Efficient polarimetric BRDF model

Ingmar G. E. Renhorn; Tomas Hallberg; Glenn D. Boreman

doi:10.1364/OE.23.031253

1. Introduction

There is an increasing interest in the propagation of polarized radiation scattered from rough surfaces for target detection and recognition applications [1]. Polarization imagery can help to discriminate manmade objects from background clutter. There is a growing appreciation that polarization is also of importance when the imaging sensor is not discriminating between different polarization states. This is due to the fact that polarization plays an important part in the scattering process and polarimetric models are therefore a natural choice. Real surfaces are however often at the crossroad where most vector theories require a rather smooth surface and therefore fail to account for the observed distributions. Approximate models are therefore still a necessity due to the complexity of realistic numerical scattering solutions [2] although they often do not predict the scattering with great accuracy. A possibility is to use measured data for materials with great variety in optical scattering phenomenology. Such measurements are however very time consuming and therefore too expensive in most cases. Good models based on scattering phenomenology are thus still in demand. In order to obtain a 10-percent accuracy for a general rough surface, the full empirical model might be needed [3]. As shown below, the semi-empirical model presented here can account for the observed scattering pattern for a wide diversity of surfaces with accuracy in general better than 10 percent.

A good model should also be able to describe in a reasonable way phenomenology outside the initial considerations. An example is the commonly used facet model [4] that also can catch some of the diffractive phenomenology. In practice, such a model is used outside its physics-based intention due to the versatility of the model. Phenomenology outside the initial intention is accounted for by the use of more or less elaborate shadowing functions.

In hyperspectral imaging, the observed spectrum is dependent on the acquisition scenario including the directional and diffuse illumination, atmospheric absorption and scattering and the bidirectional scattering with respect to surfaces. The hyperspectral data has to be corrected for all these effects prior to target classification. Ideally, this should be possible using the measurements of the scene combined with some a priori knowledge. Observation of the target from different relative positions could possibly be used together with reasonable physics- based models in order to obtain such a solution. The present model is an attempt in this direction.

Over the years, a physics-inspired model for rough surfaces and arbitrary incidence and scattering angles has been developed [5,6]. A surface transfer function is combined with a reflectivity polarization factor related to surface material properties which results in a both efficient and accurate model requiring very few measurements in order to determine the model parameters. The model has been validated against bead-blasted metallic surfaces, diffuse paint and clear-coated paint.

In similarity with the facet model, only linear polarization is treated here and depolarization in the scattering process is not taken into account in this manuscript. In the smooth surface limit the depolarization is also assumed to be small. In the region between these two extremes, the depolarization phenomenology is more complex. For rough surfaces involving multiple scattering, depolarization is certainly an effect that should be considered. Since depolarization is not measured, it is here disregarded and assumed to be small and therefore lumped into the linearly polarized parameter set both at the DHR measurement and at the BRDF measurement. This is a quite common approximation but the issue may deserve more detailed consideration in future work.

With respect to surface characteristics, many models assume a Gaussian autocorrelation function due to its simplicity. It was however observed quite early that many surfaces exhibited autocorrelation functions that deviate from a Gaussian distribution [7]. In practice, different types of surfaces can be classified into typical autocorrelation shapes. This has led to development of BRDF solutions applicable to specific classes of surfaces. This points to the need to develop a theory where the shape of the autocorrelation function of the surface, or rather the shape of its Fourier transform can be varied. In the model presented here, a generalized Gaussian distribution is used. The shape is varied using a single shape parameter that can be fitted with respect to the experimental observation.

The BRDF is used in a multitude of applications varying from 3-D graphics applications to the study of infrared signatures. The relationship between BRDF, DHR and emissivity is well established [8].

Based on the established connection between radiometry and polarized scattering, the polarimetric BRDF is represented by a 2 × 2 matrix. Only the diagonal elements are non-zero in the present treatment (thus ignoring depolarization). The BRDF is described by a four-parameter model. Two of the parameters are the effective complex index of refraction, n and k. These parameters are not trivial to determine and care has to be taken in order to obtain realistic values. For the purpose of the present model, it is natural to use the DHR as a function of angle of incidence for s- and p-polarized irradiance when determining n and k. In this process, it is important to correctly calibrate the DHR instrument. Possible diffuse scattering bias, not included in the Fresnel equations, must also be taken into account. The ultimate usability metric of an approximate theory is the agreement with observations. The model does not necessarily distinguish between surface scatter and volume scatter. Only the resulting effective parameters for the scattered field above the surface have to be adequately determined. Contributions from different phenomena are modelled as a sum of weighted basic BRDF models with different effective parameters. The basic BRDF is described by the surface or volume reflectance, an effective Fresnel-based function and a distribution function determined by the statistical properties of the surface and the volume scattering. Although the model can handle an anisotropic material such as brushed metal, only the isotropic solution is explicitly treated here.

The advantage of using spectral DHR measurements is that they are relatively easy to perform and they give three crucial model parameters as a function of wavelength. This forms the basis for the hyperspectral pBRDF results.

For rough surfaces, a narrow specular reflection is growing at very large angles of incidence. This is interpreted as a phenomenon where near-surface micro-scale variation becomes small due to shadowing effects at grazing incidence [9]. The grazing incidence properties are often of importance in target-detection applications. Using the pBRDF model also for this phenomenon, a set of parameters representative for the near-grazing incidence surface properties is obtained from measurements.

Scattering from rough inhomogeneous media can be separated in a surface contribution of the binder and a volume contribution from scattering particles. Pigmented films of the top layer of paint coatings, sometimes with a clear coat for high gloss is modelled by an incoherent combination of the basic surface and volume models [10]. The clear coat is causing a specular reflection that depends only on the optical properties of the clear-coat and is not affected by volume scattering. Radiation penetrating the clear-coat layer is here assumed to be scattered many times by the pigments collectively resembling a Lambertian surface and thus the radiation will finally leave the surface through the clear-coat at a different location than the entrance point. The pigment scattering will cause substantial depolarization although the clear-coat layer will reinstate some polarization effects. It is therefore obvious that the different layers also exhibit different spectral and polarimetric properties. Knowing the scattering properties of the different layers, it is possible to predict the surface appearance under different illumination conditions.

Once a material has been characterized with respect to BRDF measurements, similar materials can be characterized using fewer measurements in combination with a reliable model. Those measurements should focus on details that are differentiating properties within the surface class. These differentiating properties are often wavelength dependent. For materials in which bulk scattering predominates, the wavelength dependence of the angular scattering properties varies differently from what is observed for surface-scattering materials. This variation can also be observed for the same sample, when comparing visible-wavelength behavior (bulk scattering phenomena) with infrared-wavelength behavior (surface scattering).

The polarimetric description is thus approximate but can be further extended as needed. The reason is that in practice, the polarimetric information about the scene object is limited and the imaging application often does not resolve the full Mueller matrix. The model is here adapted to that the typical situation where the target is illuminated using polarized radiation, while the receiver typically does not separate the different polarizations or vice versa. This applies to the DHR and BRDF measurements as well as many other practical applications. Therefore the physical meaning of the parameters can be somewhat compromised. Depolarization is not being considered here but if depolarization is small, the deviations from the “true” values will also be small. The parameters and the model can be refined as new measurements and more sophisticated instrumentation allows for more elements to be populated in the Mueller matrix.

2. Initial considerations

In the previous work it was observed that the surface scattering could be modelled using a surface scattering factor combined with a modified bidirectional Fresnel reflectance. It was possible to describe a painted surface using only a four-parameter description. The multitude of pBRDF models is based on the need to describe surfaces with very different scattering properties. There is obviously a need of a BRDF model with larger flexibility but still with a firm connection to the underlying physics. The facet model [1] gains its popularity from a connection to Fresnel reflectance and some flexibility in the description of the facet normal distribution, which is typically assumed to be Gaussian. Invoking more or less ad hoc masking/shadowing functions, differences between the basic facet model and real surface scattering distribution can be accounted for.

As experimentally observed [11] and also supported by theory [8] modeling of pBRDF is most effective using direction cosines. In order to be able to account for both surface and volume scattering, the model has to be independent of the local properties on a wavelength scale. The scattering properties are therefore described by a scattering factor that is related to the geometric properties of the scattering element and a polarimetric factor that is related to the polarimetric properties of the material. The assumption is that the material is linear and satisfies reciprocity. It is also assumed that the scattered radiation is represented by a quasi-homogeneous source [12]. This means that the angular distribution of the scattered radiation is independent of the shape of the scattering patch. For rough surfaces, an increased forward scattering angle is observed. The present model does not capture the off-specular forward scattering phenomenon which has to be added and determined from observations. The facet model does capture this phenomenon but relies on the non-diffractive properties of the scattering process and therefore often deviates from the observed value. Real surfaces also exhibit diffractive properties and other surface phenomenon such as surface plasmon polariton scattering. The magnitude of forward scattering thus depends on the detailed scattering phenomenology.

The present model has been validated against three types of scattering materials. The first type shown in Fig. 1 is characterized by surface scattering represented by bead-blasted aluminum. Different bead sizes and different blasting pressure has been applied. There are two competing phenomena depending on the amount of blasting being performed. For no bead blasting, the surface acts as a mirror. For extensive bead blasting, the surface is diffuse and can at specific conditions also exhibit surface plasmon polariton contributions to the polarimetric scattering. Most surfaces of interest are in between these extremes and will be described by the sum of these two contributions. This will be further explained below.

Fig. 1 Surface scattering materials.

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The second type is a diffuse painted surface, shown in Fig. 2. Here, the scattering is due to pigments close to the surface in combination with some surface scattering. The model is similar to the previous one but the parameters are taking the different physics into account.

Fig. 2 Surface and volume scattering materials.

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The third type is represented by automotive paint shown in Fig. 3. Here, the pigmented volume is scattering radiation transmitted through the clear coat layer. The optical properties of the clear coat are similar to the paint binder. The surface scattering component will be more specular than for the radiation scattered by the volume; and the two components will have very different spectral and polarimetric properties.

Fig. 3 Clear-coat covered scattering materials.

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The materials are here assumed to be uniform but it is straight forward to extend the model to non-uniform materials. Even structured materials could be incorporated into the scattering factor. A key component to the success of the model is the adoption of a generalized Fresnel equation. Fresnel equations formally only apply to flat extended surfaces. Here it is used as a heuristic approach for fitting polarimetric observations of materials. It can be extended to materials that are diffusely scattering or to some extent depolarizing. The parameters can thus only be regarded to be “effective” and equal to Fresnel’s parameters only for flat and extended surfaces. The values that are obtained are however physically plausible. The bidirectional properties adopted here are ad hoc but have served the purpose in a useful way, supporting an extensive range of predictive capabilities. Further theoretical study of this approach is therefore of interest.

3. Theory for bidirectional reflectivity

The fundamental concept of radiometry is the spectral radiance, L_λ(r,s) which has units of W m⁻³ sr⁻¹. The quantity L_λ is defined by

L_{λ} (r, s) = \frac{d^{2} ϕ_{λ}}{d A s_{z} d Ω}

i.e. power per unit area, and per projected solid angle and per unit wavelength. The parameter r is the coordinate point in the element dA and s is a unit vector in the direction of the propagating flux. The coordinates of s are the direction cosines and s_z is the Z component of s. The radiant spectral intensity or flux of energy per projected solid angle per unit wavelength integrated over the surface is given by

I_{λ} (s) = \int^{​} L_{λ} (r, s) s_{z} d A .

The BRDF describes the scattered distribution of the incident spectral radiance and is defined by [13,14]

L_{λ}^{r} (r, s) = \int^{​} ρ (r, s, S) L_{λ}^{i n c} (r, S) S_{z} d Ω

Observe that the connection between the incident spectral radiance and the reflected spectral radiance is local. The local properties can be allowed to vary with the location but have to be homogeneous over an area defined by the coherence properties of the radiation.

There is a need to connect radiometric models to electromagnetic theory [8]. For a planar source, the cross-spectral density is defined by [12]

W (r_{1}, r_{2}; λ) = 〈 U^{*} (r_{1}; λ) U (r_{2}; λ) 〉

where r is a two-dimensional position vector. The spectral density is consequently given by

M (r; λ) = W (r, r; λ) .

For an extended quasi-homogeneous source which is a sub-class of Schell-model sources, the treatment simplifies considerably and the spectral density function can be factorized and approximated by

W^{(0)} (r_{1}, r_{2}; λ) \approx M^{(0)} (\frac{r_{1} + r_{2}}{2}; λ) μ^{(0)} (r_{2} - r_{1}; λ)

where W⁽⁰⁾ is the corresponding cross-spectral density at z = 0 and µ⁽⁰⁾ is the spectral degree of coherence depending only on the difference of the two points r₁ and r₂. The spectral density

M^{(0)} (\frac{r_{1} + r_{2}}{2}; λ)

is assumed to be approximately constant over the coherence area

μ^{(0)} (r_{2} - r_{1}; λ)

. The far-zone coherence properties can be determined from this relation. The Fourier transform of the approximate cross-spectral density is given by

{\tilde{W}}^{(0)} (s_{⊥}, s_{⊥}^{'}; λ) \approx {\tilde{M}}^{(0)} (k s_{⊥}^{'}) {\tilde{μ}}^{(0)} (k s_{⊥})

where k = 2π/λ. Using the stationary-phase approximation and the van Cittert-Zernike theorem [15], the spectral radiance is after some algebra given by

L_{λ} (r, s) = k^{2} | s_{z} | M^{(0)} (r) {\tilde{μ}}^{(0)} (k s_{⊥}) .

The angular distribution of the spectral radiance is now separated into a position dependent spectral density, $S^{(0)} (r)$ , and angular distribution, ${\tilde{μ}}^{(0)} (k s_{⊥})$ , governed only by the coherence properties of the source.

The separation of position and angular distribution in Eq. (7) suggests a pBRDF of type

f_{B R D F}^{p o l} (s, S; λ) = n o r m^{p o l} g (s, S; λ) f_{D H R}^{p o l} (S; λ) Q^{p o l} (s, S; λ)

The first function, $g (s, S; λ)$ , is the scattering factor that depends on the geometric structure of the material. The parameters are determined from the observed angular distribution of the scattered radiation, i.e. BRDF measurement.

The parameter $f_{D H R}^{p o l} (S; λ)$ is determined by the DHR measurements. Most commonly, the measured signal depends only on the polarized input represented by the vector S and does not take depolarization into account. The parameter $f_{D H R}^{p o l} (S; λ)$ is introduced in order to obtain a convenient normalization procedure. For a flat extended surface, the measurements can be fitted to the Fresnel equations, $f_{D H R}^{p o l} (S; λ) = R_{α}^{p o l} (S, n, k)$ . For a rough surface, the observed values can still often be fitted to the Fresnel equations, but now the parameters have to be interpreted as effective parameters,

f_{D H R}^{p o l} (S; λ) = σ_{r e l} R_{α}^{p o l} (S, n_{e f f}, k_{e f f}) + D_{α}^{p o l}

where

σ_{r e l}

has been introduced since the reflectance of these materials do not have to approach one at grazing incidence. To obtain a good fit, sometimes a bias has to be included in the model due to the partly diffuse scattering properties. For more complex surfaces, such as surfaces with plasmon polariton contributions, a more complex angular dependent bias formulation might be needed.

The definition of the parameter Q^pol depends on the definition of DHR and is the polarization-dependent reflectance part of the surface that depends on both the input direction S and the reflected direction s and is normalized at the angle of incidence. In the Rayleigh-Rice scattering theory, the quantity Q for s-polarized radiation is proportional to [16]

Q^{s} \propto \sqrt{R_{s} (s) R_{s} (S)}

This approximation is however not valid for rough surfaces or for other polarization states. An ad hoc approximation that fits observations rather well has been adopted according to

\begin{array}{l} Q^{s} = \frac{f_{D H R}^{s} (\frac{α + α_{0}}{2}; λ) f_{D H R}^{p} (\frac{β}{2}; λ)}{f_{D H R}^{s} (α_{0}; λ) f_{D H R}^{p} (0; λ)} \\ Q^{p} = \frac{f_{D H R}^{p} (\frac{α + α_{0}}{2}; λ) f_{D H R}^{s} (\frac{β}{2}; λ)}{f_{D H R}^{p} (α_{0}; λ) f_{D H R}^{p} (0; λ)} \end{array}

where the direction cosines $s = {α, β, \sqrt{1 - α^{2} - β^{2}}}$ has been introduced and $S_{0} = {α_{0}, 0, - \sqrt{1 - α_{0}^{2}}}$ is the direction of the incident radiation.

The Fresnel equations are given by

\begin{array}{l} R^{s} (α, n, k) = \frac{((n - i k) \sqrt{1 - \frac{α^{2}}{{(n - i k)}^{2}}} - \sqrt{1 - α^{2}}) ((n + i k) \sqrt{1 - \frac{α^{2}}{{(n + i k)}^{2}}} - \sqrt{1 - α^{2}})}{((n - i k) \sqrt{1 - \frac{α^{2}}{{(n - i k)}^{2}}} + \sqrt{1 - α^{2}}) ((n + i k) \sqrt{1 - \frac{α^{2}}{{(n + i k)}^{2}}} + \sqrt{1 - α^{2}})} \\ R^{p} (α, n, k) = \frac{((n - i k) \sqrt{1 - α^{2}} - \sqrt{1 - \frac{α^{2}}{{(n - i k)}^{2}}}) ((n + i k) \sqrt{1 - α^{2}} - \sqrt{1 - \frac{α^{2}}{{(n + i k)}^{2}}})}{((n - i k) \sqrt{1 - α^{2}} + \sqrt{1 - \frac{α^{2}}{{(n - i k)}^{2}}}) ((n + i k) \sqrt{1 - α^{2}} + \sqrt{1 - \frac{α^{2}}{{(n + i k)}^{2}}})} \end{array}

The angular distribution of the scattered radiation is governed by the statistical scattering characteristics. The mathematical treatment is simplified if the reflected field auto-covariance (ACV) function is normally distributed. This is however rarely the case in practice. The ACV depends on the material properties [17]. Many materials, e.g. painted surfaces, often exhibit a non-Gaussian height distribution [7]. The power spectral density (PSD) is the Fourier transform of the ACV. It is convenient to normalize all spatial variables by the wavelength of the incident radiation. A key problem is to build a statistical model that fits the observed data. A widely used distribution for PSD is used here, namely the generalized Gaussian distribution (GGD). A single shape parameter, ν, is adopted. The diffuseness parameter $\hat{ρ}$ is related to random amplitude and phase variations in the surface scattering process. It is of course possible to define functions with larger flexibility, but the GGD provides a useful balance between simplicity and flexibility. For the generalized Gaussian distribution, the scattering factor g is given by

g_{G} (α, β, α_{0}, \hat{ρ}, ν; λ) = E x p [- \frac{{| α - α_{0} |}^{ν} + {| β |}^{ν}}{{((\sqrt{1 - α_{0}^{2}} + \sqrt{1 - α^{2} - β^{2}}) \hat{ρ})}^{ν}}]

This scattering distribution has the flexibility to cover a large variety of materials. The distribution is illustrated in Fig. 4 for the shape parameter ν varying from 1 to 4. For most materials, the shape parameter ν varies between 1 and 2 but can sometimes occasionally take on more extreme values.

Fig. 4 Scattering factor using the generalized Gaussian distribution with shape factors 1, 2 and 4 where the factor equal to 2 is the normal distribution.

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Special care has to be taken when normalizing the distribution function $f_{B R D F}^{p o l} (s, S; λ)$ since values of the direction cosines are constrained by the evanescent wave values.

The normalization is given by

\int_{α = - 1}^{α = 1} \int_{β = - \sqrt{1 - α^{2}}}^{β = \sqrt{1 - α^{2}}} n o r m^{p o l} g (α, β, α_{0}, \hat{ρ}) Q^{p o l} (α, β, α_{0}, n, k) d α d β = 1.

This integral is rather involved and does not have an analytical solution, but is conveniently solved numerically. One has a choice of several commercial programs that can solve the integral numerically with high accuracy. For large angles of incidence, the function to be integrated varies drastically over a small angular region and the numerical solution may have to take this into account.

3.1 Diffuse scattering

Scalar diffuse scattering is described by

D (α, β) = σ_{D} {(1 - α^{2} - β^{2})}^{ν_{D} / 2}

for diffuse surface scattering and the corresponding polarimetric expressions are

\begin{array}{l} D^{s s} (α, β) = σ_{D} {(1 - α^{2} - β^{2})}^{ν_{D} / 2} (1 - R^{s} (α_{0}, n, k)) \frac{1 - R^{s} (\sqrt{α^{2} + β^{2}}, n, k)}{2} \\ D^{s p} (α, β) = σ_{D} {(1 - α^{2} - β^{2})}^{ν_{D} / 2} (1 - R^{s} (α_{0}, n, k)) \frac{1 - R^{p} (\sqrt{α^{2} + β^{2}}, n, k)}{2} \\ D^{p p} (α, β) = σ_{D} {(1 - α^{2} - β^{2})}^{ν_{D} / 2} (1 - R^{p} (α_{0}, n, k)) \frac{1 - R^{p} (\sqrt{α^{2} + β^{2}}, n, k)}{2} \\ D^{p s} (α, β) = σ_{D} {(1 - α^{2} - β^{2})}^{ν_{D} / 2} (1 - R^{p} (α_{0}, n, k)) \frac{1 - R^{s} (\sqrt{α^{2} + β^{2}}, n, k)}{2} \end{array}

for diffuse scattering under a clear coat assuming complete depolarization in the subsurface scattering process. The parameter ν_D varies as

- 1 < v_{D} \leq 1

where

v_{D} = 0

corresponds to a Lambertian surface with a source-coherence dimension of the order of the wavelength; the value of

v_{D} = 1

corresponds to an incoherent source [18]

If the subsurface scattering is to preserve some of the directional and polarization information, the diffuse model has to be further developed using combinations of the models given above.

3.2 Total BRDF expression

The total BRDF is given by a series expansion of the specular generalized Gaussian model plus a diffuse contribution of the generalized Lambertian model.

f_{B R D F}^{p o l} (α, β, α_{0}, \hat{ρ}; λ) = \sum_{i = 1}^{N} a_{i} f_{B R D F}^{p o l} (α, β, α_{i}, {\hat{ρ}}_{i}; λ) + D^{p o l} (α, β)

where α_i is a modified α_o angle due to forward scattering effects. It is always possible to obtain a good fit by adding many parameters. It is however important to minimize the complexity of the model consistent with the identified scattering phenomenology. Here, two main contributions to the scattering are observed, resulting in two different scattering lobes superimposed. The parameter N is here set to 2. There are still some deviations observed in the scattering tail about two orders of magnitude below the main peak.

3.3 Directional hemispheric reflectance

The directional-hemispherical reflectance is the integral of the pBRDF over all viewing angles

f_{D H R}^{p o l} (α_{0}, \hat{ρ}; λ) = \int_{α = - 1}^{α = 1} \int_{β = - \sqrt{1 - α^{2}}}^{β = \sqrt{1 - α^{2}}} f_{B R D F}^{p o l} (α, β, α_{0}, \hat{ρ}; λ) d α d β .

3.4 Off-specular angle correction

As shown using the facet model, an offset towards larger angles is expected compared to the specular angle for rough surfaces. An offset correction is therefore needed for rough surfaces. The magnitude of this correction is found empirically for each type of material. A simple linear relation of the type

α_{C} = \frac{E r f [k_{α} α_{0}]}{E r f [k_{α}]}

can be used. The parameter

k_{α}

is expected to be related to the roughness parameters, σ_rel and

\hat{ρ}

.

4. Polarization ray-tracing with surface normal along z-axes

Only in-plane results are shown experimentally here. For future use, the change in polarization due to out-of-plane scattering is derived assuming no depolarization. This phenomenon will be further explored in subsequent papers.

The polarization state varies with the source polarization, surface orientation and the scattering angle. Here, only linear polarization is treated assuming no depolarization in the scattering process. Extension to other polarization states and inclusion of depolarization is rather straightforward. This is similar to the approach in the popular facet model. The propagation of polarization in three dimensions is obtained by generalizing a two-by-two Jones matrix into a three-by-three matrix [19]. The local coordinate system is defined by three orthogonal vectors, {s,p,k}, where s is the direction of the s-polarized vector, p is the direction of the p-polarized vector and k is the propagation direction. The assumption of no depolarization means that the polarization directions changes as if reflected by a perfect mirror.

The output complex amplitude E_out after reflecting the input complex amplitude, E_in is given by

E_{o u t} = O_{o u t} J_{s u r f a c e} O_{i n}^{- 1} E_{i n}

where

O_{i n}^{- 1} = [\begin{matrix} s_{x, i n} & s_{y, i n} & s_{z, i n} \\ p_{x, i n} & p_{y, i n} & p_{z, i n} \\ k_{x, i n} & k_{x, i n} & k_{x, i n} \end{matrix}]

O_{o u t} = [\begin{matrix} s_{x, o u t} & p_{x, o u t} & k_{x, o u t} \\ s_{y, o u t} & p_{y, o u t} & k_{y, o u t} \\ s_{z, o u t} & p_{z, o u t} & k_{z, o u t} \end{matrix}]

and

s_{i n} = \frac{k_{i n} \times k_{o u t}}{| k_{i n} \times k_{o u t} |}

p_{i n} = k_{i n} \times s_{i n}

s_{o u t} = s_{i n}

p_{o u t} = k_{o u t} \times s_{o u t}

The polarization transformation is described with the scattering surface normal oriented along the z-axes. In this coordinate system, the source radiation can be described using the direction cosines. The normalized vectors are given by

s_{0} = {0, - 1, 0}

p_{0} = {- \sqrt{1 - α_{0}^{2}}, 0, - α_{0}}

k_{0} = {α_{0}, 0, - \sqrt{1 - α_{0}^{2}}}

The vectors in the direction of scattering are given by

s_{f} = \frac{1}{\sqrt{α^{2} + β^{2}}} {β, - α, 0}

p_{f} = \frac{1}{\sqrt{α^{2} + β^{2}}} {α \sqrt{1 - α^{2} - β^{2}}, β \sqrt{1 - α^{2} - β^{2}}, - α^{2} - β^{2}}

k_{f} = {α, β, \sqrt{1 - α^{2} - β^{2}}}

Similarly, the input and output matrices are given by

s_{i n} = \frac{{\sqrt{1 - α_{0}^{2}} β, - α \sqrt{1 - α_{0}^{2}} - α_{0} \sqrt{1 - α^{2} - β^{2}}, α_{0} β}}{\sqrt{α_{0}^{2} β^{2} + (1 - α_{0}^{2}) β^{2} + {(- α \sqrt{1 - α_{0}^{2}} - α_{0} \sqrt{1 - α^{2} - β^{2}})}^{2}}}

p_{i n} = \frac{{- α + α α_{0}^{2} - α_{0} \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}}, - β, - α α_{0} \sqrt{1 - α_{0}^{2}} - α_{0}^{2} \sqrt{1 - α^{2} - β^{2}}}}{\sqrt{α^{2} + α_{0}^{2} - 2 α^{2} α_{0}^{2} + β^{2} - α_{0}^{2} β^{2} + 2 α α_{0} \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}}}}

k_{i n} = k_{0}

and

s_{o u t} = s_{i n}

p_{o u t} = \frac{{α_{0} - α^{2} α_{0} + α \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}}, - α α_{0} β + \sqrt{1 - α_{0}^{2}} β \sqrt{1 - α^{2} - β^{2}}, - α^{2} \sqrt{1 - α_{0}^{2}} - \sqrt{1 - α_{0}^{2}} β^{2} - α α_{0} \sqrt{1 - α^{2} - β^{2}}}}{\sqrt{α^{2} + α_{0}^{2} - 2 α^{2} α_{0}^{2} + β^{2} - α_{0}^{2} β^{2} + 2 α α_{0} \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}}}}

The scattered power-normalized polarized radiation for s-polarized radiation is given by

P_{s s} = {(s_{f} . O_{o u t} . O_{i n}^{- 1} . s_{0})}^{2}

and

P_{s p} = 1 - P_{s s}

where the normalization requires the surface to be a mirror with J_surface equal to an identity matrix.

Similarly

P_{p s} = 1 - P_{s s}

P_{p p} = P_{s s}

The final result is given by

P_{s s} = \frac{{(α^{3} + α α_{0}^{2} - 2 α^{3} α_{0}^{2} + α β^{2} + α_{0} β^{2} - 2 α α_{0}^{2} β^{2} + 2 α^{2} α_{0} \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}} + α_{0} \sqrt{1 - α_{0}^{2}} β \sqrt{1 - α^{2} - β^{2}})}^{2}}{(α^{2} + β^{2}) {(α^{2} + α_{0}^{2} + β^{2} + 2 α α_{0} \sqrt{1 - α_{0}^{2}} \sqrt{1 - α^{2} - β^{2}} - α_{0}^{2} (2 α^{2} + β^{2}))}^{2}}

The functions P_ss and P_sp are illustrated in Fig. 5 for the input directional cosine α₀ = 0.5.

Fig. 5 P_ss (left) and P_sp (right) for input direction α₀ = 0.5.

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5. Experimental validation

Model validation of scattering behavior from diffuse surfaces requires extensive measurements of polarimetric bidirectional reflectance distributions. Such measurement programs are expensive and it is therefore attractive to find and validate models that can be applied to classes of surfaces. Here, three classes of surface are studied, scattering from different types of surface structures, volume scattering involving pigments and clear coated surfaces with diffusively scattering pigmented particles. They are all considered to be spatially uniform statistically and the scattering pattern is dependent only on the surface properties and not the coherence properties of the radiation source. The model described above has been applied with remarkable effectiveness to all these surfaces where also details in the s- and p-polarized model are observed. The accuracy is satisfactory over two orders of magnitude.

Because of the bistatic angle dependence, measurements of bidirectional reflectance distributions are very labor intensive. With a good understanding of the general behavior of the different types of surfaces, simplified procedures can be introduced that reduce the measurement burden. Directional hemispheric measurements of surfaces are less expensive and are often available. Several of the model parameters can be determined from a DHR measurement. It is also possible to exclude specular scattering in a DHR measurement. As will be shown below, it is not possible to separate different semi-diffuse contributions that are dominant in the distribution of the scattered radiation from most diffuse surfaces. This separation could possibly be obtained by a simplified BRDF measurement. Extensions of this methodology remain to be further developed.

There are material variabilities that result in differences for nominally similar surfaces. This results in different sets of fitting parameters for different samples of the same type but coming from e.g. different suppliers. For general applicability, such deviations have to be averaged out in the model. This is done here by fitting such sets of parameters to simple analytic functions. As shown here, this could also average out important polarization enhancement effects at specific angles of incidence caused by e.g. surface plasmon polaritons. Such enhancement phenomena are also observed as deviations from Fresnel’s equations in the DHR measurement. In many applications, this is acceptable. This is especially true in applications where the receiver is not separating the different polarizations.

5.1 DHR measurement of diffuse green paint

The directional hemispherical reflectance is measured for s- and p-polarization using a series of angles of incidence. Fitting the observations to Eq. (9), the bias parameter b, the complex index of refraction (n_eff,k_eff) and the scaling parameter σ_rel can be obtained. The scaling parameter σ_rel can vary between zero and one since the reflectance has a maximum value of one. This determines four of the five parameters needed for predicting the scattering properties of the material. The fifth parameter, $\hat{ρ}$ , is determined from a possibly simplified BRDF measurement. Most materials of the type discussed above can be fitted to this equation with the exception of materials exhibiting surface plasmon polaritons.

An example of a DHR measurements of diffuse green paint for s- and p-polarization as a function of wavelength is shown in Fig. 6 at an angle of incidence of 60 degrees.

Fig. 6 Hemispherical (DHR) data for s- and p-polarizations at 60 degrees angle of incidence as a function of wavelength for green paint.

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These measurements are repeated for a number of angles of incidence. Using data from angles of incidence at 12, 30, 40, 50, 60, 70, 75 and 80 degrees, the four parameters b, (n_eff,k_eff) and σ_rel can be calculated. The results are evaluated for the wavelength 3.39 µm which is the wavelength where the scattering parameter, $\hat{ρ}$ , is measured. At this wavelength, the parameter values are b = 0.035, σ_rel = 0.627, n_eff = 1.575 and k_eff = 0.190 determined from a non-linear least square fit. The model is compared to measurements in Fig. 7. Observe that the scattering is enhanced for p-polarized radiation with approximately 25% at angles of incidence around 50 to 60 degrees compared to the theoretical curve.

Fig. 7 Hemispherical (DHR) data for s- and p-polarizations at the wavelength 3.39 µm as a function of angle of incidence for green paint. A small bias has been included when fitting to the Fresnel’s equations.

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DHR measurements have to be interpreted with some caution. The bias does not necessarily mean that there is a diffuse scattering contribution. It could still represent a specular behavior but appears due to a deviation from an ideal Fresnel surface. On the other hand, the scattering can be a combination of a specular and a diffuse scattering with similar angular reflectance behavior. This is not always easy to distinguish from a simple specular to diffuse ratio measurement. For a layered material such as automotive paint, the different layers can also have very different spectral properties. The BRDF measurements can give substantial insight into the various types of phenomenology in play for a given sample.

Fitting of Eq. (9) can be repeated for each wavelength resulting in wavelength- dependent fitting parameters. The bias is observed to correlate with the total level of reflectance. The absorption coefficient, k_eff, of the complex index of refraction is zero except at a few absorption bands. The real index of refraction, n_eff, and the scaling factor, σ_rel, seem to anti-correlate. The result is shown in Fig. 8.

Fig. 8 The fitting parameter of the DHR model are shown as a function of wavelength. A) Bias correlates strongly with the total reflectance. B) The scaling parameter σ shows anti-correlation to the total reflectance values. C) The index of refraction n is mostly centered on a value of 1.6 except in the 4-5.5 µm spectral region where the reflectance is high. D) The absorption coefficient k is zero except at a few narrow spectral regions.

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5.2 BRDF measurements at 3.39 µm

Three types of materials are measured and used for illustration of the model performance. The first material is bead-blasted aluminum with a variation in bead size and blasting pressure. The second material is diffuse green paint. The third material is automotive paint with clear coating. Quite a large variety in behavior is observed.

The parameters obtained in the model fit are shown in more detail for the first sample presented. Other samples will show deviations in separate effects on s- and p-polarized radiation attributed to influence of surface plasmon polaritons. For samples with clear coat a slightly different model as described above is invoked.

All pBRDF measurements are carried through at the angles of incidence of 5, 20, 40, 60 and 80 degrees. The roughness parameter $\hat{ρ}$ and the shape parameter ν can be determined from any of the angles of incidence but as will be shown below, there are different semi-specular peaks that are dominant at different angles of incidence.

5.3 Bead-blasted aluminum

For bead-blasted aluminum, the literature values of complex index of refraction is used, i.e. n_eff = 5.131 and k_eff = 32.911.

Bead-blasted aluminum is not necessarily an ideal test target since the material can under some circumstances be influenced by surface plasmon polaritons. The magnitude of this phenomenon depends on the size of the micro structure relative to the wavelength. Two examples is given here, one when the influence is weak, here called sample A, and one when the influence is strong, here called sample B.

5.3.1 Bead-blasted aluminum sample A

This sample is a bead blasted aluminum surface using bead size 100-140 µm and a pressure of 1 kp. There are three major contributions to the scattering at normal incidence. One part is a broad specular contribution containing about 93% of the scattered radiation. A diffuse contribution constitutes about 4% and the narrow center peak contains only about 2% of the scattering. Even if the narrow peak does not contain much energy, it is dominating at specular angles.

In the log-scale figures, a deviation can be seen in Fig. 9 at the tail of the broad specular distribution. This tail behavior could be taken into account by adding additional parameters. This has however not been pursued here but could be added if the present model is not found suitable for a given purpose.

Fig. 9 pBRDF measurements and model fit to a bead-blasted aluminum surface. Diffuse scattering levels are different for s- and p-polarized radiation. Red is s-polarized and blue is p-polarized.

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The variation in the parameters σ_narrow, $\hat{ρ}$ _narrow, σ_broad and $\hat{ρ}$ _broad have been fitted to a function $p a r a m e t e r = k_{0} E r f [k_{1} (1 - α_{0})] + b i a s$ where each parameter has its own set of constants k₀, k₁, bias. The general shape of this function can be seen in Fig. 10. The function stays approximately constant at low angles of incidence and then decreases at larger angles of incidence. They all show similar functional behavior, indicating a common physical cause to the variation. This is a variation that is reminiscent of the shadowing effect in the facet model.

Fig. 10 Figures shows the variation in magnitude (upper) and spread (lower) of the two peaks, one narrow (left) and one broad (right), as a function of angle of incidence.

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The shape parameter ν varies between 1.8 and 1.2 for the broad specular peak but stays constant at 1.6 for the narrow specular peak. This indicates that the scattering statistics vary with angle for the broad scattering but stay the same for the narrow scattering. It is not surprising that the scattering parameter $\hat{ρ}$ for the broad scattering varies differently compared to the corresponding parameter for the narrow scattering.

The s-polarized diffuse scattering is decreasing with larger angles of incidence but is also compensating for the long tail distribution of the broad specular scattering at large angles. Diffuse p-polarized scattering stays approximately constant with angle of incidence.

5.3.2 Bead-blasted aluminum sample B

This sample is a bead blasted aluminum surface using bead size 100-140 µm and a pressure of 3 kp. As can be seen in Fig. 11, there are two major contributions to the scattering at normal incidence. One part is a broad specular contribution containing about 85% of the scattered radiation. A diffuse contribution constitutes about 15%, similar to the levels of sample A. At large angles of incidence, there is an onset of a narrow specular peak. The most notable difference from above is the larger magnitude of p-polarized scattering compared to s-polarized scattering, except at 5 degrees of incidence. This phenomenon disappears at angles of incidence larger than approximately 50 degrees. It is interpreted as a resonant surface plasmon polariton effect.

Fig. 11 pBRDF measurements and model fit to a bead-blasted aluminum surface. Scattering levels of p-polarized radiation are stronger than s-polarized radiation. Red is s-polarized and blue is p-polarized.

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5.4 Green diffuse paint at 3.39 µm

From the DHR measurements at this wavelength, the following parameters are determined: b = 0.035, σ = 0.627, n = 1.575 and k = 0.190. The resulting fit to the observations are shown in Fig. 12.

Fig. 12 pBRDF measurements and model fit to diffuse green paint.

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5.5 BRDF measurements at 633 nm of diffuse green paint

Because of local instrumentation limitations, there are no angular dependent DHR measurements available at this wavelength. The parameters are then determined from fittings to BRDF measurements. The complex index of refraction has not been explicitly measured and the commonly adopted values n = 1.65, k = 0.0 is used. The fit to the observed scattering is still quite satisfactory as shown in Fig. 13.

Fig. 13 pBRDF measurements and model fit to diffuse green paint.

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Also for diffuse paint, enhanced scattering at angles of incidence around 40 to 60 degrees is observed for p-polarized radiation.

5.6 BRDF measurements at 633 nm of clear coat automotive paint

Figure 14 shows results from the measurement of automotive paint with a clear coat. A value of n_eff = 1.5 for the index of refraction of the clear coat was adopted. In the present model, the radiation transmitted through the clear coat is assumed to be completely depolarized when scattered by the paint particles. The scattered angular dependence is also assumed to be close to a Lambertian surface. A polarization is reintroduced when the radiation is transmitted back through the clear coat. The narrow peak comes from specular reflection of the clear coat. This peak also exhibits a small slightly broadened base that is well described using the imposed index of refraction.

Fig. 14 BRDF results for the red automotive paint. The specular peak at normal incidence is blocked. At other angles the peak is partly limited by the angular resolution of the instrument.

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The diffuse scattering level of the green and blue paint is very low at 633 nm. The specular reflection from the clear coat is the same for both paints. Scattering from the green car paint measured at 633 nm is shown in Fig. 15.

Fig. 15 The diffuse scattering level is much lower for the green paint compared to red paint at 633 nm. The low scattering level reveals the deviation at the main peak tail. The specular peak is off scale by several orders of magnitude.

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6. Conclusions

An important issue is the functional form of the BRDF. A very flexible but still simple approach is presented here, where the functional form can be varied. More general shapes e.g. structured patterns can also be introduced but are not shown here. The generalized Gaussian distribution has proven successful in many applications and is the one used here. The series expansion of the four parameter model and the generalized Lambertian distribution presented here show good agreement with observations and also give insight into the physical nature of the scattering process.

The basic four parameter model presented here depends on three parameters obtained from DHR measurements and one parameter, the optical roughness, obtained from a BRDF measurement. The three DHR parameters are the complex index of refraction (two parameters, n and k) and the limited reflectance at grazing incidence due to surface roughness (σ_rel). In the classic Fresnel model, reflectance at grazing incidence is always one. Reflectance for rough surfaces with scattering centers of the order in size of the wavelength, do not have to approach one at grazing incidence as is experimentally shown. The BRDF measurement sets the parameter describing the diffuseness of the surface ( $\hat{ρ}$ ). A correspondence to the masking/shadowing correction in the facet model is also introduced for large angles of incidence by varying the parameters at large angles of incidence. The normalization of BRDF is obtained by separating the DHR contribution from the BRDF expression and requiring the resulting expression to integrate to one. The accuracy in this procedure depends of course on how well the BRDF model describes the real situation.

Determination of multispectral properties are greatly simplified by using DHR measurements in the determination of three of the four parameters of the basic model. The spread parameter does not follow a simple wavelength dependence since the material properties change with wavelength.

Some glossy surfaces exhibit a surface coating and scattering pigments below the surface. These surfaces have to be described by a combination of the four-parameter model and a generalized Lambertian model combined with the transmission properties of the coat. This applies e.g. to automotive-paints.

The complex mixture of sources was here described by a summation of two generalized Gaussian/Fresnel models and a single generalized Lambertian model. Only at the tails of strong peaks, deviations were observed, at a magnitude several orders of magnitude lower than the peak. For general modeling and simulations where there also are a natural variation in similar materials due to manufacturing processes and influence of weather and wear, simplified models with approximated parameters varying as simple functions of the angle of incidence can be used. It is of interest to develop common parameter expressions for similar materials.

Although the physics inspired model has some ad hoc features, the predictive power of the model is impressive. The model has been applied successfully to painted surfaces, both dull and glossy and also on metallic bead-blasted surfaces.

Since measurements have only been performed at in plane angles, the model has not been validated for out-of-plane angles. There is consequently a need for polarimetric BRDF measurements also at the out of plane angles. As has been pointed out many times, BRDF measurements are quite expensive but for the future performance modeling of polarimetric remote sensing systems including depolarization, such measurements are invaluable. Of special interest are materials that are common in remote sensing and where knowledge of the variation in BRDF would be useful in signature analyses. A systematic study of these classes of materials might help to lower the cost of such endeavors.

For the implementation of the BDRF model in scene rendering applications, the transformations for a randomly oriented scattering surface has to be added. A ray tracing approach for this application is being developed and the practical implementation will be presented in a subsequent paper.

References and links

1. R. G. Priest and T. A. Germer, “Polarimetric BRDF in the microfacet model: Theory and measurements,” in Military Sensing Symposia (MSS) Specialty Group Meeting on Passive Sensors, (Infrared Information Analysis Center, Ann Arbor, MI, 2000). Available at www.dtic.mil, approved for public release, distribution unlimited.

2. T. M. Elfouhaily and C. A. Guerin, “A critical survey of approximate scattering wave theories from random rough surfaces,” Waves Random Media 14(4), R1–R40 (2004). [CrossRef]

3. M. A. Culpepper, “Empirical bidirectional reflectivity model,” Proc. SPIE 2469, 159–168 (2001).

4. R. G. Priest and S. R. Meier, “Polarimetric microfacet scattering theory with applications to absorptive and reflective surfaces,” Opt. Eng. 41(5), 988–993 (2002). [CrossRef]

5. I. G. E. Renhorn, T. Hallberg, D. Bergström, and G. D. Boreman, “Four-parameter model for polarization-resolved rough-surface BRDF,” Opt. Express 19(2), 1027–1036 (2011). [CrossRef] [PubMed]

6. I. G. Renhorn and G. D. Boreman, “Analytical fitting model for rough-surface BRDF,” Opt. Express 16(17), 12892–12898 (2008). [CrossRef] [PubMed]

7. J. M. Elson and J. M. Bennett, “Relation between the angular dependence of scattering and the statistical properties of optical surfaces,” J. Opt. Soc. Am. A 69(1), 31–47 (1979). [CrossRef]

8. J.-J. Greffet and M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15(10), 2735–2744 (1998). [CrossRef]

9. J. Qiu, W. J. Zhang, L. H. Liu, P. F. Hsu, and L. J. Liu, “Reflective properties of randomly rough surfaces under large incidence angles,” J. Opt. Soc. Am. A 31(6), 1251–1258 (2014). [CrossRef] [PubMed]

10. S. Shafer, “Using color to separate reflection components,” Color Res. Appl. 10(4), 210–218 (1985). [CrossRef]

11. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. Dissertation, University of Arizona (1976).

12. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).

13. F. E. Nicodemus, “Directional reflectance and emissivity of an opaque surface,” Appl. Opt. 4(7), 767–773 (1965). [CrossRef]

14. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9(6), 1474–1475 (1970). [CrossRef] [PubMed]

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18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

19. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50(18), 2855–2865 (2011). [CrossRef] [PubMed]

Efficient polarimetric BRDF model

Abstract

1. Introduction

2. Initial considerations

3. Theory for bidirectional reflectivity

3.1 Diffuse scattering

3.2 Total BRDF expression

3.3 Directional hemispheric reflectance

3.4 Off-specular angle correction

4. Polarization ray-tracing with surface normal along z-axes

5. Experimental validation

5.1 DHR measurement of diffuse green paint

5.2 BRDF measurements at 3.39 µm

5.3 Bead-blasted aluminum

5.3.1 Bead-blasted aluminum sample A

5.3.2 Bead-blasted aluminum sample B

5.4 Green diffuse paint at 3.39 µm

5.5 BRDF measurements at 633 nm of diffuse green paint

5.6 BRDF measurements at 633 nm of clear coat automotive paint

6. Conclusions

References and links

Cited By

Figures (15)

Equations (44)

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