Improved atmospheric effects elimination method for pBRDF models of painted surfaces

Ying Zhang; Yi Zhang; Huijie Zhao; Zeying Wang

doi:10.1364/OE.25.016458

1. Introduction

Polarization is an inherent characteristic of light. Any object on Earth can produce a specific polarization of light based on the surface configuration, inner configuration, and the angle of incident light. Compared to traditional remote sensing, polarization remote sensing describes the electromagnetic wave in more detail and produces more target information, which greatly improves the accuracy of target detection and recognition [1]. Thus, polarization has become a valuable tool in many applications, including clouds and atmospheric aerosol detection, geological exploration, soil analyses, and medical diagnoses [2,3].

The polarization information obtained from polarization remote sensing has been used for image defogging, target classification, and other qualitative research. However, most quantitative research has been limited to laboratory research and non-imaging methods. The outdoor quantitative imaging research is mainly affected by two external factors [4]: target surface reflection and atmospheric effects of light propagation. Target surface reflection is determined by the inherent features of the target such as its structural features, dielectric properties, and roughness. On the other hand, atmosphere effects of light propagation in the earth-atmosphere system are influenced by the size and shape of atmospheric molecules, aerosol optical thickness, and the azimuth and zenith of the sun. Under different atmospheric conditions, variations in skylight could seriously affect the polarization detection of ground objects.

The polarized state of light reflected by the painted surface of artificial targets carries information about the surface shape, material, thickness, and roughness. It is valuable for quantitatively identifying and classifying the target to get such information by modeling the painted surface and analyzing the reflected light [5]. Usually, the bidirectional reflectance distribution function (BRDF) defined by Nicodemus [6] is used to evaluate the reflectance property of natural or artificial targets. Ideally, BRDF measurements in a laboratory environment use a point illumination source and a corresponding radiometer. The detector location and the source location move in the hemisphere of sample. Improvements in imaging technology have greatly increased the BRDF measurement accuracy and reduced the workload.

When considering the polarization properties, the BRDF needs to be expanded to the polarimetric BRDF (pBRDF). The most well-known pBRDF model for painted surfaces was proposed by Priest and Meier [1]. The Priest–Meier model, which is based on the microfacet model, has good accuracy and is easy to calculate. However, this model does not account for wavelength variations. To compensate for this shortcoming, Renhorn et al. [7] and Butler et al. [8] proposed different models.

In the pBRDF model, the reflective pBRDF Mueller matrix is quantified by pBRDF measurements. Outdoor measurements of multiple incident polarization states provide the most generalized means of acquiring the pBRDF matrix elements. Polarimetric imaging polarimeters have been proposed by expanding the principles of the point radiometer to two dimensions using polarizers and retarders [9], such as interferometric configurations [10], polarization gratings [11, 12], Fourier transform [13], birefringent elements [14, 15] and liquid crystal variable retarders (LCVRs) [16–19].

In polarimetric remote sensing, outdoor measurements are unavoidable. Several field measurement experiments have been performed to detect polarization information. Litvinov et al. [20] measured the BRDF and bidirectional polarization distribution function (BPDF) for bare soil and vegetation surfaces using multi-angle, multi-spectral photopolarimetric airborne measurements of the Research Scanning Polarimeter (RSP) for the Aerosol Polarimetry Sensor instrument of the NASA Glory Project. Kupinski et al. [21] captured images using the Jet Propulsion Laboratory’s ground-based Multiangle SpectroPolarimetric Imager (Ground-MSPI) and compared the results with a model for the surface polarized bidirectional reflectance distribution matrix (BRDM). Shell [9] developed an imaging system consisting of a 12-bit silicon charge-coupled device (CCD), a linear polarization filter, and different bandpass filters. Bartlett et al. [22] used a spectro-polarimetric imager (SPI), which was based on a liquid-crystal tunable filter (LCTF), to capture spectro-polarimetric imagery outdoors. Diner et al. [23] recorded the polarization state of light reflected by objects using a ground-based multiangle spectro-polarimetric imager. In current outdoor polarized imaging detection experiments, the vast majority of imaging data is only used for qualitative research, and there is no quantitative inversion of the surface properties. Furthermore, the impact of atmospheric conditions in the accuracy of ground target detection has rarely been considered.

Measuring and modeling skylight are useful for eliminating the impact of atmospheric conditions. Outdoor skylight polarimetric measurements are greatly influenced by the rapid movement of atmospheric aerosols and clouds; therefore, Pust and Shaw [24,25] designed and deployed a full-sky imaging polarimeter based on LCVRs that can produce a full Stokes image of incoming skylight in less than 0.1 s. Comparisons between measurements and model MODTRAN-P calculations showed strong agreement [26]. However, instruments that measure skylight would shade the target, and the error caused by this shadow is much larger when the surroundings have higher shelters such as trees or buildings. Shell [9] proposed a shadowing and comparison method for acquiring pBRDF after eliminating the atmospheric effects. This method can also be used to eliminate the effect of downwelled skylight, but only preserves high accuracy in sunny conditions.

We have previously proposed a LCVR-based imaging polarimeter that can be utilized to measure the full Stokes vector of each pixel. Our polarimeter is a small, compact, high-precision device for performing polarimetric measurements without mechanically rotating elements.

In this study, a new atmospheric effects elimination method of combining shadowing method and radiative transfer (RT) model for field polarization measurement experiments is proposed. To assess the effectiveness of our method, the polarization characteristics of certain painted surfaces are measured using our LCVR-based imaging polarimeter. The pBRDF of the painted surface is calculated through Priest’s model. Moreover, we apply both our atmospheric effects elimination method and that of Shell to remove atmospheric effects. And then the surface root-mean-square (RMS) slope σ is inverted from the data before and after applying the atmospheric effects elimination methods. Finally, comparisons between these results indicate that our method effectively eliminates the atmospheric effects and improves the inversion accuracy of σ. Furthermore, it shows that our method is more accurate than that of Shell.

2. Principle

2.1 Modeling of polarimetric BRDF

To study the physical properties of an object, pBRDF models of the target are necessary for predicting the polarization state of reflected light for a given incident state. These models are a prerequisite to the quantitative analysis of polarimetric images in remote sensing.

A BRDF is defined to characterize optical scattering from surface reflections [24]:

f (θ_{i}, ϕ_{i}, θ_{r}, ϕ_{r}, λ) = \frac{d L_{r} (θ_{r}, ϕ_{r})}{d E (θ_{i}, ϕ_{i})} s r^{- 1}

Where the incident zenith and azimuth angles are given by θ_i and ϕ_i, respectively, and the reflected zenith and azimuth angles are given by θ_r and ϕ_r, respectively_. Here, λ indicates the wavelength, L_r is the radiance reflected from the surface with units of watts per square meter per steradian (w/m²/sr), and E is the incident irradiance with units of watts per square meter (w/m²), resulting in the BRDF with units of inverse steradian (sr⁻¹). Figure 1 illustrates the geometry of the BRDF definition.

Fig. 1 BRDF geometry

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The polarimetric BRDF is a more generalized case of the scalar BRDF in which the four-component Stokes vector replaces the usual scalar radiance. Since a Mueller matrix relates the incident and reflected Stokes vectors, the polarimetric relationship between the incident and reflected light can be given by

d L_{r} (θ_{r}, ϕ_{r}) = F (θ_{i}, ϕ_{i}, θ_{r}, ϕ_{r}, λ) d E (θ_{i}, ϕ_{i})

where L_r and E are the reflected Stokes vector and incident Stokes vector, respectively, and F is the Mueller matrix of pBRDF.

Here, we chose painted surfaces as our research target. As popular military camouflage targets, studying the polarized reflectance characteristics is of great importance in the monitoring and identification of painted surfaces.

Painted surfaces are usually described by the microfacet theory. This theory assumes that a rough surface with the roughness larger than or equal to the wavelength is composed of a collection of randomly oriented microfacets. Each microfacet acts as a specular reflector obeying Fresnel’s equations [1].

The commonly used facet distributions are the uniform distribution, the Breon distribution [20], and the Gaussian model distribution [27].

For the Gaussian distribution, the microfacet distribution function is given by

p (θ) = \frac{1}{2 π σ^{2} \cos^{3} (θ)} \exp (\frac{- \tan^{2} (θ)}{2 σ^{2}})

Where σ² is the slope variance related to the surface roughness or texture, and θ is the orientation angle of microfacets relative to the object surface normal. This model has commonly been used to describe the passive multiangle polarimetric characteristics of rough surfaces for determining the complex refractive index of materials [23].

The orientation angle of microfacets θ can be denoted as

\cos (θ) = \frac{\cos (θ_{i}) + \cos (θ_{r})}{2 \cos (β)}

\cos (2 β) = \cos (θ_{i}) \cos (θ_{r}) + \sin (θ_{i}) \sin (θ_{r}) \cos (ϕ_{i} - ϕ_{r})

Figure 2 illustrates the definitions of angles in a microfacet coordinate system [24], where OB is the surface normal, OC is the target microfacet normal, AO is the incident direction, and OD is the detection direction.

Fig. 2 Definitions of angles in a microfacet coordinate system.

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We generalize the work of Priest and Meier [1], the pBRDF can be written as:

F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r}) = \frac{G (θ_{i}, θ_{r}) p (θ)}{4 \cos (θ_{r}) \cos (θ_{i})} M (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})

Where

F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})

denotes the pBRDF Mueller matrix,

p (θ)

is the Gaussian distribution function, Geometrical attenuation factor G(θ_i, θ_r) is the attenuation effect of the reflected light caused by the ups and downs of micro-surfaces shadowing the incident light or reflected light [23]. The calculation method was proposed by Liu [29]

M (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})

is the Mueller matrix for the Fresnel reflection from each facet with the same size as

F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})

.

The incident light and reflected light can also be related by a Jones matrix:

(\begin{matrix} E_{s}^{r} \\ E_{p}^{r} \end{matrix}) = (\begin{matrix} T_{s s} & T_{p s} \\ T_{s p} & T_{p p} \end{matrix}) (\begin{matrix} E_{s}^{i} \\ E_{p}^{i} \end{matrix})

where the subscripts s and p denote the electric field components perpendicular to and parallel to the plane of incidence, respectively. The superscripts i and r denote the incident and reflected components, respectively. Jones matrix T is determined by the four angles fixing the observation geometry and the complex index of the sample

ε = n - i k

, which indicates the inherent characteristic of the target’s attenuation of incident light. The calculation method of Jones matrix T, as given by [8, 27], can be summarized as follows:

(\begin{matrix} E_{s}^{r} \\ E_{p}^{r} \end{matrix}) = (\begin{matrix} \cos (η r) & \sin (η r) \\ - \sin (η r) & \cos (η r) \end{matrix}) (\begin{matrix} a_{s s} & 0 \\ 0 & a_{p p} \end{matrix}) (\begin{matrix} \cos (η i) & - \sin (η i) \\ \sin (η i) & \cos (η i) \end{matrix}) (\begin{matrix} E_{s}^{i} \\ E_{p}^{i} \end{matrix})

\cos (η i) = \frac{\frac{\cos (θ i) + \cos (θ r)}{2 \cos (β)} - \cos (θ i) \cos (β)}{\sin (θ i) \sin (β)}

\cos (η r) = \frac{\frac{\cos (θ i) + \cos (θ r)}{2 \cos (β)} - \cos (θ r) \cos (β)}{\sin (θ r) \sin (β)}

a_{s s} = \frac{ε \cos (θ_{i}) - \sqrt{ε - \sin^{2} (θ_{i})}}{ε \cos (θ_{i}) + \sqrt{ε - \sin^{2} (θ_{i})}}

a_{p p} = \frac{\cos (θ_{i}) - \sqrt{ε - \sin^{2} (θ_{i})}}{\cos (θ_{i}) + \sqrt{ε - \sin^{2} (θ_{i})}}

Where

η i

represents the angle between the plane composed of the incident direction and the microfacet normal and the plane composed of the incident direction and the target surface normal, and

η r

represents the angle between the plane composed of the detection direction and the microfacet normal and the plane composed of the detection direction and the target surface normal.

According to the relation between the Jones matrix and the Mueller matrix [28], we have described M as

M = [\begin{matrix} \frac{1}{2} ({| T_{s s} |}^{2} + {| T_{s p} |}^{2} + {| T_{p s} |}^{2} + {| T_{p p} |}^{2}) & \frac{1}{2} ({| T_{s s} |}^{2} + {| T_{s p} |}^{2} - {| T_{p s} |}^{2} - {| T_{p p} |}^{2}) & Re (T_{s s} T_{p s}^{*} + T_{s p} T_{p p}^{*}) & - Im (T_{p s} T_{s s}^{*} + T_{p p} T_{s p}^{*}) \\ \frac{1}{2} ({| T_{s s} |}^{2} - {| T_{s p} |}^{2} + {| T_{p s} |}^{2} - {| T_{p p} |}^{2}) & \frac{1}{2} ({| T_{s s} |}^{2} - {| T_{s p} |}^{2} - {| T_{p s} |}^{2} + {| T_{p p} |}^{2}) & Re (T_{s s} T_{p s}^{*} - T_{s p} T_{p p}^{*}) & - Im (T_{p s} T_{s s}^{*} - T_{p p} T_{s p}^{*}) \\ Re (T_{s s} T_{s p}^{*} + T_{p s} T_{p p}^{*}) & Re (T_{s s} T_{s p}^{*} - T_{p s} T_{p p}^{*}) & Re (T_{s s} T_{p p}^{*} + T_{p s} T_{s p}^{*}) & - Im (T_{p s} T_{s p}^{*} - T_{s s} T_{p p}^{*}) \\ - Im (T_{s s} T_{s p}^{*} + T_{p s} T_{p p}^{*}) & - Im (T_{s s} T_{s p}^{*} - T_{p s} T_{p p}^{*}) & - Im (T_{s s} T_{p p}^{*} + T_{p s} T_{s p}^{*}) & Re (T_{s s} T_{p p}^{*} - T_{p s} T_{s p}^{*}) \end{matrix}]

Where the four parameters T_ss, T_ps, T_pp and T_sp can be calculated using Eqs. (7)-(12) and the asterisk indicates the complex conjugate.

Therefore, the pBRDF Mueller matrix can be expressed as:

F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r}) = \frac{1}{2 π} \frac{1}{4 σ^{2}} \frac{1}{\cos^{4} (θ)} \frac{\exp (- \frac{\tan^{2} (θ)}{2 σ^{2}})}{\cos (θ_{r}) \cos (θ_{i})} M (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r}) G (θ_{i}, θ_{r})

In this model, three target parameters are involved, including the real and imaginary parts of the index of refraction n and κ, respectively, and the surface roughness σ. Moreover, $F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})$ in Eq. (14) can be calculated using Eqs. (7)–(13). The pBRDF Mueller matrix also has properties of anisotropy related to the observation geometry.

Therefore, if we want to study the pBRDF properties of painted surfaces, we need to establish certain geometry conditions. The changes of f₀₀ with θ_r are illustrated in Fig. 3 and Fig. 4. Other parameters are σ = 0.30, 0.20, 0.1, 0.07 and 0.04, respectively, θ_i = 60°, $ε = 1.35 - 0.008 i$ , in-plane $ϕ_{i} - ϕ_{r} = 180 °$ in Fig. 3 and non-in-plane $ϕ_{i} - ϕ_{r} = 90 °$ in Fig. 4, respectively.

Fig. 3 Change of f₀₀ of in-plane pBRDF with θ_r when σ = 0.30, 0.20, 0.10, 0.07 and 0.04.

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Fig. 4 Change of f₀₀ non-in-plane of pBRDF with θr when σ = 0.30, 0.20, 0.10, 0.07 and 0.04.

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In Fig. 3, when the surface is rougher σ = 0.30, f₀₀ slowly increases with the increase of θr. When σ = 0.07, the peak value appears, indicating that the surface presents specular reflection characteristics. With σ further reduced to 0.04, the specular reflection characteristics become more obvious. In Fig. 3 and Fig. 4, the diffuse reflection characteristics of painted surfaces are also obvious.

2.2 Polarized form of the radiance composition reaching the imaging polarimeter

When we use the imaging polarizer to detect surface physical properties of an object, three main sources contribute to the radiance reaching the imaging polarimeter, as shown in Fig. 5. These contributions are reflected direct solar radiance from the target, target-reflected downwelled radiance, and upwelled atmospheric radiance resulting from solar scattered along the target-to-sensor path. Here, the scalar form of the governing equation that includes atmospheric scattering terms is discussed.

Fig. 5 Schematic of polarization calibration.

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1. Reflected direct solar radiance from the target (L_r)

The exoatmospheric solar irradiance E_s, which is non-polarization, propagates through the atmosphere along the solar-to-target path with a transmittance of τ_i. Then, it is reflected by the target surface and attenuated by the atmosphere along the ground-to-sensor path by τ_r. The bidirectional reflectance distribution function f_r is used to relate the reflected radiance to the incident irradiance. Therefore, L_r can be expressed as

L_{r} = τ_{r} (θ_{r}) f_{r} (θ_{i}, θ_{r}, ϕ) \cos θ_{i} τ_{i} (θ_{i}) E_{s} (θ_{i})

2. Target-reflected downwelled radiance from the sky (L_d)

Similarly, the total irradiance contribution to the target from the sky is calculated by integrating the downwelled radiance from every direction $L_{d}^{Ω_{i}} (θ_{i}, ϕ_{i})$ over the entire hemisphere, which related to the cosine of the incident angle θ_i from the surface normal. Each irradiance contribution is reflected by the surface BRDF f_r. Then, the total irradiance is attenuated by the target-to-sensor atmospheric transmittance τ_r.

Therefore, an appropriate expression for L_d is

L_{d} = τ_{r} (θ_{r}) \iint_{Ω_{i}} f_{r} (θ_{i}, θ_{r}, ϕ) \cos θ_{i} L_{d}^{Ω_{i}} (θ_{i}, ϕ_{i}) d Ω_{i}

3. Upwelled atmospheric radiance resulting from solar scatter along the target-to-sensor path (L_u)

The upwelled atmospheric radiance can be rather complicated, and the full representation will not be given here. The target reflectance and atmospheric conditions greatly influence their relative values. Thus, an approximate expression for L_u can be given as

L_{u} = L_{u} (θ_{r}, ϕ_{r})

To fully describe the sources of radiance reaching the polarimeter, Eqs. (15)-(17) are transformed into the polarized representation using the Mueller–Stokes formalism introduced in Section 2.1. Moreover, the three contributions are rewritten based on the polarized representation. The two scalars f_r and $L_{d}^{Ω_{i}} (θ_{i}, ϕ_{i})$ are replaced by stokes vector F_r and ${\vec{L}}_{d}^{Ω_{i}} (θ_{i}, ϕ_{i})$ respectively. The contributions can be written as

\vec{L_{r}} = τ_{r} (θ_{r}) F_{r} (θ_{i}, θ_{r}, ϕ) τ_{i} (θ_{i}) \cos θ_{i} E_{s} (θ_{i})

\vec{L_{d}} = τ_{r} (θ_{r}) \iint_{Ω_{i}} F_{r} (θ_{i}, θ_{r}, ϕ) \cos θ_{i} {\vec{L}}_{d}^{Ω_{i}} (θ_{i}, ϕ_{i}) d Ω_{i}

\vec{L_{u}} = \vec{L_{u}} (θ_{r}, ϕ_{r})

where F_r is polarimetric BRDF.

Finally, the total polarized radiance reaching the imaging polarimeter is expressed as

\vec{L_{s}} = \vec{L_{r}} + \vec{L_{d}} + \vec{L_{u}}

2.3 Theory of Shell’s atmospheric effects elimination method

In polarimetric remote sensing of object features, given the polarized radiance reaching the sensor $\vec{L_{s}}$ , $F_{r}$ can be estimated by the following procedure:

\vec{L_{r}} = \vec{L_{s}} - \vec{L_{d}} - \vec{L_{u}}

Substitution and rearrangement of Eq. (22) yields

τ_{r} F_{r} τ_{i} \cos θ_{i} E_{s} = \vec{L_{s}} - τ_{r} \iint_{Ω_{i}} F_{r} \cos θ_{i} {\vec{L}}_{d}^{Ω_{i}} d Ω_{i} - \vec{L_{u}}

With the atmospheric irradiance randomly polarized, only the first column of the pBRDF Mueller matrix can be obtained. In fact, overhead polarimetric remote sensing is usually restricted to the first column because solving for the other matrix elements requires illumination by varying the polarization states. In field measurements, this is impossible.

From Eq. (16), the first column of the pBRDF is given by:

[\begin{matrix} f_{00} \\ f_{10} \\ f_{20} \\ f_{30} \end{matrix}] = \frac{\vec{L_{s}} - τ_{r} \iint_{Ω_{i}} F_{r} \cos θ_{i} {\vec{L}}_{d}^{Ω_{i}} d Ω_{i} - \vec{L_{u}}}{τ_{r} τ_{i} \cos θ_{i} E_{s}}

The polarized representations $\vec{L_{d}}$ and $\vec{L_{u}}$ have high spatial variability with atmospheric conditions. The atmospheric effects are so complex that it is difficult to quantify these effects in field measurements. Therefore, the method illustrated in Fig. 6 is used for outdoor pBRDF measurements.

Fig. 6 Shell’s atmospheric effects elimination method

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Under certain observation geometries, measurements will be referred to as image C in Fig. 6, when the target surface is illuminated by both the sun and downwelled sky radiance. Moreover, when shelter the sun, namely imaging the shadowed surface, measurements will be referred to as image D in Fig. 6. Image D data quantifies the effects resulting from the downwelled radiance from the sky [28]. Therefore, it is obvious that [9]:

L_{r} = (L_{r} + L_{d} + L_{u}) - (L_{d} + L_{u}) \propto C - D

Using the diffuse reflectance standard (DFS) as a calibration target, the pBRDF of any painted surface can be calculated with the comparison method. The DFS has a highly Lambertian reference surface and an approximately angular-invariant BRDF of ρ/π with a nearly randomly polarized reflectance of ρ. Using the same method as that for the target surface, images of the DFS are taken both in sun and in shadow, as denoted by image A and B in Fig. 6, respectively. Over a short time period with certain observation geometry by considering τ_i, τ_r, and E_s to be invariant, the following relationship can be determined:

\frac{L_{r}^{c a l}}{L_{r}^{s u r}} = \frac{τ_{r} f_{r}^{c a l} τ_{i} \cos θ_{i} E_{s}}{τ_{r} f_{r}^{} τ_{i} \cos θ_{i} E_{s}} = \frac{f_{r}^{c a l}}{f_{r}^{}} = \frac{\frac{ρ}{π}}{f_{r}} \to f_{r} = \frac{ρ}{π} \frac{L_{r}^{s u r}}{L_{r}^{c a l}}

This relationship can also be expressed as

f_{r} = \frac{ρ}{π} [\frac{C - D}{A - B}]

Further, the scalar formula is transformed into the polarized formula as

f_{r} = [\begin{matrix} f_{00} \\ f_{10} \\ f_{20} \\ f_{30} \end{matrix}] = \frac{ρ [(\vec{L_{r}} + \vec{L_{d}} + \vec{L_{u}}) - (\vec{L_{d}} + \vec{L_{u}})]}{\frac{1}{2} π (A_{0} - B_{0})} = \frac{ρ \vec{L_{r}}}{\frac{1}{2} π (A_{0} - B_{0})}

where A₀ and B₀ denote the measured the first elements of the Stokes vectors of the DFS in sun and in shadow, respectively.

\vec{L_{r}} + \vec{L_{d}} + \vec{L_{u}}

and

\vec{L_{d}} + \vec{L_{u}}

denote the measured Stokes vectors of the target surface in sun and in shadow, respectively. The data can be obtained by employing the LCVR-based imaging polarimeter discussed earlier.

The core of atmospheric effects elimination method is that incident light in sun direction is non-polarization and DFS is used to normalize the radiation of incident light. After steps of Eqs. (25)-(28), f_r can be treated as the stokes vector of the reflected light from the target corresponding to the incident light’s stokes vector [1 0 0 0]. That is also the four components of the first column of the target pBRDF matrix in Eq. (14).

Shell’s method directly measures the first column of the pBRDF matrix of the target without any other empirical functions. This method can effectively eliminate the atmospheric effects caused by $\vec{L_{d}}$ in most sunny weather and is suitable for targets with small area. Theoretically, Shell’s method is a very good solution for eliminating atmospheric effects in field experiments of pBRDF measurements.

2.4 Theory of the proposed atmospheric effects elimination method

Shell’s method requires that all targets and DFS should be in shadow; therefore, at one point, part of downwelled radiance could be blocked by a larger shelter. The error in Shell’s method is positively correlated with the target area and the proportion of scattered light in skylight, and the error in Shell’s method is negatively correlated with sun's zenith and the height from the shelter to the target [30]. To reduce the error in Shell’s method during field experiments, we can choose the appropriate shelter and smaller targets while controlling the height of the shelter. However, we cannot obtain real-time and accurate proportions of scattered light in skylight. Therefore, we propose an atmospheric effects elimination method to eliminate the impact of these errors.

We propose a parameter x, that means the proportion of the error caused by $\vec{L_{d}}$ in the calculation results $f_{r}$ in Shell’s method. Thus, considering the shadow area, Eq. (25) can be modified as

(\vec{L_{r}} + \vec{L_{d}} + \vec{L_{u}}) - (\iint_{1 - Ω_{0}} {\vec{L}}_{d}^{Ω_{0}} d Ω_{i} + \vec{L_{u}}) \propto C - D

Where

Ω 0

represents the shadow area. The Eq. (28) should be written as follows:

f_{r}^{s} = [\begin{matrix} f_{00}^{s} \\ f_{10}^{s} \\ f_{20}^{s} \\ f_{30}^{s} \end{matrix}] = = \frac{ρ (\vec{L_{r}} + \iint_{Ω_{0}} F_{r} \cos θ_{i} {\vec{L}}_{d}^{Ω_{0}} d Ω_{0})}{\frac{1}{2} π (A_{0} - B_{0})} = \frac{ρ \vec{L_{0}}}{\frac{1}{2} π (A_{0} - B_{0})}

The superscript s in Eq. (30) represents the vector

f_{r}^{}

calculated by the Shell’s method. The

\vec{L_{0}}

in Eq. (30) means the total reflected radiation from target received by polarimeter in Shell’s method. And then we can calculate the parameter x as:

x = \frac{\iint_{Ω_{0}} F_{r} \cos θ_{i} {\vec{L}}_{d}^{Ω_{0}} d Ω_{0}}{\vec{L_{0}}}

Where the

F_{r}

is equal to

F (θ_{i}, θ_{r}, ϕ_{i} - ϕ_{r})

in Eq. (14) for target and

ρ / π

in Eq. (26) for DFS. The three parameters

F_{r}

,

Ω 0

and

{\vec{L}}_{d}^{Ω_{0}}

can be estimated by several steps in red dashed box in Fig. 7. Finally, the corrected

f_{r}^{}

can get as:

f_{r}^{} = [\begin{matrix} f_{00}^{} \\ f_{10}^{} \\ f_{20}^{} \\ f_{30}^{} \end{matrix}] = = R [\begin{matrix} f_{00}^{s} \\ f_{10}^{s} \\ f_{20}^{s} \\ f_{30}^{s} \end{matrix}]

where the R in Eq. (32) is the ratio of x for target and x for DFS.

Fig. 7 Proposed atmospheric effects elimination method.

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Our proposed atmospheric effects elimination method is shown in Fig. 7.

In Fig. 7, most of steps in the blue and green dashed boxes are the same as those in the process of Shell’s method in Fig. 6. The red dashed box shows the process of using RT model to eliminate the error from Shell’s method and calculate the parameter x. Red solid boxes indicate the added steps in our proposed method. The RT model in Fig. 7 requires additional input parameters, namely the local aerosol conditions and weather conditions.

Step 5 in red dashed box creates a hypothetical target using Eq. (14) to reflect the downwelled skylight in the detection direction, which is blocked by the shelter in Image D and simulated by the RT model, to compensate for the downwelled skylight radiance in C–D in Eq. (29) caused by a larger shelter. The input parameters in Eq. (14) can refer to the results of Shell’s method or to other measurements for the same material target.

2.5 Genetic algorithm for inversion

In the atmospheric effects elimination method, the first line of pBRDF matrix is obtained under different directional behavior. For painted surfaces, there are three unknown parameters in microfacet pBRDF model, namely the real part n and imaginary part k of the complex refractive index and the surface RMS slope σ. Here, the surface roughness parameter is inverted through a genetic algorithm (GA), and the objective function is set as

\min f (n, k, σ) = \frac{\sum_{θ_{i}} \sum_{θ_{r}} \sum_{ϕ} {[f_{s}^{m} (θ_{i}, θ_{r}, ϕ) - f_{s} (θ_{i}, θ_{r}, ϕ, n, k, σ)]}^{2}}{\sum_{θ_{i}} \sum_{θ_{r}} \sum_{ϕ} {[f_{s}^{m} (θ_{i}, θ_{r}, ϕ)]}^{2}}

where

f_{s} (θ_{i}, θ_{r}, ϕ, n, k, σ)

is the simulation value with Eq. (14), and

f_{s}^{m} (θ_{i}, θ_{r}, ϕ)

is the measured data. The GA will try different groups of the three values, including n, k, and σ, within the default range to find the minimum

f (n \min, k \min, σ \min)

. The

n \min

,

k \min

and

σ \min

are the inversion results, which are the most consistent values in the pBRDF model with measurements.

3. Experimental validation

3.1 Outdoor polarization imaging experiments

The polarimetric images of targets were captured using an LCVR-based imaging polarimeter. Outdoor measurements of pBRDFs of certain target surfaces were performed in our university stadium (E116.34° N39.98°) under different weather conditions, including cloudy and partly cloudy. Figure 8 shows the entire system placed on a tripod mount with a compass, level, and 180° rotating platform. Using this platform, the system could be rotated, and the viewing geometry could be changed. Under certain atmospheric conditions, targets with several surface roughnesses were imaged at different elevation angles and azimuthal angles at different wavelengths. During the short period of measurement, the solar position was considered to be invariant so that we could study the change of pBRDF elements with the change of surface roughness and viewing elevation angle. In this way, the polarization state reflected by the targets and the DFS were measured under two illumination conditions (in sun and in shadow), and the first line of the pBRDF matrix could be calculated with the our method.

Fig. 8 The imaging polarimeter placed on a tripod mount.

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We chose six panels with different surface roughnesses as targets for research, and we attached these panels to board to fix their relative position as shown in Fig. 9. Each panel area should be sufficiently large to average out high spatial frequency inhomogeneity or texture in the target material. Second, the viewing angle of the targets and standards were assumed to be the same for they showed an insignificant difference between sets of images. In addition, two DFS were used separately with reflectance of 30% and 70%. During the experiment, an opaque shelter cloth is used for shadowing, and it is adjusted with the real-time position of the sun.

Fig. 9 Experimental scene and painted target

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Figure 10 shows the scene at the 476-nm band acquired at 12:00 on 28 October 2016, with respect to the intensity and degree of polarization (DOP). Different targets with different surface roughnesses clearly have different DN (Digital Number) values and DOP values.

Fig. 10 The 476-nm measurement data acquired 28 October 2016.

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There are also other objects in the scene, which are of no concern in this study. From the DOP image, the two DFS in the Fig. 10(b) are almost unpolarized, as expected. Multi-angle data were acquired by collecting images at varying viewing elevation angles of 9°, 12°, 15°, 20°, 24°, and 27° on October 28, 2016, in cloudy conditions and 8°, 10°, 13°, 17°, 20°, 26°, 29°, and 34° on October 29, 2016, in partly cloudy conditions.

Using the measurements data of Target 1 as an example, the change of f₀₀ with the increase of viewing elevation angle at 476 nm with different elimination method are shown in Figs. 11(a)-11(c), respectively. Blue and red curves represent cloudy and partly cloudy, respectively. The solid line is the theoretical value of f₀₀ with $n = 1.7$ , $k = 1.2$ and $σ = 0.25$ , and the dotted line is the measurements.

Fig. 11 The change in f₀₀ of Target 1 at 476 nm with respect to the viewing elevation angle using different atmospheric effects elimination methods.

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The calculation results of parameter x in Eq. (32) and direction parameters for each group of data is list in Table 1.

Table 1. Direction parameters for each group of measurements data

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The difference between the solid line and the dotted line in Figs. 11(a)-11(c) indicates the effect of the L_d. The experimental results show that pBRDF matrix elements calculated using our proposed elimination method have a smaller difference under different weather conditions than those of other methods. Furthermore, we can see in Fig. 11(b), the f₀₀ is far more less than 0.01 and Shell's method show the elimination effect for the error caused by L_d in such a cloudy weather is not good. The f₀₀ results after our method in Fig. 11(c) are closer to the theoretical value than results after Shell’s method.

Comparing between the two curves in the Fig. 11(a)-11(c), polarimetric characteristic measurements of targets can be greatly affected by bad weather conditions. Our proposed atmospheric effects elimination method together with LCVR-based polarization detection can effectively eliminate the errors caused by different weather conditions.

3.2 Results and errors of surface roughness inversion

Using the GA, the surface RMS slope σ is obtained under different circumstances using the corrected data. The parameters of GA were set as follows: the elite selection operation is chosen, the population size was set to 1000, the chromosome size was set to 46, the number of iterations was set to 200, the crossover probability was 0.2, and the mutation probability was 0.8.

Both f₁₀ and f₂₀ are very sensitive to the type and size of atmospheric particles [31], which move rapidly in the sky and are difficult to measure. We use f₀₀ as the main parameter for the inversion. The real surface parameters σ of targets was measured using a Taylor Hobson Talysurf 30, which has measuring accuracy of four decimal places. Then, σ, n and k were inverted using the GA.

Target 1 and 2 are green paint on wood and aluminum plates, respectively. Their inverted and measured values of σ, n and k are shown in Table 2. Table 2 also lists the results obtained using the data without correction, after atmospheric correction with Shell’s method and our method. After atmospheric effects elimination method, the inversion accuracy of σ has improved significantly. The Target 1 error is reduced from 26.50% to 6.81% and 4.82% using Shell’s method and our method, respectively, and the Target 2 error is reduced from 18.38% to 7.05% and 4.23% using Shell’s method and our method, respectively.

Table 2. The inverted σ and complex refractive index of Targets 1 and 2 and their Real σ

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Thus, by using data obtained through an atmospheric effects elimination method, the inversion accuracy of σ is greatly improved. Our new method is shown to be more accurate than Shell’s method. The main sources of error in our method could be a result of

a. changes in the sun’s position during detection,
b. changes in the weather conditions during detection,
c. neighborhood effects of targets,
d. instrument calibration errors, and artificial errors during measurement.

These errors will need to be minimized in a future study.

4. Conclusions

Measurements of the polarimetric BRDF characteristics of painted surfaces with a LCVR-based imaging polarimeter were performed, and an atmospheric effects elimination method using the RT model was proposed in this paper. The microfacet model for pBRDF was analyzed to study the characteristics of pBRDF elements. Then, the polarized form of the composition of radiance reaching the imaging polarimeter was analyzed, and the theory behind Shell’s atmospheric effects elimination method and that of the proposed atmospheric effects elimination method was introduced. Finally, the GA for surface roughness and complex refractive index inversion using calculated data was introduced. These theories were analyzed using the results of outdoor experiments with data collected from an LCVR-based imaging polarimeter. After obtaining the pBRDF elements of painted targets, the surface RMS slope σ and complex refractive index $ε = n - i k$ were inverted through GA. The inverted results showed that the inversion accuracy of σ is greatly improved by using data obtained after applying atmospheric effects elimination methods, indicating the effectiveness of the atmospheric effects elimination method. Finally, a comparison of the inversion σ using data after applying Shell’s method and our proposed method showed that our method was more effective at eliminating the atmospheric effects to reduce the error of the inversion σ from 6.81% to 4.82% for Target 1 and 7.05% to 4.23% for Target 2.

Thus, our proposed method for eliminating atmospheric effects can effectively improve the quantitative accuracy of the target physical parameters in polarimetric imaging remote sensing, which improves the accuracy of target detection and material classification. A follow-up study will include determination of the complex refractive index of targets and the surface roughness in the polarization detection.

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) (61571029).

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October 28, 2016
Group	$θ_{i}$ (°)	$ϕ_{i}$ (°)	$θ_{r}$ (°)	$ϕ_{r}$ (°)	G	X
1	54	−14.	81	88	0.94	0.9985
2	56	−22	78	88	0.96	0.9825
3	58	−30	75	88	0.98	0.9455
4	63	−35	70	88	0.99	0.8976
5	57	−26	66	88	1.0	0.7047
6	64	−42	63	88	1.0	0.5887

		Real σ	Inverted σ	Errors	Inverted complex refractive index
No correction	Target 1	0.257	0.3251	26.50%	1.9878-1.1879i
No correction	Target 2	0.234	0.2770	18.38%	2.0992-1.2241i
Shell’s method	Target 1	0.257	0.2745	6.81%	2.0882-1.1253i
Shell’s method	Target 2	0.234	0.2505	7.05%	2.1250-1.3277i
Proposed method	Target 1	0.257	0.2694	4.82%	2.1144-1.2687i
Proposed method	Target 2	0.234	0.2439	4.23%	2.1309-1.2412i

October 28, 2016
Group	$θ_{i}$ (°)	$ϕ_{i}$ (°)	$θ_{r}$ (°)	$ϕ_{r}$ (°)	G	X
1	54	−14.	81	88	0.94	0.9985
2	56	−22	78	88	0.96	0.9825
3	58	−30	75	88	0.98	0.9455
4	63	−35	70	88	0.99	0.8976
5	57	−26	66	88	1.0	0.7047
6	64	−42	63	88	1.0	0.5887

		Real σ	Inverted σ	Errors	Inverted complex refractive index
No correction	Target 1	0.257	0.3251	26.50%	1.9878-1.1879i
No correction	Target 2	0.234	0.2770	18.38%	2.0992-1.2241i
Shell’s method	Target 1	0.257	0.2745	6.81%	2.0882-1.1253i
Shell’s method	Target 2	0.234	0.2505	7.05%	2.1250-1.3277i
Proposed method	Target 1	0.257	0.2694	4.82%	2.1144-1.2687i
Proposed method	Target 2	0.234	0.2439	4.23%	2.1309-1.2412i

Improved atmospheric effects elimination method for pBRDF models of painted surfaces

Abstract

1. Introduction

2. Principle

2.1 Modeling of polarimetric BRDF

2.2 Polarized form of the radiance composition reaching the imaging polarimeter

1. Reflected direct solar radiance from the target (L_r)

2. Target-reflected downwelled radiance from the sky (L_d)

3. Upwelled atmospheric radiance resulting from solar scatter along the target-to-sensor path (L_u)

2.3 Theory of Shell’s atmospheric effects elimination method

2.4 Theory of the proposed atmospheric effects elimination method

2.5 Genetic algorithm for inversion

3. Experimental validation

3.1 Outdoor polarization imaging experiments

3.2 Results and errors of surface roughness inversion

4. Conclusions

Funding

References and links

Cited By

Figures (11)

Tables (2)

Equations (33)

Optics Express

October 29, 2016
1	55	14	82	83	0.93	0.9999
2	54	7	80	83	0.95	0.9999
3	53	-1	77	83	0.97	0.9991
4	54	-9	73	83	0.99	0.9847
5	57	-23	70	83	0.99	0.8926
6	58	-28	64	83	1.0	0.7441
7	62	-37	61	83	1.0	0.5987
8	64	-42	56	83	1.0	0.4660

Abstract

1. Introduction

2. Principle

2.1 Modeling of polarimetric BRDF

2.2 Polarized form of the radiance composition reaching the imaging polarimeter

1. Reflected direct solar radiance from the target (Lr)

2. Target-reflected downwelled radiance from the sky (Ld)

3. Upwelled atmospheric radiance resulting from solar scatter along the target-to-sensor path (Lu)

2.3 Theory of Shell’s atmospheric effects elimination method

2.4 Theory of the proposed atmospheric effects elimination method

2.5 Genetic algorithm for inversion

3. Experimental validation

3.1 Outdoor polarization imaging experiments

3.2 Results and errors of surface roughness inversion

4. Conclusions

Funding

References and links

Cited By

Figures (11)

Tables (2)

Equations (33)

Optics Express

1. Reflected direct solar radiance from the target (L_r)

2. Target-reflected downwelled radiance from the sky (L_d)

3. Upwelled atmospheric radiance resulting from solar scatter along the target-to-sensor path (L_u)