intensity. In addition to direct sunlight interference, the underwater imaging polarized sensor is also affected by environmental reflections and multiple atmospheric scatterings. However, relative to the traditional method, it still significantly improves orientation accuracy.

4.3 Computational efficiency experiment

The computational efficiency experiment of the direct sunlight compensation algorithm is further conducted. Utilizing the six sets of experimental data for underwater orientation mentioned above, the processing time for direct sunlight compensation of single-frame image using different methods is statistically analyzed, as shown in Table 6. The traditional method adopts an optimization approach based on least squares to solve error parameters, requiring iterative approximation using the Jacobian matrix, resulting in low algorithm efficiency. The average processing time for single-frame images across the six sets was 0.7826 seconds. In contrast, the proposed method enhances algorithmic computational efficiency significantly by numerically solving the direct sunlight weight parameters through constructing a linearized equation for output light intensity. The average processing time for single-frame images across the six sets is 0.0017 seconds, validating the practical application value of the proposed algorithm.

Table 6. Time statistics (s) of direct sunlight compensation processing for single frame images.

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5. Conclusions

This paper focuses on the high-precision and fully autonomous orientation requirements for underwater unmanned platforms and addresses the issue of decreased polarization orientation accuracy caused by underwater direct sunlight interference. An underwater polarization orientation method based on direct sunlight compensation is proposed. The polarization intensity and direct sunlight weight coefficients are utilized to mitigate the impact of direct sunlight on the detected results of polarization channels. The augmented Stokes vector is utilized to construct linear equations for solving the direct sunlight weight coefficients. In the underwater orientation experiments with different solar elevations, the proposed method exhibits an average reduction of 92.53${\% }$ in heading error compared to the traditional method. The results indicate that the proposed method can effectively separates the scattered light intensity detected by the underwater IPS polarization channels from the direct sunlight intensity, improves the accuracy of AOP detection and consequently enhances underwater polarization orientation precision, which demonstrates high practical value in real-world applications.

Further research will be conducted on the integrated navigation method of underwater IPS and other sensors to improve the navigation accuracy and robustness of the integrated system.

Funding

National Natural Science Foundation of China (52071121, 52271311); Heilongjiang Provincial Postdoctoral Science Foundation (LBH-Z22161).

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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Parameters	Values
$I_{i n}$	220 $L x$
$β_{k}$	[1.027, 1.000, 1.038, 1.005]
$η_{k}$	[1, 1, 1, 1]
${\tilde{α}}_{k}$	[0 $^{\circ}$ ,45 $^{\circ}$ ,90 $^{\circ}$ ,135 $^{\circ}$ ]
$k_{m}$	0.3
$[k_{c, x}, k_{c, y}]$	[622.493, 515.056]
$[f_{c, x}, f_{c, y}]$	[259.102, 259.203]
$[D_{0}, D_{1}, D_{2}, D_{3}]$	[0.029 274, −0.013 300, 0.009 758, −0.004 919]
Longitude	126.726 $^{\circ}$
Latitude	45.623 $^{\circ}$
$α_{s}^{n}$	[0 $^{\circ}$ ,60 $^{\circ}$ ,120 $^{\circ}$ ,180 $^{\circ}$ ,240 $^{\circ}$ ,300 $^{\circ}$ ]
$θ_{s}^{n}$ .	[45 $^{\circ}$ ,55 $^{\circ}$ ,60 $^{\circ}$ ,65 $^{\circ}$ ,70 $^{\circ}$ ,85 $^{\circ}$ ]

	1	2	3	4	5	6	Average
Traditional	2.959	0.203	0.169	0.165	0.089	0.068	0.609
Proposed	0.299	0.082	0.07	0.071	0.013	0.040	0.096
Ratio	89.90%	59.61%	58.58%	56.97%	85.39%	41.18%	84.24%

Device name	Device model	Parameters
Lucid polarization camera	PHXET050S-P	Resolution: 2048x2448,Frame rate: 22FPS
Fujinon Fisheye Lens	FE185C057HA-1	Focal length: 1.8mm,view: 185.0 $^{\circ}$ x 185.0 $^{\circ}$
		Bias: 0.02 mg(1 $σ$ ) ( Accelerometer);
FINS	—	Bias: 0.02 $^{\circ}$ / $h$ (1 $σ$ ),
		ARW: 0.002 $^{\circ}$ / $\sqrt{h}$ (Gyro).

	1	2	3	4	5	6	Average
Time	10:15	11:09	14:29	15:27	16:24	17:12	-
Traditional	11.520	11.815	12.250	0.735	0.606	0.726	6.275
Proposed	0.811	0.903	0.540	0.302	0.152	0.105	0.469
Ratio	92.96%	92.36%	95.59%	58.91%	74.92%	85.54%	92.53%

	1	2	3	4	5	6	Average
Traditional	0.7874	0.7706	0.7806	0.8160	0.7783	0.7628	0.7826
Proposed	0.0013	0.0012	0.0020	0.0011	0.0013	0.0032	0.0017

Underwater biomimetic orientation method using imaging polarization sensor based on direct sunlight compensation

Abstract

1. Introduction

2. Traditional underwater polarization detection model

3. Underwater polarization orientation based on direct sunlight compensation

3.1 Improved detection model of underwater IPS

3.2 Analytical solution for polarization state based on the improved detection model

3.3 Underwater polarization orientation

4. Experimental results and discussion

4.1 Simulation

4.2 Underwater polarization orientation experiment

4.3 Computational efficiency experiment

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (6)

Equations (32)

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