Modified polarized geometrical attenuation model for bidirectional reflection distribution function based on random surface microfacet theory

Hong Liu; Jingping Zhu; Kai Wang

doi:10.1364/OE.23.022788

1. Introduction

Reflecting properties of different surfaces have been studied and applied in many areas as 3D graphics simulation [1], biological sensing [2, 3] and prediction of vegetation [4, 5]. The scattering of surfaces is typically modeled by BRDF which was introduced by Nicodemus in 1960s [6] and actively researched for nearly half a century. BRDF models can be classified in empirical models and analytic models. Empirical models are obtained by fitting of BRDF experimental data [7], while analytic models are derived by physical optics theory or geometrical optics theory. Geometrical optics BRDF models have simple expressions but are limited that the surface roughness should be larger than the incident wavelength. In fact, physical optics models tend to be more accurate but its mathematical form is too complex. For this reason, the application and development of BRDF focus mainly on the geometrical optics models. As the most famous geometrical optics model, Torrance-Sparrow (T -S) model [8] has been widely studied and expanded to polarized BRDF (pBRDF) models [9–14].

The geometrical attenuation factor G caused by masking of incident light and shadowing of reflect light in rough surfaces is an important term of BRDF. So there have been researchers focused on the term of G of BRDF models, from physical models to geometrical models. The G of physical models is calculated by the random height distribution of surface. In geometrical optics models, G is developed based on the slope angles of surfaces. Considering the geometrical relationship between microfacets and light rays, T -S model gave out a complicated expression of G [8]. Blinn simplified the expression of G of T -S model to a piecewise trigonometric function form [15]. For its briefness, Blinn’s G expression was chosen to be the geometrical attenuation factor term of many well-known geometrical optics BRDF models as Cook-Torrance model [16] and pBRDF models as Hyde model [14], and some physical BRDF models as He model [17].

Although Blinn’s piecewise trigonometric G expression has a simple mathematical form, the shortcoming in its physical assumption is significant. The symmetrical V-grooves composition of surfaces Blinn assumed is often not the case for practical surfaces. And the piecewise curve of Blinn’s G function doesn’t follow physical common sense and experimental result [18]. Further, as an important part of reflect properties, the effect of polarization must be taken into account. However, it is limited that the ray scalar theory used by Blinn is out of consideration of polarized effect and variation of reflectance. For these reasons, a modification of polarized geometrical attenuation model and new G expressions are shown in this paper.

2. T -S BRDF model and Blinn’s G model

BRDF is defined as the ratio of the scattered radiance dL_r to the incident irradiance dE_i. The geometry of BRDF is shown in Fig. 1.

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

Fig. 1 Geometry of BRDF. Illuminated surface is in the X − Y plane, surface normal is in Z direction. θ_i and θ_r are the zenith angles of incident and reflect light respectively, ϕ_i and ϕ_r are their azimuth angles, and ϕ = |ϕ_i −ϕ_r|. α is the polar angle from the mean surface normal Z to the microfacet normal n, θ is the incident angle as measured from the microfacet.

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T -S BRDF model is based on the microfacet theory, in which the surface is composed by large number of small plants called microfacets which follow Snell’s law like small mirrors. Since the slope angles distribution of most natural surfaces are random Gaussian, the microfacet slope angles distribution utilized in most geometrical optics BRDF models [5–10] is:

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

Where C is the normalization coefficient which makes the integral value of p(α) over the entire hemisphere is 1. BRDF value f can be divided to specular component f_s and diffuse component f_d which is assumed to be a constant in different reflect angles.

f_{s} (θ_{i}, θ_{r}, ϕ) = \frac{1}{2 π} \frac{1}{4 σ^{2}} \frac{1}{{cos}^{4} α} \frac{1}{cos θ_{r} cos θ_{i}} exp (- \frac{{tan}^{2} α}{2 σ^{2}}) G (θ_{i}, θ_{r}, ϕ)

Where α and θ are derived using spherical trigonometry:

cos α = (cos θ_{i} + cos θ_{r}) / (2 cos θ)

cos 2 θ = cos θ_{i} cos θ_{r} + sin θ_{i} sin θ_{r} cos ϕ

Blinn considered a geometry of symmetrical V-shaped groove. There are two cases of geometrical attenuation effects – shadowing and masking, as shown in Fig. 2.

Fig. 2 Geometrical attenuation effect of Blinn’s symmetrical V-shaped groove assumption.

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The expression of G function given by Blinn is:

G (θ_{i}, θ_{r}, ϕ) = min (1; \frac{2 cos α cos θ_{r}}{cos θ}; \frac{2 cos α cos θ_{i}}{cos θ})

Blinn’s piecewise G function plays an important role to keep f_s a finite value when observation approaches grazing, as shown in Fig. 3.

Fig. 3 (a) Blinn’s G function curves. (b)T-S f_s curves with Blinn’s G when σ=0.25.

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3. Geometrical attenuation model modification

In this work, the two main improvements are modification of the masking and shadowing effects, and addition of polarized effect for the geometrical attenuation model. Both of the two improvements will be discussed in detail in this chapter.

3.1. Masking and shadowing effects

In random surface microfacet theory, the reflect energy distributions should be smooth curves because of the continuous Gaussian slope angles distribution. But in each BRDF curve there is a sharp turning point, as shown in Fig. 3(b). The cause of this unreasonable phenomenon is the Blinn’s symmetrical V-shaped groove assumption which assumes adjacent microfacets have a same slope angle with opposite orientations is disagree with random surface microfacet theory. For this reason, we present following five assumptions of the masking and shadowing effects based on random surface microfact theory: 1) The surface is composed by large number of microfacets with a same area; 2) The surface goes up and down in one direction from the microscopic view. Adjacent microfacets are opposite oriented but not symmetrical, slope angle of each microfacet is independent and follows semi-Gaussian distribution; 3) Only specular component of the reflection is considered; 4) Only masking is considered at large incident angles while only shadowing is considered at large reflect angles; 5) Depending on the slope angles of microfacets, masking/shadowing model falls into three submodels – passing submodel, semi-passing submodel and masking/shadowing submodel.

The slope angle α of the illuminated microfacet 1 is fixed as $\frac{1}{2} | θ_{r} - θ_{i} |$ .The slope angle γ of microfacet 2 which is next to microfacet 1 and close to the observer follows semi-Gaussian distribution as Eq. (2) with a half angular range. The possibility of passing submodel is:

P_{1} (θ_{r}, σ) = \int_{0}^{\frac{π}{2} - θ_{r}} p (γ) d γ

The bounds of the integral above are got easily from the passing condition. The portion of passed light in the passing submodel and masking submodel can be written easily as Eq. (8) and Eq. (9) respectively:

g_{M 1} (θ_{r}, σ) = \int_{0}^{\frac{π}{2} - θ_{r}} 1 \cdot p (γ) d γ

g_{M 2} (θ_{r}, σ) = \int_{π - 2 θ_{r} + α}^{\frac{π}{2}} 0 \cdot p (γ) d γ = 0

In semi-passing submodel, a portion of the incident light is blocked and the other is passed when $\frac{π}{2} - θ_{r} < γ < π - 2 θ_{r} + α$ . As is shown in Fig. 4(c), the portion of passed and blocked light can be translated into the length of a and b in the semi-passing submodel. The passed proportion of incident light $\frac{a}{a + b}$ is got by geometric operations:

\frac{a}{a + b} = \frac{sin α tan θ_{r} + cos α + cos γ - sin γ tan θ_{r}}{sin α tan θ_{r} + cos α}

Fig. 4 Three submodels of the masking factor assumption.

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As a result, the portion of passed light in the semi-passing model is:

\begin{array}{l} g_{M 3} (θ_{r}, σ) & = \int_{\frac{π}{2} - θ_{r}}^{π 2 - 2 θ_{r} + α} P (γ) \cdot \frac{a}{a + b} d γ \\ = \int_{\frac{π}{2} - θ_{r}}^{π 2 - 2 θ_{r} + α} P (γ) \cdot \frac{sin α tan θ_{r} + cos α + cos γ - sin γ tan θ_{r}}{sin α tan θ_{r} + cos α} d γ \end{array}

Only considering masking factor, the passed portion of incident light g_M is the sum of g_M1, g_M2, and g_M3:

\begin{array}{l} g_{M} (θ_{r}, σ) & = g_{M 1} + g_{M 2} + g_{M 3} \\ = \int_{- \frac{π}{2}}^{\frac{π}{2} - θ_{r}} p (γ) d γ + \int_{\frac{π}{2} - θ_{r}}^{π 2 - 2 θ_{r} + α} P (γ) \cdot \frac{sin α tan θ_{r} + cos α + cos γ - sin γ tan θ_{r}}{sin α tan θ_{r} + cos α} d γ \end{array}

Similar with the masking model, the expression of shadowing model can be derived from three submodels, including passing submodel, shadowing submodel and semi-passing sub-model shown in Fig 5. The portion of passed light in the passing submodel and shadowing submodel can be expressed as Eq. (13) and Eq. (14).

g_{S 1} (θ_{i}, σ) = \int_{0}^{\frac{π}{2} - θ_{i}} 1 \cdot p (γ) d γ

g_{S 2} (θ_{i}, σ) = \int_{π - 2 θ_{i} + α}^{\frac{π}{2}} 0 \cdot p (γ) d γ = 0

Fig. 5 Three submodels of the shadowing factor assumption.

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As is shown in Fig. 5(c), the passed proportion is expressed as Eq. (15), and the portion of passed light in semi-passing submodel of shadowing model is expressed as Eq. (16).

\frac{a}{a + b} = \frac{cos α cot θ_{i} + cos γ cot θ_{i} + sin α - sin γ}{cos α cot θ_{i} + sin α}

\begin{array}{l} g_{S 3} (θ_{i}, σ) & = \int_{\frac{π}{2} - θ_{i}}^{π - 2 θ_{i} + α} P (γ) \cdot \frac{a}{a + b} d γ \\ = \int_{\frac{π}{2} - θ_{i}}^{π - 2 θ_{i} + α} P (γ) \cdot \frac{cos α cot θ_{i} + cos γ cot θ_{i} + sin α - sin γ}{cos α cot θ_{i} + sin α} d γ \end{array}

The passed portion in shadowing model g_S is:

\begin{array}{l} g_{S} (θ_{i}, σ) & = g_{S 1} + g_{S 2} + g_{S 3} \\ = \int_{- \frac{π}{2}}^{\frac{π}{2} - θ_{i}} p (γ) d γ + \int_{\frac{π}{2} - θ_{i}}^{π - 2 θ_{i} + α} P (γ) \cdot \frac{cos α cot θ_{i} + cos γ cot θ_{i} + sin α - sin γ}{cos α cot θ_{i} + sin α} d γ \end{array}

Generally, only masking effect works at large incident angles and only shadowing effect works at large reflect angles. So g should be the less one of g_M and g_S.

g (θ_{i}, θ_{r}, σ) = min [g_{M} (θ_{r}, σ), g_{S} (θ_{i}, σ)]

It is easy to understand that surface roughness strongly affects that is obvious for rough surfaces and is unapparent for smooth surfaces. Obviously, there should be no masking and shadowing effect in an ideal smooth surface whose σ is zero. However, Blinn’s G function is irrelevant with roughness σ. As an important improvement, the modified g function given in this paper contains the variable of σ.

As is shown in Fig. 6, Blinn’s G function curve is fixed and the modified g shows different distributions with changing values of σ. Surfaces with different roughness are described by the modified g function which has a lower value for a surface with a larger σ. It depicts the property of the masking and shadowing effects more precisely. Besides, the modification improves masking and shadowing function from a broken line to a smooth curve, which is more physically reasonable.

Fig. 6 Blinn’s G and modified G curves with different roughness when θ_i=0°.

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3.2. Effects of polarization

According to Maxwell’s equation and the boundary conditions for the electric and magnetic field components, Fresnel introduced the formalism to describe the interaction of a polarized beam with a reflective or transmissive medium [19], as is shown in Fig. 7. Fresnel’s equation shows that for radiation incident onto a planar dielectric surface (i.e., an optically flat surface), the reflectivities of perpendicular and parallel polarized electric field r_ss and r_pp are functions of the index of refraction of the two media and incident angle expressed as Eq. (20) and Eq. (21).

n = \frac{n_{2}}{n_{1}}

r_{s s} = \frac{E_{s}^{r}}{E_{s}^{i}} = \frac{{(n^{2} - {sin}^{2} θ)}^{\frac{1}{2}} - cos θ}{{(n^{2} - {sin}^{2} θ)}^{\frac{1}{2}} + cos θ}

r_{p p} = \frac{E_{p}^{r}}{E_{p}^{i}} = \frac{n^{2} cos θ - {(n^{2} - {sin}^{2} θ)}^{\frac{1}{2}}}{n^{2} cos θ + {(n^{2} - {sin}^{2} θ)}^{\frac{1}{2}}}

Fig. 7 Fresnel’s reflection.

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Where n₁ is the index of refraction in the medium in which the wave is propagating (often air) and n₂ is the index of refraction of the second medium (the reflecting surface), n is the relative index of refraction and θ is the incident angle. For electric conductor materials, the expressions of r_ss and r_pp can be given by replacing index of refraction n in Eq. (20) and Eq. (21) by complex index of refraction n^*=n−ik in which the real part n is the refractive index and indicates the phase velocity while the imaginary part k is called the extinction coefficient. The intensity reflectance of perpendicular, parallel polarized and unpolarized light R_s, R_p and R_unpol are expressed in Eq. (22) – Eq. (24).

R_{s} = | {r_{s s}}^{2} |

R_{p} = | {r_{p p}}^{2} |

R_{unpol} = \frac{1}{2} (R_{s} + R_{s})

According to Fresnel’s equation, the simulated R_s, R_p and R_unpol of dielectric with n=1.8 and electric conductor with n^*=1.5+3i are shown in Fig. 8.

Fig. 8 (a) Fresnel reflectance for n=1.8. (b) Fresnel reflectance for n^*=1.5+3i.

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According to the geometry of BRDF shown in Fig. 1, the incident angle θ measured from the reflecting microfacet can be expressed as:

θ = \frac{1}{2} | θ_{i} + θ_{r} |

The Fresnel’s reflection factors on microfacet R_s, R_p and R_unpol for different polarization are related to the incident angle θ_i, reflect angle θ_r, and index of refraction n or complex index of refraction n^* of the material. Thus R_s(θ_i, θ_r, n), R_p(θ_i, θ_r, n) and R_unpol(θ_i, θ_r, n) can be given from Eq. (26) – Eq. (28) for arbitrary incident and reflect angles.

R_{s} (θ_{i}, θ_{r}, n) = {[\frac{{(n^{2} - {sin}^{2} \frac{| θ_{i} + θ_{r} |}{2})}^{\frac{1}{2}} - cos \frac{| θ_{i} + θ_{r} |}{2}}{{(n^{2} - {sin}^{2} \frac{| θ_{i} + θ_{r} |}{2})}^{\frac{1}{2}} + cos \frac{| θ_{i} + θ_{r} |}{2}}]}^{2}

R_{p} (θ_{i}, θ_{r}, n) = {[\frac{n^{2} cos \frac{| θ_{i} + θ_{r} |}{2} - {(n^{2} - {sin}^{2} \frac{| θ_{i} + θ_{r} |}{2})}^{\frac{1}{2}}}{n^{2} cos \frac{| θ_{i} + θ_{r} |}{2} + {(n^{2} - {sin}^{2} \frac{| θ_{i} + θ_{r} |}{2})}^{\frac{1}{2}}}]}^{2}

R_{unpol} (θ_{i}, θ_{r}, n) = \frac{R_{s} (θ_{i}, θ_{r}, n) + R_{p} (θ_{i}, θ_{r}, n)}{2}

It can be seen in Fig. 9 that the R factors of s-polarized light are much larger than that of p-polarized light for both dielectric and electric conductor materials, especially in large reflect angles. Unpolarized R factor R_unpol is the average of R_s and R_p. And electric conductor has higher R factors than dielectric. The polarization of light and the index of refraction of reflecting material strongly affect the intensity reflectance, thus R factor is non-negligible for BRDF models.

Fig. 9 R_s, R_p and R_unpol factors when θ_i=30° for (a) n=1.8 and (b) n^*=1.5+3i.

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3.3. G expression

The geometrical attenuation factor G given in this paper is the combination of the modified masking and shadowing factor g(θ_i, θ_r, n), and the Fresnel’s reflection factors R_s(θ_i, θ_r, n), R_p(θ_i, θ_r, n) and R_unpol(θ_i, θ_r, n) we discussed above. There is no interaction between these two effects and they work independently. So the combined factors G_s/G_p/G_unpol should be the product of g(θ_i, θ_r, n) and R_s(θ_i, θ_r, n)/R_p(θ_i, θ_r, n)/R_unpol(θ_i, θ_r, n), and are related to four parameters θ_i, θ_r, σ and n, as are shown below.

G_{s} (θ_{i}, θ_{r}, σ, n) = g (θ_{i}, θ_{r}, σ) \cdot R_{s} (θ_{i}, θ_{r}, n)

G_{p} (θ_{i}, θ_{r}, σ, n) = g (θ_{i}, θ_{r}, σ) \cdot R_{p} (θ_{i}, θ_{r}, n)

G_{unpol} (θ_{i}, θ_{r}, σ, n) = g (θ_{i}, θ_{r}, σ) \cdot R_{unpol} (θ_{i}, θ_{r}, n)

Fig. 10 and Fig. 11 show the simulation of the normalized combined G factors [G/G(0°)] of s-polarized, p-polarized and unpolarized light in different incident angles for dielectric and electric conductor materials compared with Blinn’s G factor.

Fig. 10 Normalized G_s, G_p, G_unpol and Blinn’s G curves when σ=0.5 and n=1.8.

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Fig. 11 Normalized G_s, G_p, G_unpol and Blinn’s G curves when σ=0.5 and n^*=1.5+3i.

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It shows in the discussion above that, Blinn’s G function curve is only related to incident and reflect angles. The modified G given in this paper has a smooth function curve, which is controlled by surface roughness σ. Besides, the modified G factors are related to the index of refraction of the reflecting material and has different distribution behaviors for different polarization.

4. Comparison of simulation and measurement

In order to validate the modified G expression, the comparison of simulated specular component of BRDF f_s with G factors and experimental result is present. We measured two ZnO coating samples with a same index of refractive n=1.98 and different surface roughness. The values of surface roughness of sample 1 and sample 2 are 0.2 and 0.08 respectively. Each f_s of the two samples was measured with s-polarized, p-polarized and unpolarized radiation under the condition of θ_i=0° and θ_i=30°. The simulated and measured normalized f_s [f_s/f_s−unpol(θ_i)] curves are shown in Fig. 12 and Fig. 13.

Fig. 12 Simulated and measured BRDF of ZnO sample 1.

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Fig. 13 Simulated and measured BRDF of ZnO sample 2.

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The simulation and experimental results show that for the two samples the normalized f_s curves with modified G factors of s-polarized, p-polarized and unpolarized radiation given in this paper fit well with the measured data. It shows that, especially for sample 1, the sharp turning point caused by Blinn’s G is eliminated by the modified G function which turns unpolarized f_s curves from broken lines to smooth curves, and a better prediction of BRDF is given by the modified G which makes BRDF closer to the experimental results.

The error of theoretical f_s in the comparison shown above is believed to come mainly from the diffused component of reflection that is not taken account into the f_s calculation. Also, the measured roughness σ of the two samples may be different from the real result. The next step in our research is to focus on establishing polarized BRDF model which is more generally and precisely to depict the polarized properties of reflection.

5. Conclusion

In this paper, a modified geometrical attenuation model based on random surface microfacet theory is presented compared with Blinn’s G model which relies on a symmetrical V-shape groove assumption. The modified G factor possesses two main improvements. The first is the integral expression of the masking and shadowing effects based on Gaussian slope angle distribution of microfacets. The second is the addition of polarized effect by considering Fresnel’s reflectance of different polarized radiation. The modified G expressions are given by combining the masking and shadowing effects and the polarized effect. Compared with Blinn’s G, it is more physically reasonable that the modified G factor contains the variable of surface roughness σ and eliminates the sharp turning points in BRDF curves. The effect of polarization is considered thus the G factors are related to index of refraction n of the reflecting material and the polarization of radiation. BRDF of two ZnO samples with different roughness was measured with different polarized radiation to validate the modified G factors. It showed that the simulated BRDF curves with modified G factors fit well with the experimental results and the precision of BRDF model is improved.

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Modified polarized geometrical attenuation model for bidirectional reflection distribution function based on random surface microfacet theory

Abstract

1. Introduction

2. T -S BRDF model and Blinn’s G model

3. Geometrical attenuation model modification

3.1. Masking and shadowing effects

3.2. Effects of polarization

3.3. G expression

4. Comparison of simulation and measurement

5. Conclusion

References and links

Cited By

Figures (13)

Equations (31)

Optics Express