1. Introduction

Polarization imagery can be used to help identify objects or to discriminate man-made objects from natural clutter [1–5]. Most naturally occurring materials do not have a significant polarimetric signature, however, many man-made objects do. Recent works on polarimetric measurement systems have shown the enormous advantage of polarimetric measurement in practical applications. Wang [6] proposed a matrix diffractive deep neural networks design framework that can rapidly and robustly facilitate the deployment of polarization information into meta-devices. In Ref. [7], Wang proposed a reflective terahertz (THz) metalens with four focal points for polarization detection of THz beams. Liu [8] proposed an end-to-end network model called a semantic-guided dual discriminator generative adversarial network to solve the polarization image fusion problem. Cheng [9] presented a physically based method using dual-polarization features and saliency information to solve the problem of reflection removal. Realistic modeling material appearance via the polarimetric bidirectional reflectance distribution function (pBRDF) is a vital prerequisite of physically-based synthetic polarized image simulation [10]. However, previous work in this area mainly focuses on the visible (VIS) and near-infrared (NIR) regions of the spectrum [11–14], where the polarization signatures are dominated by reflection radiance. In the mid-wave infrared (MWIR) and long-wave infrared (LWIR), the relative contributions and complementary polarization orientation between the thermally emitted and reflected background radiance make the passive polarimetric detection exceeding complicated [15,16]. In addition, the combination of both reflected and emitted polarized radiation tends to cancel or reduce the overall polarimetric signature, which makes it quite difficult to measure polarized model parameters. Therefore, it is necessary to model the polarimetric infrared radiance accurately to observe relationships that could be leveraged in polarized model parameters measurements with further analysis.

Various bidirectional reflectance distribution function (BRDF)/pBRDF models are proposed to model how objects reflect light [17–20]. There are mainly two approaches used for modeling BRDF/pBRDF: descriptive models and analytical models. Descriptive models are typically obtained by measurements and experiences [21], such kinds of BRDFs including Phong model [22], Blinn-Phong model [23], Ward model [24], and so on. Analytical models attempt to model from either physical optics or geometrical optics. This type of model includes the Cook-Torrance model (microfacet theory) [25], Beard-Maxwell model [26], and Beckmann-Spizzichino model [27]. When considering polarimetry, it is common to extend the microfacet theory to pBRDF models. The seminal work is that Priest and Germer [28] extend this theory to model pBRDF by replacing the unpolarized Fresnel term of the original model with an expression that accounts for changes in the polarization state. Thereafter, numerous pBRDF models based on microfacet theory have been developed in literature [29,30].

The radiance in the thermal infrared is known to be composed of the reflected and emitted radiance. Hence, in addition to the polarimetric reflected radiance modeling, the polarimetric emitted radiance should also be modeled by deriving polarized emissivity from Kirchhoff’s law [31]. Although rare, the theoretical modeling and experimental work in polarimetric radiance for the infrared spectrum have received some attention recently. Wellems [32] published a paper describing their work about LWIR polarimetric modeling and measurements. Gartley [15,33] investigate polarimetric modeling and measurements of remotely sensed scenes in the LWIR. Both of their pBRDF models are based on microfacet theory, which is the same as VIS and NIR. Besides, the polarized emissivity is obtained by Kirchhoff’s law. Concerning the emission polarization measurement, the key is to increase the difference between the incident and emitted radiance. In Ref. [33], the measurements were always done at night under a starlight sky to ensure the amount of incident light was much less than the level of emitted radiance. Jordan and Lewis [34] measure the emission polarization by mounting the samples to a thermal bath to maintain the temperature of approximately 70C and to keep the thermally emitted radiance level well above the incident radiance.

In this work, we focus on modeling the polarimetric radiance for the infrared spectrum as well as the experimental measurement of the polarized emissivity model parameters. The main contribution of our polarimetric model is that temperature is introduced and the effect of temperature on the polarization signal is analyzed. Since the previous measurement experiment has strict requirements on sample temperature and incident background radiance, our goal is to find a polarized parameters measurement approach that is relatively easy to implement and insensitive to environmental effects. To this end, the samples are heated to high temperatures to reduce the reflected radiance component. After that, the observed Stokes images from the surfaces are normalized to eliminate the dependence of each Stokes image on temperature, which means there is no need to maintain a uniform temperature of the sample surfaces. This enables us to efficiently and conveniently measure the polarized emissivity. This paper is organized as follows: The microfacet theory is shown to be accurate and simple in mathematical and always used for pBRDF modeling in the prior work [32,33], the background on polarization and microfacet theory are illustrated in Section 2. Section 3.1 adopts the microfacet pBRDF models to model the reflected radiance component. The polarized emissivity is subsequently derived by the conservation of energy and Kirchhoff’s law in Section 3.2. After that, we obtain the general surface polarimetric radiance equation in the thermal infrared and also the degree of linear polarization (DOLP) model in Section 3.3. The DOLP is commonly used to evaluate the polarization information of an optical field and correlated with the sample temperature. Therefore, we quantitatively analyze the relationship between the proposed DOLP model and temperature by the simulation of DOLP varied with the object temperature to find the polarized parameters measurement approach in Section 4. Section 5 details the experiment setup and compares the modeling and measurement results in the MWIR. Finally, the conclusions are presented in Section 6.

2. Theory and background

This section is the theoretical basis of the paper. The description of polarized light relies on the Stokes vector and Mueller matrix. Microfacet theory is the central of pBRDF modeling. Hence we give a brief review of polarization formalism and microfacet theory.

2.1 Polarization formalism

The Stokes vector is a four-element vector, developed by George Gabriel Stokes in 1852 for describing polarized or partially polarized light [35]:

Mueller matrix is a $4\times 4$ matrix characterizing the optical element or system’s effect on incident light. The outgoing Stokes vector $\boldsymbol {S}_{out}$ is related by the Mueller matrix $\boldsymbol {M}$ with incident Stokes vector $\boldsymbol {S}_{in}$ as:

2.2 Microfacet theory

For perfectly or close-to-perfectly flat materials, the surface reflectance can be modeled by Fresnel equations. This reflectance is often referred to as specular reflectance and the reflected light follows the principles of geometrical optics [21]. In this case, the reflected zenith angle $\theta _{r}$ is equal to the incident zenith angle $\theta _{i}$ and the difference between reflected azimuth angle $\phi _{r}$ and incident azimuth angle $\phi _{i}$ is exactly 180 degrees. However, most material surfaces in the real world are not perfectly flat. The surface roughness scatters the light in a spatial direction ranging from perfectly flat (specular reflectance) to perfectly diffuse (Lambertian). Therefore, the BRDF needs to be used to describe this spatial reflectance of rough surfaces. The BRDF can be split into two components, specular and diffuse, that is:

The specular component can be modeled by microfact theory. This method assume that the surface consists of a series of randomly oriented microfacets, each of which follows perfectly specular reflectance. The light is reflected on the mirror-like microfacet surface of materials and therefore holds strong polarization. The geometry of reflection is shown in Fig. 1. $\gamma$ is the angle between macroscopic surface normal $z$ and microfacet normal $z_{i}$. $\beta$ is the angle between microfacet surface normal $z_{i}$ and incident light and reflected light. Angle $\eta _{i}$ quantifies the rotation angle between the macroscopic and microfacet plane of incidence about the incident direction. Similarly, $\eta _{r}$ is the angle rotating from the macroscopic plane of reflection to the microfacet plane of reflection about the reflected direction.

 

Fig. 1. Reflection geometry of a rough surface with the assumption of being composed of microfacets.

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The specular reflection is expressed as:

In Eq. (5),

The expressions for all the 16 elements of the Fresnel reflection Mueller matrix $\boldsymbol {M}$ are therefore obtained. The more detailed expressions of the elements of Mueller matrix $\boldsymbol {M}$ can be found in numerous prior Refs. [28,29].

$P(\gamma,\sigma,B)$ is the microfacet slope distribution function. It describes the probability distribution of the orientation of the microfacet surface normal. For the reason that the slope distributions of most natural surfaces follow Gaussian distribution, a Gaussian function is used here:

$G(\gamma,\beta,\delta,\Omega )$ is the geometrical attenuation factor to account for masking and shadowing. It models the incident and reflected light blocked by adjacent microfacets. This function is instrumental in keeping the BRDF bounded and thus satisfying the conservation of energy. The $G$ function derived by Beard and Maxwell [26] utilized here:

3. Polarimetric model description

In the VIS and NIR region of the spectrum, detection relies on reflected radiance. However, in the 3-14 $\mu m$ spectral range, the accurate signature calculation requires a combination of both reflected $L_{r}$ and emitted radiance $L_{e}$:

For the passive polarimetry case, the polarimetric radiance equation in the form of Stokes vectors is given by:

In the following, the polarized reflected radiance $\boldsymbol {S}_{r}$ and polarized thermal emissive radiance $\boldsymbol {S}_{e}$ are modeled based on the pBRDF.

3.1 Polarimetric reflected radiance model

Consider a pair of incident and reflected light rays. Light incident on a rough surface within solid angle $d\omega _{i}$ at zenith angle $\theta _{i}$ and azimuth angle $\phi _{i}$. The reflected light is observed in the direction of solid angle $d\omega _{r}$, zenith angle $\theta _{r}$, and azimuth angle $\phi _{r}$, the azimuth difference with respect to the incident light is $\phi$. The relation between incident and reflected light is described by BRDF for unpolarized case:

In Eq. (13), $f(\theta _{i},\phi _{i},\theta _{r},\phi _{r},\lambda )$ is the BRDF and depends on both incident and reflected direction and wavelength. $dL_{r}(\theta _{r},\phi _{r},\lambda )$ is the reflected radiance. The incident irradiance $dE_{i}$ satisfies the following equation:

Substitute Eq. (14) into Eq. (13) and integrate over the full hemisphere above the surface, the radiance reflected from a surface in direction $(\theta _{r},\phi _{r})$ is thus obtained as:

When the incident and reflected radiance are replaced by Stokes vectors, polarized reflected radiance $\boldsymbol {S}_{r}(\theta _{r},\phi _{r},\lambda )$ is associated with polarized incident radiance $\boldsymbol {S}_{i}(\theta _{i},\phi _{i},\lambda )$ by the pBRDF:

In Eq. (16), $\boldsymbol {f}(\theta _{i},\phi _{i},\theta _{r},\phi _{r},\lambda )$ represents pBRDF, which takes the form of Mueller matrix. It should be noticed that the BRDF is the element in the first row and first column of pBRDF. Similar with BRDF, the pBRDF can be formulated as the sum of a specular component and a diffuse component:

In Eq. (17), the specular component $\boldsymbol {f}_{spec}$ is a $4\times 4$ Mueller matrix, which can be modeled by microfacet theory as mentioned in Section 2.1. The specular reflection occurs on the surfaces of an object and holds strong polarization. The diffuse reflection term describes light that is absorbed into the object to be unpolarized by subsurface scattering and linearly polarized again when reflected back out to the air. The diffuse component can be modeled as [12]:

The first diffuse reflection term originates in subsurface scattering. The second diffuse reflection term is Lambertian reflection, which is totally unpolarized and originates from multiple surface reflections when the surface is rough enough. The light is scattered equally in all directions and does not depend on the location of the observer.

Therefore, the pBRDF is eventually expressed as:

Substitute Eq. (20) into Eq. (16), the polarized radiance reflected in $(\theta _{r},\phi _{r})$ direction from a surface is thus obtained:

3.2 Polarimetric emitted radiance model

The polarized self-emission radiance can be derived from the conservation of energy. According to the conservation of energy, for opaque objects (the transmissivity $\tau =0$), the flux incident on an object can be reflected or absorbed. Therefore:

It should be noted that the material properties are the function of spectral and angular. Therefore, the forms of emissivity, absorptivity and reflectivity should be kept consistent to make sure the above formula is correct. In this case, the spectral directional emissivity in observe direction $(\theta _r,\phi _r)$ is expressed as:

When extending Eq. (24) to polarization case, it should be noted that the polarized emissivity takes the form of a four-element Stokes-like vector. However, the polarized DHR is a $4\times 4$ Mueller-like matrix. Thus, to obtain the polarized emissivity $\boldsymbol {\epsilon }$, the polarized DHR should be multiplied times blackbody emissivity Stokes vector $\boldsymbol {\epsilon }_{BB}=[\begin {array}{l} 1\quad 0\quad 0\quad 0 \end {array}]^T$ as follows:

Thus, the polarized emissivity Stokes vector is:

One can obtain the polarized emitted radiance by:

In Eq. (28), $L_{BB}(\lambda,T)$ is blackbody radiance, which is related to the object temperature and wavelength. The definition of $L_{BB}(\lambda,T)$ is given by:

3.3 Summary of the polarimetric radiance model

Combining the polarized reflectance radiance (Eq. (16)) and the polarized emitted radiance (Eq. (28)), the total energy observed in direction $(\theta _{r},\phi _{r})$ is thus given by:

If we consider the incident radiation is isotropic, the reflected term is thus rewritten as:

Therefore, the expression of total radiation in observed direction $(\theta _{r},\phi _{r})$ is finally obtained:

Furthermore, when the incident radiance is unpolarized, that is $\boldsymbol {S}_i=[\begin {array}{l} L_i\quad 0\quad 0\quad 0 \end {array}]^T$, the above expression is simplified as:

From Eq. (33), we can find that the property of the polarimetric state of emitted radiance is in contrast to reflected light. The balance between the reflected and emitted radiance terms dictates what polarized signature is detected for a given target.

Generally speaking, the DOLP is commonly utilized to quantify the polarization signal. At this point, the DOLP for the total observed polarized radiance in infrared is given by:

In addition, the DOLPs for reflected and emitted radiance are also listed here:

The above formulas (Eq. (34)-Eq. (36)) indicate that both $DOLP_r$ and $DOLP_e$ only depend on the properties of materials. However, the $DOLP_t$ is also related with the difference between $L_i$ and $L_{BB}(\lambda,T)$. The quantitative analysis and simulation of $DOLP_t$ will be presented in the next section. In addition, according to the definition of the angle of polarization (AOP), the relationship between the observed, reflected and emitted AOP is easily obtained:

The above equation means that the light reflected and emitted from a surface has the same AOP when the incident radiance is unpolarized. In other words, the AOP only depends on the inherent material properties and is unrelated to the temperature.

4. Simulation

This section examines the effect of the difference between the incident background radiance $L_i$ and the blackbody radiance $L_{BB}(\lambda,T)$ (relies on temperature $T$) on polarimetric signature in the infrared. The DOLP model proposed in the last section is utilized to quantify this influence. The simulation results provide the basis for polarized emissivity measurement methods in Section 5. We use the polarized model parameters of flat black paint measured by Gartley here. The specific parameters can be consulted in Ref. [33].

Figure 2(a)–2(c) plot the $f_{00}$, $f_{10}$ and $f_{20}$ components of pBRDF as a function of relative azimuth angle (from 80 to 280 degrees) for flat black paint. In addition, the polarized emissivity versus reflected zenith angle is plotted in Fig. 2(d). In this case, assume that the observed zenith angle $\theta _{r}$ is between 0-70 degrees. The relative azimuth is 180 degrees to keep in the specular in-plane. It is common to ignore the circular polarization in most infrared polarized measurement applications, which means the $f_{30}$ component of pBRDF and $\epsilon _3$ component of polarized emissivity are equal to zero, which are not plotted here. It should be noted that although the $f_{20}$ component is non-zero, it is anti-symmetric about the specular azimuth angle 180 degrees (as shown in Fig. 2(c)) in the emissivity integral equation (Eq. (27)), resulting in the $\epsilon _2$ component down to virtually zero (as shown in Fig. 2(d)). The negative $\epsilon _1$ component of the polarized emissivity indicates that the thermally emitted radiance has a vertical polarization orientation (p-polarized), which is in contrast to that of reflected radiance (s-polarized). Both the $\epsilon _0$ and $\epsilon _1$ component falls off with the observed zenith angle increasing.

 

Fig. 2. The $f_{00}$(a), $f_{10}$(b) and $f_{20}$(c) components of pBRDF and the polarized emissivity(d) for flat black paint.

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Due to the $\epsilon _2$ component being virtually zero, the DOLPs are therefore simplified as:

Figure 3 plots the simulation results of DOLP according to Eq. (38)- Eq. (40). The relationship between $DOLP_t$ and $L_{BB}(\lambda,T)/L_i$ is shown in Fig. 3(a). We can find that the $DOLP_t$ is approximately equal to $DOLP_r$ when $L_{BB}(\lambda,T)\ll L_i$. As the $L_{BB}(\lambda,T)$ increasing, $DOLP_t$ changes from $DOLP_r$ to zero as shown in the stage $\boldsymbol {B}$ of Fig. 3(a). Specially, when $L_{BB}(\lambda,T)=L_i$, the $DOLP_t$ is equal to zero. This means a surface with strong reflection and emission properties may appear to have no polarization when the level of radiance reflected from the surface is comparable to that emitted from the surface. In this case, the total radiance observed from the surface in direction $(\theta _{r},\phi _{r})$ is $\boldsymbol {S}_{total}=[\begin {array}{l} L_i\quad 0\quad 0\quad 0 \end {array}]^T$, which is exactly equal to incident radiance Stokes vector $\boldsymbol {S}_i$. On the contrary, if $L_{BB}(\lambda,T)>L_i$, $DOLP_t$ will vary from zero to $DOLP_e$ as shown in the stage $\boldsymbol {D}$ of Fig. 3(a). Another extreme case is the $DOLP_t$ approaches to $DOLP_e$ when $L_{BB}(\lambda,T)\gg L_i$. In conclusion, the $DOLP_t$ satisfies the following equation:

 

Fig. 3. Simulation results of DOLP. (a)The $DOLP_t$ varies with $L_{BB}(\lambda,T)/L_i$ when $\theta _r=35$ degrees. The curves of $DOLP_t$, $DOLP_r$, and $DOLP_e$ at $\theta _r=0-70$ degrees when (b) $L_{BB}(\lambda,T)<L_i$, (c) $L_{BB}(\lambda,T)>L_i$, and (d) $L_{BB}(\lambda,T)\gg L_i$.

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Figure 3(b)–3(d), corresponding to stage $\boldsymbol {B}$, $\boldsymbol {D}$, and $\boldsymbol {E}$ in the Fig. 3(a) respectively, plot the DOLP curves at $\theta _r=0-70$ degrees. The green, blue, and orange curves in Fig. 3(b)–3(d) represent $DOLP_t$, $DOLP_r$, and $DOLP_e$ respectively. It is found that the green curves $DOLP_t$ vary from $DOLP_r$ to $DOLP_e$, and eventually approximately coincident with $DOLP_e$. These three images illustrate that the $DOLP_t$ for each observed angle has the same variation tendency as that at $\theta _r=35$ degrees (shown in Fig. 3(a)).

Based on the above analysis, to measure the polarization characteristics of a material, the difference between $L_i$ and $L_{BB}(\lambda,T)$ should be increased as much as possible. It is much easier to control the sample temperature T (that is $L_{BB}$) than incident radiance, hence we will change the sample temperature in the following experimental measurements to increase the difference between thermally emitted radiance and incident radiance.

In order to find out the suitable heating temperature for the laboratory measurement environment, at which the $DOLP_t$ is approximately equal to $DOLP_e$, we conduct the simulation experiments with the assumption that the sample temperature is $250K$, $300K$, $350K$ and $400K$, the incident background radiance is $1.1e^{-4} W/cm^2/sr$. The simulations are undertaken in the MWIR $(3-5\mu m)$ region. In this case, it is easy to calculate that $L_{BB}(\lambda,T)/L_i$ is 0.16, 1.47, 7.26, and 24.71 for $T=250K$, $300K$, $350K$ and $400K$ respectively. In addition, to evaluate the difference between total observed and emitted lights, the relative radiance deviation and relative DOLP deviation are defined:

Figure 4 shows the $\Delta _{radiance}$ and $\Delta _{DOLP}$ results varying with $L_{BB}(\lambda,T)/L_i$ at $\theta _{r}=$ 0, 35, and 70 degrees. $\Delta _{radiance}$ and $\Delta _{DOLP}$ are inversely proportional to $L_{BB}(\lambda,T)/L_i$. As the temperature increases, the increasing values of $L_{BB}(\lambda,T)/L_i$ lead to a virtually zero $\Delta _{radiance}$ and $\Delta _{DOLP}$. When $T=400K$, the maximum values of $\Delta _{radiance}$ and $\Delta _{radiance}$ appear at $\theta _{r}=$ 70 degrees, and the maximum values are 0.008 and 0.049 respectively. Such a small value means that the difference between total observed light and emitted light can be ignored.

 

Fig. 4. The relationship between the $\Delta _{radiance}$, $\Delta _{DOLP}$ and $L_{BB}(\lambda,T)/L_i$.

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Therefore, we can deduce that for laboratory measurement environment, if the temperature of an object is over $400K$, the thermal emitted radiance level is well above the ambient radiance that might be reflected from the sample surfaces. The total observed radiance and $DOLP_t$ are equal to thermally emitted radiance and $DOLP_e$ respectively, that is:

The depolarization effect on the observed polarimetric state from the reflected radiation can be ignored. Hence, we will utilize this property to measure the polarized model parameters of the materials in the next section.

5. Polarized emissivity measurement approach and results

In this section, a novel method for polarized emissivity measurement is proposed based on the previous analysis and simulations. The MWIR polarized measurement system includes a MWIR imaging polarimeter, a sample holder, and a turntable driven by a motor as shown in Fig. 5. The MWIR imaging polarimeter consists of a focusing lens, a rotating quarter wave plate, a linear polarizer and a $320\times 256$ HgCdTe detector. The instrument response to the $3.7-4.8 \mu m$ waveband range, and the field of view is $6^{\circ }\times 8^{\circ }$. The extinction ratio of the polarizer is better than 200:1. Three different heat-resisting materials marked as $m_a$, $m_b$ and $m_c$ are placed on the sample holder. The sizes of $m_a$ and $m_c$ are $30cm\times 30cm$, $m_b$ is $15cm\times 30cm$. All three materials to be investigated are high-temperature refractory materials. The observed angle for target polarized emissivity measurement is from 0 to 80 with an interval of 5 degrees. Four sets of measurement experiments (the linear polarizer oriented at angles of 0, 45, 90, and 135 degrees) for each viewing angle $\theta _{r}$ were completed in 6 seconds. All the $4\times 17$ measurement experiments were completed in 5 minutes.

 

Fig. 5. Photograph of our measurement system setup.

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Due to the complementary polarization orientation of reflected and emitted radiance, the emissivity measurement in the thermal infrared is difficult than that in the visible. The reflected radiance tend to reduce the total pilarimetric signature. Therefore, the key point of emissivity measurement in the infrared is either to decrease the incident radiance, which will influence the reflected radiance, or increase the emitted radiance. Gartley’s measurements [33] were done at night under a starlight sky to ensure the amount of incident light was much less than the level of emitted radiance. Besides, the imaging system and the ground surrounding the target sample were covered with aluminum foil. This measurement method is demanding for experimental conditions. Jordan and Lewis [34] measured the emission polarization by mounting the sample to a thermal bath to maintain the temperature of 70C. In the last section, we derived the sample temperature at which the reflected light can be approximatively negligible. Therefore, we can measure the parameters by heating the samples to high temperature and do not need to control the incident radiance, nor maintain the sample at a fixed temperature. The proposed measurement approach is insensitive to environmental effects.

Before measuring, the three materials to be investigated are heated by a high-temperature furnace to approximately $800K$ based on the previous analysis. It is possible to assume that the laboratory background radiation is negligible compared with thermally emitted radiance in such high-temperature. Therefore, the total observed Stokes vector is equal to the emitted Stokes vector $\boldsymbol {S}_{total}=\boldsymbol {S}_{e}$, where $\boldsymbol {S}_{total}$ is obtained by measurement radiance at polarizer orientations of 0, 45, 90 and 135 degrees as mentioned in Eq. (1). Due to the heat radiation effect and heat conduction effect, it is obvious that the temperature distribution on the materials surface is inhomogeneous as shown in the top subfigures of Fig. 6. In the $S_{t0}$ images, the middle part of the three materials on the sample holder has larger digital number (DN) than that of the margins part. Besides, the lower part of materials is brighter than the upper part. This means that the temperature is a function of coordinate position $(x,y)$ of the image. It is acceptable to ignore the time-dependence of temperature at each viewing angle because of the short measurement time (6 seconds). Therefore, $\boldsymbol {S}_{total}(x,y)$ of each observed angle only relate to position $(x,y)$:

 

Fig. 6. Schematic diagram of processing flow for obtaining $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$ images.

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It should be noted that the components of the polarized emissivity cannot be directly derived by

Equation (48) represents that the $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$ images only relate to the materials polarized emissivity and are temperature-independent. The temperature distribution on the surface of the material becomes irrelevant. It is shown in the bottom subfigures of Fig. 6 that the $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$ images have the uniform DN values of each material surface. The proposed method is shown to be fairly immune to environmental effects.

The measured $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$ data for various of observed angles is plotted in Fig. 7 and Fig. 8. For the reason of positive $\epsilon _0$ and negative $\epsilon _1$ and $\epsilon _2$, the $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$ data for three materials have negative values. Also, the plots in Fig. 9 show the measured $DOLP_t$ for the three high temperature samples at observed angles from 0 to 85 degrees. Different from $S_{t1}/S_{t0}$ and $S_{t2}/S_{t0}$, the measured $DOLP_t$ data have positive values. In order to further analysis, the mean of absolute values of $S_{t1}/S_{t0}$, $S_{t2}/S_{t0}$, and $DOLP_t$ for each sample are calculated and listed in Table 1 and plotted in Fig. 10. Among three materials, the $DOLP_t$, $\left |S_{t1}/S_{t0}\right |$, and $\left |S_{t2}/S_{t0}\right |$ of $m_b$ are the largest, whereas those of $m_a$ are the smallest. As for a specific material, it is observed that the $\left |S_{t1}/S_{t0}\right |$, $\left |S_{t2}/S_{t0}\right |$, and $DOLP_t$ increases obviously with the view angles increasing. Due to the assumption of azimuthal symmetry for the surface roughness, the $\epsilon _2$ component is always zero in theory. Although in reality, the measured $\left |S_{t2}/S_{t0}\right |$ data is non-zero, it is much less than $\left |S_{t1}/S_{t0}\right |$ and the difference between them gets larger when the view angles increase as shown in Table 1 and Fig. 10. Therefore, it is possible to ignore $\left |S_{t2}/S_{t0}\right |$. At this point, $\left |S_{t1}/S_{t0}\right |$ is approximately equal to $DOLP_t$. That is, $DOLP_t\approx \left |S_{t1}/S_{t0}\right |=\left |\epsilon _1/\epsilon _0\right |$. This conclusion is more intuitively plotted in Fig. 10. Our measured results are shown to be consistent with the theoretical analysis (see Eq. (45)).

 

Fig. 7. Images of measured $S_{t1}/S_{t0}$ for the three materials at each observed angle.

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Fig. 8. Images of measured $S_{t2}/S_{t0}$ for the three materials at each observed angles.

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Fig. 9. Images of measured $DOLP_t$ for the three materials at each observed angle.

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Fig. 10. The measured mean values of $\left |S_{t1}/S_{t0}\right |$, $\left |S_{t2}/S_{t0}\right |$, and $DOLP_t$ of three materials.

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Table 1. The measured mean values of $\left |S_{t1}/S_{t0}\right |$, $\left |S_{t2}/S_{t0}\right |$, and $DOLP_t$ for three materials.

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The polarized model parameters are unknown and can be determined by fitting experimental $\left |S_{t1}/S_{t0}\right |$ data with the modeling $\left |S_{t1}/S_{t0}\right |$ data. Before fitting, the initial values of model parameters are chosen. As discussed by Gartley in Ref. [33], the $\sigma$ value changing will influent the shape of $S_{t1}$ curves. The bias value $B$ associates with both the shape and magnitude of $S_{t1}$ curves. By varying the values of $\rho _d$ and $\rho _v$, the shape and magnitude of $S_{t0}$ curves will be changed. Based on these prior theories, we adjust these model parameters by visual assessment between the modeled and measured $\left |S_{t1}/S_{t0}\right |$ data. Therefore, the polarized emissivity model parameters results that produce the best fit of the experimentally measured $\left |S_{t1}/S_{t0}\right |$ data are obtained and listed in Table 2. The model parameters include roughness parameter $\sigma$, bias parameter $B$, complex index of refraction $(n, k)$, diffuse parameter $\rho _d$ and $\rho _v$, shadowing/masking parameters $\delta$ and $\Omega$. It should be noted that data for angle 85 degrees is plotted in Fig. 7–9 but not used for parameters fitting due to near grazing zenith angles.

Table 2. Table of the fitting polarized model parameters results.

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Figure 11 plots the modeled and measured $\left |S_{t1}/S_{t0}\right |$ data for the three materials. The predicted results are shown excellent agreement with measured data based on comparisons for angles 0-80 degrees (except for material $m_a$ at 80 degrees). The error between the modeled and measured data is of the order of 0.01 or smaller. In addition, the predicted polarized emissivity components for $m_a$, $m_b$ and $m_c$ are plotted in Fig. 12 based on the fitting parameters listed in Table 2. Among all the three materials, $m_a$ has the minimum decline rate of the polarized emissivity components, on the contrary, the decline rate of $m_b$ is the maximum. As a result, $m_a$ has the largest $\epsilon _0$ and $\epsilon _1$ (due to negative) components for different observed angles, this means $m_a$ emits the most amount of thermal radiance but has the minimum DOLP. Results show the utility of the proposed parameters acquisition technique. The problem of the dependence of the measured Stokes data on temperature is solved by the fit to the $\left |S_{t1}/S_{t0}\right |$. Hence, there is no need to maintain a consistent temperature of the sample surfaces. The environmental effects and systematic errors are eliminated by normalizing the measured data.

 

Fig. 11. Polarized emissivity model fit to experimentally measured $\left |S_{t1}/S_{t0}\right |$ for three samples.

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Fig. 12. The polarized emissivity for material $m_a$, $m_b$ and $m_c$ using the parameters in Table 2.

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6. Conclusions

In this paper, we have derived a comprehensive polarimetric radiance model and a DOLP model in the thermal infrared based on pBRDF. Both the reflected and emitted radiance is considered in the models. The polarization of a surface is found to be related to the object temperature according to the proposed polarized models. This enabled us to introduce an efficient acquisition technique of polarized emissivity model parameters by taking advantage of the temperature dependence of the surfaces polarization. The materials investigated are heated to reduce the contribution of the reflected radiance. Our measurement method does not require the precise control of the object temperature. The problem of the dependence of each Stokes image on temperature is solved by the fits to the $\left |S_{t1}/S_{t0}\right |$ instead of the $S_{t0}$ or $S_{t1}$ data. Besides, the normalizing of the data can eliminate environmental effects and systematic errors. The model parameters determined from $\left |S_{t1}/S_{t0}\right |$ fitting measurements can be used to investigate the material appearance and estimate polarimetric radiance. The utility of the polarized emissivity parameters acquisition experiments was demonstrated with three different heat-resisting materials in the MWIR. It is shown that the $\left |S_{t1}/S_{t0}\right |$ agreement between model and experiments data is excellent for the materials investigated. We believe that our work will enable further progress in the infrared appearance-from-polarization and polarimetric rendering.