Modified Model of Polarized Bidirectional Reflectance Distribution Function Used for Light Detection and Ranging (LiDAR)

Abstract

In order to analyze the performance of a light detection and ranging system based on polarization modulation, it is necessary to theoretically analyze and model the polarization scattering characteristics of common target materials. In this paper, the shortcomings of the classical Hyde pBRDF (polarization bidirectional reflectance distribution function) model are analyzed. Based on the research results of many researchers in recent years, a new six-parameter pBRDF model is proposed. To verify the accuracy of the proposed model, this paper builds a measurement system for the polarization scattering characteristics of the target surface in the laser active imaging scene, and the polarization scattering characteristics of two common materials, namely a white paint coating and an aluminum plate, are measured. Based on the measurement results of the DOP (degree of polarization) of the scattered light of the target material and the BBO-FA (biogeography-based optimization-Firefly algorithm) algorithm, we performed inversion calculations on the key parameters of the target material. Using the parameters of the target material obtained via inversion, we use the model to simulate the Stokes vectors of the target and compare the simulated values of Stokes vectors with the measured values to verify the accuracy of the model. The verification results show that the simulation results of Stokes vectors are in good agreement with the measurement results for these two materials, and the introduction of various improvements to the model can effectively improve the accuracy of the model, which provides a tool for studying the performance parameters of a laser three-dimensional imaging system based on polarization modulation.

1. Introduction

At present, there are many ways to detect space targets. Active laser three-dimensional imaging and passive polarization three-dimensional imaging are two commonly used detection methods. Among them, the laser three-dimensional imaging system based on polarization modulation generally uses an electro-optic modulation crystal as the modulator. By changing the applied voltage of the electro-optic modulation crystal, the polarization state of the reflected laser is modulated, so as to realize the correlation between the distance information and the gray information in the image of the array detector in the system, thus realizing three-dimensional imaging [1,2,3,4,5]. LiDAR based on polarization modulation has the advantages of a short exposure time and flash imaging, which is suitable for target detection [6,7,8]. In the actual application of LiDAR based on polarization modulation, the polarization characteristics of the scattering light are different due to the different surface materials of the target, and the detection effect of the target using LiDAR based on polarization is also different. For the sake of modeling, simulating, and analyzing the performance of LiDAR based on polarization modulation and to improve the performance of the system, it is particularly important to model and simulate the polarization scattering characteristics of typical target surfaces.

The pBRDF models based on the BRDF models are common models for studying the polarization scattering characteristics of targets. Flynn first proposed the idea of using the polarization bidirectional reflectance distribution function (pBRDF) to describe the polarization scattering characteristics. He proposed that the pBRDF can be derived by combining the scalar bidirectional reflectance distribution function with the Mueller matrix [9]. Later, R. G. Priest and Thomas A. G. deduced the first pBRDF model (P-G model) by combining the Mueller matrix and the BRDF model proposed by Torrance and Sparrow on the basis of the microfacet theory. However, the P-G model they proposed only considers the impact of specular reflection component on polarization and does not provide an expression for the diffuse reflection component. The diffuse reflection component needs to be measured manually, and the shadowing and masking effects of microfacets are not taken into account, resulting in significant errors during large zenith angle simulations. [10,11]. In 2002, Fetrow P.M. et al. proposed a new method for polarization simulation which can be used to examine the surface properties of materials in a laboratory environment to investigate infrared polarization characteristics in a complex environment [12]. In 2006, Wellems D. introduced a new method to calculate the refractive index of a target using a Mueller matrix model based on microfacet theory from the measurement results [13]. In 2009, Hyde M.W. et al. improved the P-G model and derived a mathematical expression for the diffuse reflection component. They also introduced a geometric attenuation factor G proposed by Blinn to describe shadowing and shading effects [14,15]. In recent years, new pBRDF theoretical models have been proposed by many researchers. In 2017 and 2018, Zhu J. et al. from Xi’an Jiaotong University proposed a three-component pBRDF model [16,17,18]. The core idea is to regard the scattered light of the target as being composed of three components, which are specular reflection, directional diffuse reflection and ideal diffuse reflection. Through their experimental verification, the simulation results of this model are in better agreement with the experimental values than the Hyde model. In 2018, Yang M. et al. from the University of Science and Technology of China proposed a multi-parameter pBRDF model for coating surface based on K-M (Kubelka–Munk) theory. This model improved the Hyde model by introducing K-M theory and the specular coefficient. Experiments were carried out on black paint and green paint coatings outdoors, and the key parameters were inverted using the experimental results. Their experimental results show that the simulation results of the model using the parameters inverted by one fixed incident angle are in good agreement with the measurements of other fixed incident angles of the coating surfaces, which verifies the accuracy of the model. In 2021, Chen H. et al. proposed a multi parameter pBRDF model that can better describe the infrared polarization scattering characteristics of coatings [19]. In 2022, Shen S. et al. from the National University of Defense Technology proposed a method for calculating the degree of polarization of laser backscattering from a typical geometric rough surface at long distance based on the pBRDF model [20]. This method calculates the degree of polarization through the characteristics of material morphology, rather than material properties, which provides a new idea for calculating the polarization scattering characteristics of targets., However, the existing pBRDF models are mostly improvements on the Hyde model proposed by Hyde M.W. et al. in 2009 [15]. The major shortcomings of the Hyde model are that it cannot describe the subsurface scattering effect, shadowing effect and masking effect of the target well. At present, no model can correct the two shortcomings of the Hyde model at the same time.

In order to establish a model that can be applied to the polarization scattering characteristics of rough surface of common target material, in this paper, the Hyde model is improved on the basis of the work of previous researchers. In our modified model, we divide the scattered light of the target into three parts, namely specular reflection, surface scattering and subsurface scattering. In the specular reflection component, we derive the integral geometric attenuation factor to replace the Blinn geometric attenuation factor that the Hyde model uses. In the surface scattering component, we introduce the diffuse reflection calculation method proposed by Le Hors et al. [21,22] and provide the modified model expression and the polarization degree expression of the target scattered light under the illumination of the polarized light source. We measure the degree of polarization data of the two materials using the experimental measurement device we built. Based on the measured data of the degree of polarization, we use the BBO-FA algorithm to invert the key parameters of the target material [23]. Using the inverted parameters, we simulate the Stokes vectors of the scattered light on the target surface. The simulation results show that the simulation values of the Stokes vectors of the model are in good agreement with the experimental values, which verifies the accuracy of the model. At the same time, we also analyze in detail the influence of the geometric attenuation factor model, the mirror reflection component coefficient and the subsurface scattering component coefficient on the degree of polarization of the model.

2. Modified pBRDF Model

2.1. BRDF Model

BRDF (bidirectional reflectance distribution function) provides the spatial distribution relationship between incident light and reflected light [24]. Its main influencing factors include the material properties of the target, the angle and wavelength of the incident light, etc. It is a classical model for studying the scattering characteristics of light and a basic model for deriving the pBRDF model. The physical meaning of BRDF is the ratio of the radiance $\mathrm{d}{L}_r({\theta }_r,{\varphi }_r)$ emitted in the $r({\theta }_r,{\varphi }_r)$ direction to the irradiance $\mathrm{d}{E}_i({\theta }_i,{\varphi }_i)$ incident on the measured surface in the $i({\theta }_i,{\varphi }_i)$ direction. The radiance L is defined as the radiation flux per unit area and unit solid angle along the radiation direction, and the dimension is $W/m^2\cdot Sr$. Radiation illuminance E is defined as the radiation flux per unit area, and the dimension is $W/m^2$. The unit of BRDF is $S{r}^{-1}$, and its definition is as follows:

$$f\left({\theta }_i,{\varphi }_i,{\theta }_r,{\varphi }_r\right)=\frac{\mathrm{d}{L}_r\left({\theta }_r,{\varphi }_r\right)}{\mathrm{d}{E}_i\left({\theta }_i,{\varphi }_i\right)}$$

(1)

As shown in Figure 1, ${\theta }_i$ and ${\theta }_r$ are the zenith angles of incident light and reflected light, respectively, and ${\varphi }_i$ and ${\varphi }_r$ are the azimuth angles of incident light and reflected light, respectively. The subscript i represents the incident direction and r represents the reflection direction.

2.2. Classical pBRDF Model

The pBRDF model is a generalization of the BRDF model, which is mainly used to describe the polarization scattering characteristics of materials. The P-G (Priest–Germer) model derived from the Torrance–Sparrow model is a classical pBRDF model. This model considers that the scattered light of the target material mainly consists of a specular reflection component and a diffuse reflection component. The P-G model gives the expression of the specular reflection component. However, due to the neglection of shadowing and masking effects, the model error is large when the light is incident at a large angle, and the expression of the diffuse reflection component is not given. The diffuse reflection component depends on measurement [9]. In 2009, Hyde et al. improved the P-G model and developed a new model for isotropic materials. Based on the original P-G model, the mathematical expression of the diffuse reflection component was derived, and the geometric attenuation factor G was introduced to describe the shadowing and masking effects. Similar to the P-G model, the Hyde model considers that the scattered light of the target is completely composed of a specular reflection component and a diffuse reflection component. Its expression is as follows:

$$F=F^s+F^d$$

(2)

where $F$ represents the Hyde pBRDF model, $F^s$ represents the specular reflection component, and $F^d$ represents the diffuse reflection component. The specular reflection component of the Hyde model is derived based on the microfacet theory. It is considered that a rough object surface is composed of a series of microfacets. Each microfacet can be regarded as a specular reflection unit that follows Fresnel’s law. The reflectivity of the microfacet can be solved by means of Fresnel’s law.

The schematic diagram of the microfacet is shown in Figure 2. In the figure, n_μ represents the normal direction of the microfacet, n represents the normal direction of the macro-surface, and $\alpha$ is the angle between n_μ and n. $\beta$ is the angle between the incident direction and the normal vector n_μ of the microfacet.

It is generally believed that the normal direction of the microfacet is approximately the same as the Z-axis direction in Figure 1. There is the following relationship between $\alpha$, $\beta$, ${\theta }_i$ and ${\theta }_r$:

$$\cos (2\beta )=\cos {\theta }_i\cos {\theta }_r+\sin {\theta }_i\sin {\theta }_r\cos \phi$$

(3)

$$\cos \alpha =\frac{\cos {\theta }_i+\cos {\theta }_r}{2\cos \beta }$$

(4)

The Hyde model uses the height standard deviation σ_h and the autocorrelation length $\mathcal{l}$ to describe the slope distribution of the rough surface. The normal distribution function is expressed as follows:

$$P\left(\alpha ;{\sigma }_h,\mathcal{l}\right)=\frac{{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\alpha /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\alpha }$$

(5)

The height standard deviation and autocorrelation function are actually equivalent to the roughness σ in the P-G model, which can be transformed using the following formula:

$$\sigma =\frac{\sqrt{2}{\sigma }_h}{\mathcal{l}}$$

(6)

In the above formula, σ represents the roughness of the target surface material. In the Hyde model, the geometric attenuation factor G is in the form of a piecewise function proposed by Blinn, and its expression is:

$$G\left({\theta }_i,{\theta }_r,\varphi \right)=\min \left(1;\frac{2\cos \alpha \cos {\theta }_r}{\cos \beta };\frac{2\cos \alpha \cos {\theta }_i}{\cos \beta }\right)$$

(7)

where $\varphi$ represents the difference between the azimuth angle of the incident direction and the observation direction, and the expression of the specular reflection component $F^s$ is:

$$\begin{array}{l}{F}_{jk}^s\left({\theta }_i,{\theta }_r,\varphi ;\sigma ,n,k\right)\\ =P\left({\theta }_i,{\theta }_r,\varphi ;\sigma \right)\\ \times G\left({\theta }_i,{\theta }_r,\varphi \right)M_{jk}({\theta }_i,{\theta }_r,\varphi ;n,k)\end{array}$$

(8)

The expression of the diffuse reflection component given by Hyde is:

$$\begin{array}{l}{F}_{00}^d\left({\theta }_i;\sigma \right)=\\ \frac{1}{\pi }\left(1-{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{\pi /2}{\int }}{f}_{00}^s\cos {\theta }_r\sin {\theta }_{r}d{\theta }_{r}d\varphi }}\right)M_{00}\left({\theta }_i,{\theta }_r,\varphi ;n,k\right)\end{array}$$

(9)

where j and k represent the j + 1 row and k + 1 column in the Mueller matrix; n and k are the real and imaginary parts of the complex refractive index of the target material, respectively. M denotes the 4 × 4 Mueller matrix derived from the Fresnel function. The detailed derivation process is shown in the literature [25].

Although the accuracy of the Hyde model is significantly improved compared with the P-G model, the Hyde model still has the following shortcomings:

1. The Hyde model assumes that the light does not absorb into the surface of the target material, and the contribution of the diffuse reflection component and the specular reflection component is equal. However, in fact, the diffuse reflection of the target material can be subdivided into subsurface scattering and surface scattering. The Lambertian diffuse reflection formula cannot be simply applied, nor can it be simply considered that the reflection of the object surface is the superposition of the diffuse reflection component and the specular reflection component. When the incident light zenith angle is small, the error of the Hyde model is within the acceptable range, and when the incident zenith angle is close to 90°, the error of the Hyde model will even reach 179.36% [16].

2. The geometric attenuation factor used in the Hyde model is derived from the model proposed by Blinn. The premise of the model derivation is to assume that the microfacet is an isosceles V-cavity. This assumption contradicts the assumption that the microfacet of the Hyde model obeys the slope distribution function, resulting in an inflection point in the Hyde model. This phenomenon is obviously not in line with physical reality [26].

2.3. Modified pBRDF Model

In view of the first shortcoming of the above Hyde model, we consider that the three-component model considering the actual physical process of material microfacet scattering is a more appropriate improvement scheme after comprehensive analysis of a number of papers.

In Figure 3, $\gamma$ refers to the inclination angle of the microfacet adjacent to the microfacet with inclination angle $\alpha$. The three-component model considers that the interaction between the incident light and the microfacets of the object can be divided into three parts, A, B, and C, as shown in Figure 3, corresponding to the specular reflection component, the surface scattering component, and the subsurface scattering component (also known as the volume scattering), respectively. According to the assumptions of the three-component model, the improved pBRDF expression is:

$$F^{\prime }=k_s{F}^{s^{\prime }}+k_d{F}^{d^{\prime }}+k_v{F}^{v^{\prime }}$$

(10)

where $k_s$, $k_d$ and $k_v$ are the coefficients corresponding to the three components, respectively.

2.3.1. Modified Geometric Attenuation Factor

The physical process of the specular reflection component $F^{s^{\prime }}$ is shown in region A of Figure 3. In region A, the angle $\alpha$ between the microfacet normal and the macro-surface normal is small. At this time, when the incident light is incident on the surface of the microfacet, specular reflection occurs and the light is reflected into the hemispherical space. This shows that the specular reflection component $F^{s^{\prime }}$ accounts for the main part of the reflected light in the vicinity of the microfacet distribution in the similar area A. Combined with the second shortcoming proposed above, based on the specular reflection component of the Hyde model, we derive the geometric attenuation factor based on the microfacet theory.

In 2014, Eric Heitz pointed out in his research report [27] that the analytical form of the geometric attenuation factor depends on the normal distribution function. For the normal distribution function with high autocorrelation used in our model, the geometric attenuation factor of the V-cavity is usually selected. The most common geometric attenuation factor of the V-cavity form is the geometric attenuation factor proposed by Blinn, which cannot correspond well with the normal distribution function. Therefore, we need to propose a new geometric attenuation factor for the Hyde model.

In order to derive the geometric attenuation factor, the following assumptions are proposed:

• Each microfacet is independent of each other, and its tilt angle obeys a Gaussian distribution.
• Because the attenuation factor is only used to correct the specular reflection component, the relationship between the light and the microfacet can be considered to comply with Fresnel’s law.
• In the derivation process, the area of all microfacets is taken as 1 to facilitate the calculation.

Since we derive the geometric attenuation factor in the case of receiving and transmitting the same path in the LiDAR application scene, the light that can be incident on the microfacet must be able to return along the original path, so there is no masking effect, and it is easy to deduce that ${\theta }_i$= ${\theta }_r$, $\varphi$ = 0. Substituting ${\theta }_i$ = ${\theta }_r$ and $\varphi$ = 0 into Formulas (3) and (4), we can obtain:

$$\beta =0$$

(11)

$$\alpha ={\theta }_i$$

(12)

which means that, under the scenario of LiDAR application, the direction of the incident light is perpendicular to the microfacet, and the angle between the zenith angle ${\theta }_i$ of the incident light and the macro-surface and the normal line of the target surface and the normal line of the microfacet is equal. There is no masking effect under this condition, which means that the masking function is constantly 1.

For the shadowing effect, we discuss it in two cases:

Figure 4 is a schematic diagram of the situation where all light can pass through. In Figure 4, for the shadowing effect, the critical case is that when the zenith angle of the incident light and the adjacent microfacet of the incident microfacet satisfy ${\theta }_i$ + $\gamma$ = $\left(\pi /2\right)$, all the incident light can be reflected without being shadowed.

Then, the probability of the partial shadowing model under this condition is:

$$G_1={\displaystyle {\int }_0^{\frac{\pi }{2}-{\theta }_i}P(\gamma )\cdot 1d}(\gamma )={\displaystyle {\int }_0^{\frac{\pi }{2}-{\theta }_i}\frac{{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }}d\gamma$$

(13)

Figure 5 is a schematic diagram of the situation where partial light can pass through. When the angle $\gamma$ between the normal of the adjacent microfacet B and the normal of the macro-surface is greater than the angle $\left(\pi /2\right)-\gamma$, the incident and reflected light will be partially blocked by the microfacet B, resulting in energy loss. If the passed part is a and the shadowed part is b, then the proportion of light passing through is $a/(a+b)$.

From the geometric relationship in Figure 5, it can be inferred that:

$$a/(a+b)=1+\cos (\alpha +\gamma )$$

(14)

Then, the probability that the light can pass through under this condition is:

$$G_2={\displaystyle {\int }_{\frac{\pi }{2}-{\theta }_i}^{\frac{\pi }{2}}P(\gamma )\cdot ad}(\gamma )={\displaystyle {\int }_{\frac{\pi }{2}-{\theta }_i}^{\frac{\pi }{2}}(1+\cos \alpha \cos \gamma -\sin \alpha \sin \gamma )}\cdot \frac{{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }d\gamma$$

(15)

Combining the two cases, the integral geometric attenuation factor used for the LiDAR application scene is:

$$\begin{array}{l}{G}_M=G_1+G_2\\ ={\displaystyle {\int }_0^{\frac{\pi }{2}-{\theta }_i}\frac{{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }}d\gamma +\\ {\displaystyle {\int }_{\frac{\pi }{2}-{\theta }_i}^{\frac{\pi }{2}}(1+\cos \alpha \cos \gamma -\sin \alpha \sin \gamma )}\\ \cdot \frac{{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }d\gamma \end{array}$$

(16)

The following seeks to verify whether the integral geometric attenuation factor proposed by us conforms to the physical reality. Heitz demonstrated that a physical constraint that all geometric attenuation factors based on physical reality must satisfy is that the visible projection area of the microfacet should be exactly equal to the geometric projection area of the macro-surface. This equivalence imposes a constraint on the masking function, which is given by the following equation:

$$\cos \beta ={\displaystyle {\int }_0^{\frac{\pi }{2}}{G}_m}\left({\omega }_o,{\omega }_m\right)〈{\omega }_o,{\omega }_m〉P\left({\omega }_m;{\sigma }_h,\mathcal{l}\right)d{\omega }_m$$

(17)

In the above formula, ${\omega }_o$ is the vector of outgoing direction, ${\omega }_m$ is the microfacet normal, $G_m$ is the masking function, and $〈-,-〉$ means the clamped dot product. The geometric attenuation factor must always satisfy this constraint. In our model under the LiDAR application scene, the vector of the outgoing direction ${\omega }_o$ is parallel to the microfacet normal ${\omega }_m$, and so there is:

$$〈{\omega }_o,{\omega }_m〉=1$$

(18)

By inserting Equations (11) and (18) into Equation (17), we can obtain:

$${\displaystyle {\int }_0^{\frac{\pi }{2}}P\left({\omega }_m;{\sigma }_h,\mathcal{l}\right)d{\omega }_m=1}$$

(19)

The integral result of the above formula depends on A and B, and is not constantly equal to 1, which means that the geometric attenuation factor we proposed does not meet the constraint. To ensure that our model satisfies the constraint, a normalization coefficient needs to be added. If the normalized coefficient is C, then we have:

$$C=\frac{1}{{\displaystyle {\int }_0^{\frac{\pi }{2}}P\left({\omega }_m;{\sigma }_h,\mathcal{l}\right)d{\omega }_m}}$$

(20)

Then, our corrected integral geometric attenuation factor which satisfies the constraint given by Heitz is:

$$\begin{array}{l}{G}_M^{\prime }={\displaystyle {\int }_0^{\frac{\pi }{2}-{\theta }_i}\frac{C{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }}d\gamma +\\ {\displaystyle {\int }_{\frac{\pi }{2}-{\theta }_i}^{\frac{\pi }{2}}(1+\cos \alpha \cos \gamma -\sin \alpha \sin \gamma )}\\ \cdot \frac{C{\mathcal{l}}^2\exp \left(-{\mathcal{l}}^2{\tan }^2\gamma /4{\sigma }_h^2\right)}{4\pi {\sigma }_h^2{\cos }^3\gamma }d\gamma \end{array}$$

(21)

The second physical constraint Heitz proposed is that the distribution of the visible normal $P\left({\omega }_m;{\sigma }_h,\mathcal{l}\right)$ is normalized, because it is used as a weighting function to average radiances:

$${\displaystyle {\int }_0^{\frac{\pi }{2}}L({\omega }_m,{\omega }_o)P_M({\omega }_m;{\sigma }_h,\mathcal{l})d{\omega }_m=L({\omega }_o})$$

(22)

In the above formula, the modified normal distribution function $P_M$ is the Hyde normal distribution function multiplying the normalized coefficient C. In the LiDAR application scene, ${\omega }_m$ is parallel to ${\omega }_o$.In this case, Formula (22) is simplified to:

$${\displaystyle {\int }_0^{\frac{\pi }{2}}{P}_M({\omega }_m;{\sigma }_h,\mathcal{l})d{\omega }_m=}1$$

(23)

Obviously, the modified normal distribution function $P_M$ satisfies the physical second constraint.

Then, the specular reflection component of the corresponding improved model is:

$$\begin{array}{l}{F}_{jk}^{s^{\prime }}\left({\theta }_i;\sigma ,n,k\right)\\ =P\left(\alpha ;{\sigma }_h,\mathcal{l}\right)\\ \times G_M^{\prime }\left({\theta }_i,\varphi \right)M_{jk}\left({\theta }_i;n,k\right)\end{array}$$

(24)

2.3.2. Subsurface Scattering Component

The physical process of the subsurface scattering component $F^{v^{\prime }}$ is shown in the C region in Figure 3. When the incident light is incident on the surface of the material, part of the incident light will directly penetrate the surface of the material into the subsurface. Whenever the photons in the incident light interact with the particles in the subsurface, the polarization direction of the scattered light might change to any random polarization direction. A part of the incident light transmitted to the subsurface will pass through the subsurface after multiple interactions with the particles. Since this process is difficult to quantify, this paper regards the light passing through the subsurface as the light synthesized by polarized light in each polarization direction—that is, completely depolarized light.

$$F^{v^{\prime }}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(25)

2.3.3. Surface Scattering Component

The physical process of the surface scattering component $F^{v^{\prime }}$ is shown in region B in Figure 3. In region B, the angle $\alpha$ between the microfacet normal and the macro-surface normal is larger. At this time, when the incident light is incident on the microfacet, the reflected light is not directly reflected to the hemisphere space but reflected to another microfacet and continues to reflect. After multiple reflections, only a portion of the repeatedly reflected light is reflected from the microfacet into the hemispherical space due to the shadowing/masking effect. This part of the reflected light can be regarded as the superposition of polarized light in each polarization direction, resulting in a depolarization effect. In order to simplify the analysis, we assume that the surface scattering light is completely depolarized. The surface scattering component accounts for the main part of the reflected light in the microfacet, where the distribution of the microfacet is similar to area B.

For most pBRDF models, establishing how to accurately express the surface scattering component is a decisive factor in the accuracy of the model. However, the microfacet morphology of different materials varies greatly. For example, the surface morphology of metal materials is usually dense, while the surface morphology of coating materials is generally porous and sparse. It is very difficult to use a general expression to describe the surface scattering components of various materials. At present, there is no pBRDF model that can accurately characterize the scattering characteristics of various materials. A common and more accurate method is to use different fitting functions to fit the surface scattering components according to the experimental measured values of different types of materials.

In this paper, we choose to use the method proposed by Hors L.L. et al. to introduce the reflectivity parameter ${\rho }_d$ (representing the ratio of the surface scattering intensity of the object to the surface scattering intensity of the Lambertian reflectance) to calculate the surface scattering component of the target [21]. The expression of ${\rho }_d$ is as follows:

$${\rho }_d=\left(1-R_i\right)\frac{\left(1-R_r\right)R_{\infty }}{1-R_r{R}_{\infty }}$$

(26)

In the formula above, $R_i$ is the Fresnel reflectivity of the incident light when the light is incident from the vacuum to the target surface; $R_r$ is the Fresnel reflectivity of the scattered light when the light is incident from vacuum to the target surface; and $R_{\infty }$ is the reflectivity of the target material when the thickness of the target material is infinite without coating and when the light is incident from the vacuum to the target surface. In general, it can be considered that $R_i$ = $R_r$, $R_i$ can be calculated by means of the Fresnel function. $R_{\infty }$ is considered to be an unknown characteristic parameter of the material, which can be fitted after measuring the DOP data. Then, the expression of the surface scattering component can be obtained as follows:

$$F_d^{\prime }=\frac{\cos {\theta }_i}{\pi }{\rho }_d\left[\begin{array}{llll}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$$

(27)

Although the accuracy of using this method to simulate the surface scattering component of the target is not as good as the simulation results of different high-order polynomial fitting models according to different materials, the simulation results of this method are better than the results of the Hyde model. Moreover, the expression given by this method is relatively fixed, which is suitable for the analysis of the polarization scattering characteristics of the target composed of various materials on the target surface. Substituting the expressions of the three components given above into Formula (8), the pBRDF model expression used in the LiDAR scene can be obtained as follows:

$$\begin{array}{l}{F}_{jk}^{\prime }\left({\theta }_i;\sigma ,n,k\right)=\\ [k_{s}P\left({\theta }_i;\sigma \right)\times G_M^{\prime }\left({\theta }_i\right)\\ +k_d\frac{\cos {\theta }_i}{\pi }{\rho }_d+k_v]M_{jk}\left({\theta }_i;n,k\right)\end{array}$$

(28)

In the LiDAR system, the light emitted by the laser source is generally approximately linearly polarized, and almost does not contain a circularly polarized component. Therefore, the Mueller matrix $M$ can be further simplified into a $3\times 3$ Mueller matrix.

$$f_s\left({\theta }_i,\varphi ;\sigma \right)=P\left({\theta }_i,\varphi ;\sigma \right)G_M^{\prime }\left({\theta }_i,\varphi \right)$$

(29)

Then, the formula is further simplified as:

$$\left[\begin{array}{ccc}{F}_{00}^{\prime }& F_{01}^{\prime }& F_{02}^{\prime }\\ F_{10}^{\prime }& F_{11}^{\prime }& F_{12}^{\prime }\\ F_{20}^{\prime }& F_{21}^{\prime }& F_{22}^{\prime }\end{array}\right]=\left[\begin{array}{ccc}{k}_s{f}_s{M}_{00}+\frac{\cos {\theta }_i{\rho }_d}{\pi }+k_v& k_s{f}_s{M}_{01}& k_s{f}_s{M}_{02}\\ k_s{f}_s{M}_{10}& k_s{f}_s{M}_{11}& k_s{f}_s{M}_{12}\\ k_s{f}_s{M}_{20}& k_s{f}_s{M}_{21}& k_s{f}_s{M}_{22}\end{array}\right]$$

(30)

When the incident light source is the laser beam with a horizontal polarization direction, the Stokes vectors of the scattered light are:

$$\left[\begin{array}{c}{L}_1\\ L_2\\ L_3\end{array}\right]=\left[\begin{array}{ccc}{F}_{00}^{\prime }& F_{01}^{\prime }& F_{02}^{\prime }\\ F_{10}^{\prime }& F_{11}^{\prime }& F_{12}^{\prime }\\ F_{20}^{\prime }& F_{21}^{\prime }& F_{22}^{\prime }\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}{F}_{00}^{\prime }+F_{01}^{\prime }\\ F_{10}^{\prime }+F_{11}^{\prime }\\ F_{20}^{\prime }+F_{21}^{\prime }\end{array}\right]$$

(31)

Under such conditions, the polarization degree of the scattered light is:

$$\begin{array}{l}{d}_{DOP}({\theta }_i;k_s,k_v,n,k,\sigma ,R_{\infty })=\frac{\sqrt{{\left(F_{10}^{\prime }+F_{11}^{\prime }\right)}^2+{\left(F_{20}^{\prime }+F_{21}^{\prime }\right)}^2}}{F_{00}^{\prime }+F_{01}^{\prime }}\\ =\frac{k_s{f}_s\sqrt{{\left(M_{10}+M_{11}\right)}^2+{\left(M_{20}+M_{21}\right)}^2}}{k_s{f}_s{M}_{00}+\frac{\cos {\theta }_i{\rho }_d}{\pi }+k_v+k_s{f}_s{M}_{01}}\end{array}$$

(32)

In the above formula, $M_{00}$, $M_{01}$, $M_{10}$,$M_{11}$, $M_{20}$ and $M_{21}$ are all elements in the Mueller matrix $M$. The incident zenith angle ${\theta }_i$ and the relative azimuth angle $\varphi$ are input variables. The real and imaginary parts of the complex refractive index, n and k, and the surface roughness $\sigma$ are the characteristic parameters of the material, which can be obtained through data review or measurement. It should be noted that the real and imaginary parts n and k of the complex refractive index of the material in this model are not necessarily the real complex refractive index of the material. Meanwhile, $k_s$, $k_d$, $k_v$ and $R_{\infty }$ are the proportional parameters in the model proposed in this paper, which need to be inverted by experimental data. Among them, the scale parameters $k_s$, $k_d$ and $k_v$ jointly determine the proportion of the target surface scattering component in the total pBRDF and have the same effect. In order to simplify the calculation, $k_d$ is regarded as 1 in this paper. In conclusion, a total of six parameters, $k_s$, $k_d$, $k_v$, $R_{\infty }$, n and k, could be obtained via the inversion of partial polarization measurement data combined with related algorithms.

3. Measurement and the Determination of Parameters

3.1. Measurement

The schematic diagram of the experimental setup is shown in Figure 6. The emission system is mainly composed of a 532 nm wavelength 10 mW continuous wave laser, a polarizer with an extinction ratio of 50 dB and a beam expander. The receiving system is mainly composed of a telescope, a collimating lens, a polarizer and an sCMOS camera. We chose an aluminum plate and a white paint coating as the targets. The rotation angle of the target material is controlled by the Zolix MC600 Motion Controller. The detector uses a Pco.edge 5.5 sCMOS camera to measure the intensity of scattered light.

In the experiment, the emission system and the receiving system are placed in parallel, and the targets are placed on the stage of the rotatable pedestal, which is perpendicular to the two targets. When the target is placed far enough away, the transmitting and receiving can be considered in the same path. At this time, the zenith angles of the incident light and the reflected light are equal. The initial value is 0°and the difference in the azimuth angle $\varphi$ is = 180°.

At the beginning of the measurement, the 532 nm solid-state continuous laser first emits a laser beam. The emitted beam is polarized by a polarizer with a horizontal polarization direction and reaches the beam expander. After the beam expander, the divergence angle of the beam can reach 0.01 radians. The target material is placed 20 m away. When the beam reaches the target material, a circular spot with a diameter of D = 20 cm is formed, which can basically cover the main area of the target material. After being reflected by the target material, the scattered light is received by the telescope, and then after being collimated by the collimator behind the telescope, it is polarized by the polarizer in a specific direction and received by the sCMOS camera. By controlling the rotation of the high-precision rotatable pedestal and the polarization direction of the polarizer in front of the sCMOS, the light intensity measurement of the reflected light of different experimental materials at four polarization directions (the angles between the four directions and the horizontal direction are 0°, 45°, 90°, 135°) can be realized. After one measurement, the rotatable pedestal rotates 5°. At this time, the zenith angles of the incident light and the reflected light increase by 5° synchronously, and then the above measurement operation is performed. Each material was measured 13 times from 0° to 60° with an increase of 5° each time.

By measuring the intensity of the reflected light in different polarization directions, the Stokes vectors of the reflected light can be obtained as follows:

$$L_0=I_0+I_{90}$$

(33)

$$L_1=I_0-I_{90}$$

(34)

$$L_2=I_{45}-I_{135}$$

(35)

In the above formula $I_0$, $I_{45}$, $I_{90}$ and $I_{135}$ represent the light intensity of the image when the angle between the polarization direction and the horizontal direction in the front polarizer of the sCMOS camera is 0°, 45°, 90°, and 135°, respectively.

With the Stokes vectors, the degree of polarization of the reflected light can be calculated:

$$DOP=\sqrt{\frac{L_0^2+L_1^2}{L_2^2}}$$

(36)

3.2. Determination of Parameters

In order to verify the accuracy of the model, we need to first use DOP to invert the parameters of the target material, and then use the inverted parameters to simulate the Stokes vectors of the target material and compare them with the measured values of the Stokes vectors. In this paper, a BBO-FA algorithm combining the BBO algorithm (biogeography-based optimization algorithm) and the FA algorithm (FireFly algorithm) is used to invert the parameters. The BBO algorithm is a newly developed natural heuristic algorithm based on biogeography theory. In many experimental scenarios, the BBO algorithm shows better optimization strategy and effect than other meta-heuristic algorithms in the initial iteration process [28]. However, the disadvantage of the algorithm is that it does not have the competitiveness to find the optimal solution when the number of iterations increases. The FireFly algorithm is a natural optimization algorithm based on fireflies’ flicker patterns and behavior. Compared with other optimization algorithms, it has stronger ability to find the optimal solution [29]. The core idea of the BBO-FA algorithm is to combine the fast global search ability of the BBO algorithm with the ability of the FireFly algorithm to find the optimal solution, and to maximize the comprehensive performance of the algorithm. In order to perform the inversion, an objective function needs to be designed. In this paper, the goal of our inversion is to make the simulated values of the inverted model conform to the experimental measurements. The optimization function is constructed by using the mean square error of the simulation value and measurement value:

$$\begin{array}{l}\mathrm{\Delta }\min E\left(n,k,\sigma ,k_s,k_v,R_{\infty }\right)=\hfill \\ \frac{{\displaystyle \sum _{{\theta }_i}^N{\left[DOP\left({\theta }_i\right)-DO{P}_m\left({\theta }_i\right)\right]}^2}}{N}\hfill \end{array}$$

(37)

where $DOP\left({\theta }_i\right)$ represents the simulation value of the degree of polarization of the model when the incident zenith angle (turntable rotation angle) is ${\theta }_i$, and $DO{P}_m\left({\theta }_i\right)$ represents the measured value of the degree of polarization when the incident zenith angle is ${\theta }_i$. N represents the number of measurements.

The BBO-FA algorithm is used to invert the parameters of the model. The parameters of the two materials are shown in

Table 1. Parameters of two samples inverted using the BBO-FA algorithm.

Sample	n	k	$\mathit{\sigma }$	${\mathit{R}}_{\mathit{\infty }}$	${\mathit{k}}_{\mathit{s}}$	${\mathit{k}}_{\mathit{v}}$
Aluminum plate	4.32	0.65	0.22	0.33	5.76	1.0
White painted coating	0.92	0.54	0.052	0.93	0.63	0.021

.

Figure 7a,b shows the comparison between the measured DOP and simulated DOP using the inverted parameters. From the results, when the incident angle varies from 0 to 60°, the simulated values of the two materials are in good agreement with the measured values.

To evaluate the goodness of fit of the simulation results, we also calculate the determination coefficient $R^2$ of the simulated values of the fitted model. The expression of $R^2$ is as follows:

$$R^2=\frac{{\displaystyle \sum _{{\theta }_i}^N{(M-S)}^2}}{{\displaystyle \sum _{{\theta }_i}^N{(M-\overline{M})}^2}}$$

(38)

In the above formula, $M$ represents the measured value of a polarization characteristic parameter of scattered light, which can be the DOP or the Stokes vector. $S$ represents the simulation value of the corresponding parameter, while $\overline{M}$ represents the mean value of this set of measured values.

Calculated by Formulas (37) and (38), the RMSE (root mean squared error) and $R^2$ of the simulated values of the DOP of the aluminum plate are 0.0306 and 0.9939, respectively, and the RMSE and $R^2$ of the white paint coating are 0.0307 and 0.9342, respectively. From the comparison of the DOP simulation values of the two materials, we can see that the simulation value of the DOP of the aluminum plate is in better agreement with the measurement than the white paint coating. The reason for this result might be because the aluminum plate has a higher reflectivity. When the rotation angle of the rotatable pedestal rotates to a large angle, the scattered light intensity of the white paint coating received by the sCMOS is smaller than the corresponding light intensity of the aluminum plate. This leads to a relatively low signal-to-noise ratio of the light intensity signal detected by the white paint coating at a large angle, and the accuracy of the polarization measurement results is also relatively poor, which in turn affects the experimental measurement, resulting in a smaller DOP corresponding to the white paint coating.

In order to illustrate the effectiveness of our model, we use the parameters of the two materials to simulate the Stokes vectors of the scattered light. The RMSE and $R^2$ of the Stokes vectors of the two materials are calculated by using Formulas (37) and (38), respectively. The results are shown in the

Table 2. Calculation results of the RMSE and $R^2$ of the Stokes vectors for the two materials.

RMSE/R2	RMSE			R2
Sample	L0	L1	L2	L0	L1	L2
Aluminum plate	0.0225	0.0288	0.0204	0.9918	0.9931	−0.2279
White painted coating	0.0215	0.0473	0.0408	0.9865	0.9176	−4.3576

.

It should be noted that the calculation results of L2 for both samples are negative because the simulation result of the model for L2 is approximately 0 at all incident angles, and it is not meaningful to evaluate the goodness of fit of the simulation results of the model.

The simulation results of the Stokes vectors are shown in Figure 8. Both the simulated values and the measured values are normalized. It can be clearly seen from Figure 8 that the Stokes vector simulation value of the scattered light of the aluminum plate is in better agreement with the measured value compared to the white painted coating. At the same time, the RMSE and $R^2$ of the Stokes vectors of the two materials also support this conclusion. The L0 of the coating is in good agreement with the measured value, but the L1 and L2 simulated values and the measured values are relatively poor, especially at small angles. The reason why the simulated Stokes vectors of the white paint coating do not fit the measurement well enough might because the normalization of the simulated values reduces the error of L0 while enlarging the error of L1 and L2.

On the whole, the Stokes vectors of the target simulated by using the parameters inverted by DOP measurements are in good agreement with the measured values, which demonstrate the accuracy of our proposed model.

4. Simulation

4.1. Simulation and Analysis of the Modified Geometric Attenuation Factor

As shown in Figure 9, the curves of the five colors in Figure 9 represent the simulation curves of the integral geometric attenuation factor when the surface roughness of the target material is 0, 0.1, 0.3, 0.5, and 0.7, respectively. It can be seen from the simulation curve that when the roughness of the target material is zero, which means that the surface of the target material is completely smooth, the geometric attenuation factor is fixed to 1, indicating that the smooth material has no masking and shadowing effect. With the increase in the surface roughness of the target material, the influence of the geometric attenuation factor simulation curve is more obvious, which is more in line with the physical reality.

4.2. Analysis of the Influence of Specular Reflection Coefficient on the Model

Through the above inverted parameters in Table 1, it can be seen that the specular reflection component coefficients obtained via the inversion of different material surfaces are quite different. We think this might be positively correlated with the reflectivity of the target itself.

Figure 10 shows the change in the DOP curve of the target when the specular reflection component coefficient $k_s$ increases. From the curve, we can see that the polarization range of the target increases with the increase in the target specular reflection component coefficient $k_s$. The larger the specular reflection component coefficient of the target, the higher the degree of polarization of the target scattered light in the same zenith angle of the incident light.

Table 3. Parameters of aluminum inverted by using our model with $k_s$ and our model without $k_s$, respectively.

Model	n	k	$\mathit{\sigma }$	${\mathit{R}}_{\mathit{\infty }}$	${\mathit{k}}_{\mathit{s}}$	${\mathit{k}}_{\mathit{v}}$
Our model with $k_s$	4.32	0.65	0.22	0.33	5.76	1.0
Our model without $k_s$	4.82	0.63	0.22	0.27	1	0.31

presents the parameters determined by using the modified pBRDF model with the specular reflection component coefficient $k_s$ and the model without the specular reflection component coefficient ($k_s$ = 1). Using the two sets of parameters in the table below, the DOP of the scattering light of the aluminum plate is simulated and shown in Figure 11. In Figure 11, the orange curve is the DOP curve drawn by using the parameters of the aluminum plate inverted by using the model with $k_s$, and the yellow dotted line is the DOP curve drawn by using the parameters of the aluminum plate inverted by using the model without $k_s$. Using Formula (37) for calculation, the RMSE of the two is 0.0306 and 0.0340, respectively. Using Formula (38) for calculation, the $R^2$ of the two is 0.9939 and 0.9726, respectively. It can be seen from Figure 11 that the simulated DOP obtained by using the parameters determined by using the model with $k_s$ is relatively better fitted with the measured values, which indicates that $k_s$ is beneficial to improve the simulation accuracy.

4.3. Analysis of the Influence of Subsurface Scattering Coefficient on the Model

As mentioned above, the subsurface scattering component is a completely depolarized component, which means that the larger the scattering ratio of the subsurface, the lower the overall polarization degree of the target.

As shown in Figure 12, among the five curves in the figure, the blue curve is the curve of DOP simulated using the parameters of aluminum plate in Table 1, and its corresponding subsurface scattering component is about 1. The other four curves are the simulation of the DOP with a subsurface scattering components coefficient of 2, 3, 4, and 5, respectively, while the other parameters are constant. It can be seen from the simulation results that as the scattering component of the subsurface increases, the overall degree of polarization of the model decreases, which verifies our theory.

Figure 13 is a comparison of the simulated and measured values of the degree of polarization of our model with or without the subsurface scattering component coefficient $k_v$. When the model has no $k_v$, the curve of the DOP increases. When the model with the subsurface scattering component is used for simulation, the overall DOP of the model is reduced, and the simulation value is in good agreement with the measured value. Using Formula (37) for calculation, the RMSE of the two is 0.271 and 0.0306, respectively. Using Formula (38) for calculation, the $R^2$ of the two is 0.5241 and 0.9939, respectively, and the difference is very obvious. This shows that the introduction of $k_v$ helps to improve the simulation accuracy of the model.

5. Discussion

To prove the effectiveness of our improvement, we compare our performance with the classical Hyde model in the LiDAR application scenario (${\theta }_i$ = ${\theta }_r$). Firstly, we compare the geometric attenuation factor that Hyde used with the proposed modified geometric attenuation factor.

Figure 14 is a comparison of the Blinn geometric attenuation factor used by Hyde [15] and the integral geometric attenuation factor in the case of LiDAR application. In the figure, the blue dotted line represents the simulation curve of the Blinn model, and the orange solid line represents the simulation curve of the improved model when the surface roughness $\sigma$ of the material is set to 0.5.

According to Formula (7), the geometric attenuation factor proposed by Blinn is actually only related to the incident angle of the surface of the target material and has nothing to do with the roughness of the surface of the target material. In the case of LiDAR application, the inflection point appears and begins to decay when fixed at 45°, which is obviously not in line with physical reality.

Compared with the Blinn geometric attenuation factor, our model shows a continuous decreasing function rather than a sudden decline like the Blinn geometric attenuation factor, meaning that our model is more in line with the characteristics of the microfacet theory and also more in line with the physical reality.

Figure 15 is a comparison of the DOP simulated by Hyde pBRDF and our pBRDF model. We use the BBO-FA algorithm and the Hyde model to invert the characteristic parameters of the aluminum plate before comparison. The inverted parameters by using different pBRDF models are shown in

Table 4. Parameters of aluminum plate inverted by using the Hyde model and our model.

Model	n	k	$\mathit{\sigma }$	${\mathit{R}}_{\mathit{\infty }}$	${\mathit{k}}_{\mathit{s}}$	${\mathit{k}}_{\mathit{v}}$
Hyde model	5.83	1.35	0.22	-	1	0
Our model	4.32	0.65	0.22	0.33	5.76	1.0

.

Compared to our model, the Hyde model has no specular reflection component coefficient $k_s$ (the specular reflection component coefficient of Hyde model is fixed to 1) and has no subsurface scattering component coefficient $k_v$.

In Figure 15, we can see that the simulated DOP of our model suits the measurement better than that of the Hyde model, especially at small rotation angles. With the increase in the rotation angle, the fitting degree of the simulated DOP of the Hyde model to the measurement results gradually becomes better.

Table 5. Calculation results of RMSE and $R^2$ of DOP for two materials.

RMSE/R2	RMSE	R2
Models
Hyde model	0.0332	0.9898
Our model	0.0306	0.9939

gives the RMSE and $R^2$ of the DOP simulated using the two models. Both RMSE and $R^2$ indicate that the simulation result of our model fits the measurement better.

We think that the possible reason for this phenomenon is mainly due to the different geometric attenuation factors that the two models used. As we can see from Figure 9 and Figure 14, it can be easily concluded that according to our modified geometric attenuation factor, when the $\sigma$ of the aluminum plate is 0.22, the shadowing effect has little effect and begins to decline slowly at a large rotation angle. As for the Hyde model, the Blinn geometric attenuation factor begins to decline more rapidly at a fixed rotation angle of 45°, resulting in a faster decline in DOP. From this conclusion, we can infer that when the roughness $\sigma$ of the material is greater, the difference between the fitting effects of the two models will be greater too.

However, our modified model has its limitations as well. As Hyde mentioned in his previous work [15], the DHR of a pBRDF model must be $\le 1$, otherwise the pBRDF violates the conservation of energy. Unfortunately, although our model is improved based on the Hyde pBRDF model, we are unable to calculate the DHR. Our focus is to describe the polarization scattering properties of target materials in LiDAR application scenarios. The geometry attenuation factor we proposed is only defined when $\varphi =\pi$. Under this condition, the DHR of our model cannot be calculated.

6. Conclusions

In this paper, we first analyze the existing classical Hyde pBRDF model and summarize the shortcomings of the Hyde model. In order to improve these shortcomings and better characterize the polarization scattering characteristics and Stokes vectors of the target surface, we propose an improved multi-parameter model suitable for LiDAR application scenarios. This model draws on the previous work and divides the scattering component into three parts, namely the specular reflection component, the surface scattering component and the subsurface scattering component. In this model, the specular reflection component coefficient is introduced to characterize the specular reflection component in the scattered light on the surface of the material together with the original specular reflection component in the Hyde model. Due to the Hyde specular reflection component part being derived based on the microfacet theory, and the Blinn geometric attenuation factor used in this part being derived based on the V-cavity theory rather than the microfacet theory, which leads to the self-contradiction of the model, we have improved the geometric attenuation factor of the model. Based on the microfacet theory, an integral geometric attenuation factor suitable for the model is derived, and the later simulation proves that the geometric attenuation factor is more in line with the actual physical phenomenon. In terms of the surface scattering component, we introduce the work of Hors, L. L. et al. to describe the surface scattering component of the target and introduce a material parameter $R_{\infty }$ to calculate the diffuse reflection component of the target. As for the subsurface scattering component, we introduce the subsurface scattering component coefficient $k_v$ to represent this component of the target. Through the two parameters $k_v$ and $k_s$, we can represent the contribution of the three components of the target. However, it should be noted that due to the large differences in the characteristics of various materials, it is difficult to construct a polarization scattering model that can comprehensively describe the scattering characteristics of various materials. In addition, some parameters of the model depend on the inversion of the existing polarization data. Therefore, the accuracy of the polarization measurement of the target has a great influence on the accuracy of the model simulation. The validity of the integral geometric attenuation factor we proposed is only demonstrated by numerical simulation instead of actual measurement. This is because the difference in the Blinn and our integral geometric attenuation factors mainly occurs when the incident zenith angle is large, and it is difficult to measure the DOP of the target at a large rotation angle of the rotatable pedestal. Establishing how to measure the DOP of the target at a large incident zenith angle to demonstrate our integral geometric attenuation factor is a problem we need to solve in future.