Comparison of microfacet BRDF model to modified Beckmann-Kirchhoff BRDF model for rough and smooth surfaces

Samuel D. Butler; Stephen E. Nauyoks; Michael A. Marciniak

doi:10.1364/OE.23.029100

1. Introduction

The bidirectional reflectance distribution function (BRDF) defines the ratio of the reflected radiance to the incident irradiance for materials, and is commonly used in remote sensing [1,2], scene generation [3–5], and computer graphics [6–9]. Nicodemus formally defined the BRDF as [10]

f_{r} = \frac{d L_{s} ({\hat{ω}}_{i}, {\hat{ω}}_{s}, λ)}{d E_{i} ({\hat{ω}}_{i}, λ)},

where

{\hat{ω}}_{i}

is the incident unit vector and

{\hat{ω}}_{s}

is the scattered (outgoing) unit vector, where in both cases

\hat{ω} = {[1, θ, ϕ]}^{T}

is a unit vector pointing in the incident (or scattered) direction, with spherical coordinates θ (representing elevation angle) and ϕ (representing azimuthal angle) as measured from the surface normal; see Fig. 1(a). λ represents the wavelength of light.

Fig. 1 Relevant coordinate systems for the microfacet (geometric) BRDF model. (a) BRDF coordinate system orientation relative to the surface normal $\hat{z}$ . (b) Microfacet coordinates relative to the coordinates defined in (a). In all cases, $\hat{ω}$ has a θ and ϕ component in spherical coordinates, and is normalized to a magnitude of 1 [11].

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We are interested in using the BRDF to render scenes which are both radiometrically and spectrally accurate, yet are computationally inexpensive to implement. Therefore, in this paper, two different approaches to modeling BRDF will be compared: microfacet model BRDFs and scalar wave optics model BRDFs. The former assumes geometric optics only, but results in several easy-to-use closed-form expressions for the BRDF that can be written in a common form [11]. The latter uses linear systems theory to develop a scalar BRDF based on wave optics, but is much more cumbersome to apply to resource-constrained problems. The purpose of this paper is to understand how the physics of a more complete scalar wave optics model may be related to microfacet closed-form BRDF models. All components of microfacet models are directly related to components of a scalar wave optics model through the development of novel relationships between differing elements in these models. Surface reflection is fundamentally a physical optics problem, but understanding how a closed-form geometric optics model relates to a more complete, but more computationally complex, physical optics model may lead to development of models that are more radiometrically and spectrally accurate for applications that require a closed-form approximation, as is desirable for resource-constrained applications of the BRDF such as remote sensing.

To accomplish this task, first these models are introduced. Then, the rough and smooth surface limits of the wave optics model are compared to the common form used by many microfacet models. Relationships are then developed between the different components of the models, suggesting a different approach to modeling the geometric attenuation (shadowing and masking) term in microfacet models, as well as explaining limitations in microfacet models by quantifying differences in the angular terms in the models. Finally, the relationships developed in this paper are used to explain the role of the original Beckmann-Kirchhoff geometric term as compared to both models, and to explain experimental results in [12]. No BRDF model can be reasonably simple and simultaneously perfect, but comparing the microfacet class of BRDF models to a scalar wave optics model and developing relationships between all components of these models may lead to future development of closed-form BRDF models that correct for deficiencies in microfacet models.

2. Background

This section discusses two classes of BRDF models: the microfacet class of models and the Modified Beckmann-Kirchhoff (MBK) scalar wave optics model. These models form the basis of comparison in the remainder of the paper. Prior work in [11–15] has set the stage for comparing the simplicity of a closed-form approximation given by a geometric optics BRDF model with the accuracy of a more robust scalar wave optics model such as MBK BRDF theory. A summary of these key results is presented as background.

2.1. Microfacet BRDF models

The microfacet class of BRDF models includes several popular models, such as Torrance-Sparrow [3], Blinn-Phong [6], Cook-Torrance [7], Ward-Duer [16, 17], Hyde [18], Priest-Germer [19], and Beard-Maxwell [4]. These microfacet models for isotropic surfaces can all be written in a general form as [11]

f_{μ} ({\hat{ω}}_{i}, {\hat{ω}}_{s}) = ρ_{s} F (θ_{d}) D_{μ} (θ_{h}) G ({\hat{ω}}_{i}, {\hat{ω}}_{s}) σ (θ_{i}, θ_{s}) + ρ_{v} V ({\hat{ω}}_{i}, {\hat{ω}}_{s}) + ρ_{d} / π,

where the first group of terms is due to surface reflection, the second to volumetric scattering V, and the third to Lambertian scattering. ρ_s, ρ_v, and ρ_d are fit parameters that are allowed to vary with wavelength and signify the amount of surface, volume, and diffuse (Lambertian) reflectance for a material. For surface-reflecting materials in a single bounce model, ρ_v = 0 and ρ_d = 0. F is the Fresnel reflectance as derived in many optics and electromagnetism textbooks such as [20], with the usual parameters of complex refractive index ñ = n + iκ. D_μ is the microfacet surface normal statistical distribution. G is a geometric attenuation (shadowing and masking) term. σ is a conversion from scattering cross section to BRDF, defined as

σ (θ_{i}, θ_{s}) = \frac{1}{4 \cos θ_{i} \cos θ_{s}} .

Overviews of popular microfacet models are given in [11, 21]. In this paper, all surfaces are assumed to be isotropic; ϕ_i = 0 is chosen as the orientation of the ϕ axis. Microfacet coordinates θ_h (representing the half angle, or angle of the bisector) and θ_d (representing the incident angle in the microfacet’s frame of reference) have been used since the Torrance-Sparrow BRDF in 1967 [3], but were only recently formally defined as [22]

\begin{array}{l} {\hat{ω}}_{h} = \frac{{\hat{ω}}_{i} + {\hat{ω}}_{s}}{| | {\hat{ω}}_{i} + {\hat{ω}}_{s} | |}, \\ {\hat{ω}}_{d} = ℝ_{y} (- θ_{h}) ℝ_{z} (- ϕ_{h}), \end{array}

where ℝ_a is a rotation about some axis a. This formal definition has the advantage of specifying the ϕ dependence of

{\hat{ω}}_{h}

and

{\hat{ω}}_{d}

in addition to the θ dependence, which is of particular importance for anisotropic surfaces. In the isotropic case, only the θ_h and θ_d angles are computed using equations that appear in many papers in microfacet literature [3, 4, 7, 16, 18, 19, 23], and are given as

\cos (2 θ_{d}) = \cos θ_{i} \cos θ_{s} + \sin θ_{i} \sin θ_{s} \cos ϕ_{s},

\cos θ_{h} = \frac{\cos θ_{i} + \cos θ_{s}}{2 \cos θ_{d}} .

The Gaussian distribution D_g is used in the Torrance-Sparrow [3], Cook-Torrance [7], Ward-Duer [16, 17] (in the small angle approximation), Priest-Germer [19], and Hyde [18] BRDF models as well as being approximated by the popular cosine lobe distribution used in Blinn-Phong [6], and is given by [11]

D_{g} (θ_{h}) = \frac{1}{2 π σ_{g}^{2} \cos^{4} θ_{h}} \exp [- \frac{\tan^{2} θ_{h}}{2 σ_{g}^{2}}],

where σ_g is commonly thought of as statistically describing the microfacet orientation, but is actually a fit parameter which will be shown here to vary with wavelength, indicating a breakdown of the geometric optics model. From geometric optics, σ_g represents the probability distribution of microsurface normals. Mathematically, the microfacet surface normal probability distribution can be calculated from the surface height profile by calculating the normal to the derivative of the surface height profile. The angular dependence of the distribution function is primarily driven by tan² θ_h. The remaining terms, and in particular the 1/cos⁴ θ_h term, normalize the Gaussian distribution to ensure it is a proper probability distribution function in projected area (as Trowbridge indicates is the proper normalization in [24]); that is,

\int_{0}^{2 π} \int_{0}^{π / 2} D_{g} (θ_{h}) \cos θ_{h} \sin θ_{h} d θ_{h} d ϕ_{h} = 1 .

Unlike linear systems models, the closed-form expression for microfacet models presented in Eq. (2) is used for both very rough and smooth surfaces.

2.2. Modified Beckmann-Kirchhoff model

Several different physical optics models have been developed, deriving from the Kirchhoff tangent plane approximation, small perturbation approximation, and other unifying methods; for a 2004 survey of over 30 of these models, see [25]. Elfouhaily noted all of these models had deficiencies. Krywonos then developed Generalized Harvey-Shack (GHS) theory and MBK theory to correct some of these deficiencies.

More recently, a method for estimating surface statistics using the tangent plane approximation and assuming a Gaussian distribution of surface properties was developed in [26]. Although this method did not include out-of-plane scatter, its success in predicting surface statistics lends credence to using a physical optics model based on the tangent plane approximation, which is used by the Beckmann-Kirchhoff (BK) scattering model.

The BK scattering model was formulated in 1963 [27]. Similar to [26], the BK model uses the tangent plane approximation and assumes a Gaussian distribution of surface properties. However, the original BK theory was formulated prior to formal definition of the BRDF in 1977 [10]. This BK model was modified when the linear systems diffraction BRDF theory was generalized to all incident and scattered angles, as well as for a variety of surfaces, by Krywonos [15]. Linear systems diffraction BRDF theory was originally formulated for paraxial reflection in the Harvey-Shack BRDF model, then was extended to arbitrary angles in the GHS BRDF model [14, 15, 28]. The GHS BRDF model developed by Krywonos does not possess a known closed-form solution in general. GHS theory is more physically rigorous, but is more computationally intensive and unfortunately, this method fails to distinguish between scattering due to surface reflection versus scattering due to volumetric effects. In both cases, the surface is treated as adding an overall phase offset, regardless of whether the scatter is due to the surface or due to impurities in the material. Krywonos also proposed MBK BRDF theory in [15,28,29] that is equal to his GHS theory in the limit of either a polished or very rough surface [28, 29]. This formulation assumes Gaussian statistics, but results in an infinite series expression for the BRDF that more readily enables comparison to the microfacet class of models without sacrificing the accuracy of the GHS model, from which the modifications to the original BK theory were derived.

In GHS and MBK theory, the BRDF is calculated in direction cosine space (α, β) instead of spherical space (θ, ϕ). This space is defined as

\begin{array}{l} Δ α = α_{s} - α_{i} = λ ν_{y} = \sin θ_{s} \sin (ϕ_{s} - π) \\ Δ β = β_{s} - β_{i} = λ ν_{x} = \sin θ_{s} \cos (ϕ_{s} - π) - \sin θ_{i}, \end{array}

where the extra π terms here arise from a difference in defining the ϕ = 0 location in the microfacet model as compared to the linear systems model. For in-plane forward scatter, ϕ_s = π in both models as presented here (backscatter is ϕ_s = 0); however, in-plane forward scatter data is represented by ϕ_s = 0 in some linear systems papers [14, 15]. For isotropic samples, the angular dependence simplifies to

η_{r}^{2} = {(Δ α)}^{2} + {(Δ β)}^{2} = \sin^{2} θ_{i} + \sin^{2} θ_{s} + 2 \sin θ_{i} \sin θ_{s} \cos ϕ_{s} = {(\frac{λ ν_{x y}}{2 π})}^{2},

where ν_xy is the notation used by Beckmann [27].

The MBK formulation presented in Eq. (11) is the Angle Spread Function (ASF), f_a, which is related to the radiance L; the ASF is equal to the BRDF if F = 1 (total Fresnel reflectance). Assuming the surface statistics follow a Gaussian autocovariance, and taking into account the 1/λ² term present in MBK as shown in [29], the ASF is calculated as

f_{a} = \frac{π K l_{c}^{2}}{λ^{2}} \exp (- g) \sum_{m = 1}^{\infty} \frac{g^{m}}{m! m} \exp (- \frac{ν_{x y}^{2} l_{c}^{2}}{4 m}),

where K is a renormalization term that ensures conservation of energy, l_c is the correlation length of the surface, and g is related to the surface height σ_s and given as [28]

g (θ_{i}, θ_{s}) = {(\frac{2 π σ_{s}}{λ})}^{2} {(\cos θ_{i} + \cos θ_{s})}^{2} .

When written in terms of scattered radiance L_s, K is defined as [15, 28, 29]

K (β_{i}) = \frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} L_{s} (α_{s}, β_{s} - β_{i}) d β_{s} d α_{s}}{\int_{- 1}^{1} \int_{- \sqrt{1 - α_{s}^{2}}}^{\sqrt{1 - α_{s}^{2}}} L_{s} (α_{s}, β_{s} - β_{i}) d β_{s} d α_{s}} .

L_s ≥ 0 everywhere, so K≥1 for ensures all incident angles. Physically, K is a unitless quantity that energy is not lost to evanescent waves by redistributing energy that may fall outside the unit circle of real space defined by the direction cosines. For some BRDF applications, this term can be problematic to compute because it depends on knowing the surface distribution shape, and involves computation of a double integral expression that does not possess an analytic solution for most surface distributions.

For a smooth surface, σ_s << λ and as such only the m = 1 term of the sum in Eq. (11) is significant. In [29], Krywonos defines a smooth surface as g(θ_i, θ_s) < 0.025, which results in less than 1% error compared to the infinite series summation when θ_i ≤70°. By using Eqs. (10)–(12) and multiplying by the polarization factor Q to convert the ASF to a BRDF, this polished approximation f_p can be written as

f_{p} = \frac{4 π^{3} l_{c}^{2} σ_{s}^{2} Q {(\cos θ_{i} + \cos θ_{s})}^{2}}{λ^{4}} \exp [- {(\frac{π l_{c} η_{r}}{λ})}^{2}],

where exp(−g) ≈ 1 since σ_s << λ for a polished surface, and K ≈ 1 for a polished surface, as shown in [15].

At the other extreme, if a surface is very rough, the BRDF f_vr as given in [29] can be rewritten using Eqs. (10)–(12) as

f_{v r} = \frac{K Q l_{c}^{2}}{4 π σ_{s}^{2} {(\cos θ_{i} + \cos θ_{s})}^{2}} \exp [- {(\frac{l_{c}}{2 σ_{s}})}^{2} {(\frac{η_{r}}{\cos θ_{i} + \cos θ_{s}})}^{2}],

where again we have multiplied by the polarization factor Q to convert from ASF to BRDF. In [29], Krywonos defines very rough as g(θ_i, θ_s) > 800, to result in less than 1% maximum error as compared to the infinite summation when θ_i≤70°.

The appropriate Fresnel term to convert ASF to BRDF is the polarization factor Q as described by Stover in [30]. The more popular unpolarized Fresnel reflectance term F derived in many electromagnetism and optics texts assumes plane wave incidence and an infinitely smooth surface. The derivation of the polarization factor Q in [30] relaxes the infinite plane wave and perfectly smooth assumptions in the derivation of F. For s-polarized light incident on a surface, Q_s = Q_ss + Q_sp. For p-polarized light incident on a surface, Q_p = Q_ps + Q_pp. For unpolarized light incident on a surface, Q = Q_s + Q_p. If θ_i = θ_s and ϕ_s = 180° (specular reflection), the equations for Q_s and Q_p reduce to the standard Fresnel equations F_s and F_p.

In [15, 31], careful measurements of material surface statistics were made by Stover. The measured surface statistics were used to compute the Power Spectral Density (PSD) of each surface. Stover then measured the BRDF of these surfaces. The GHS BRDF model was used to compute the PSD from the BRDF data. The computed PSD using GHS theory closely matched the measured surface statistics of the surfaces. Similarly, in [32], Schröder et al. used GHS theory to predict the scatter off an arbitrary (but known) surface of arbitrary roughness, both rough and smooth. Since GHS matches MBK in the very rough or very smooth approximation, f_p and f_vr from MBK theory are considered to be accurate physical models in the limit of a polished or very rough surface, respectively.

3. Analysis

To compare these BRDF models, the ratio of microfacet BRDFs to the MBK BRDF is taken in both the very rough approximation and the smooth surface approximation, assuming an isotropic sample. We also assume surface statistics (surface height and autocorrelation length) follow a Gaussian distribution.

Since MBK theory can be used to predict accurate surface statistics, the ratios f_μ/f_p and f_μ/f_vr would equal 1 if the microfacet model also modeled physical truth. Differences from 1 indicate deviations in the microfacet model from the more physical MBK model, and may suggest how to alter the microfacet model to improve its accuracy while still maintaining a closed-form approximation to the BRDF.

3.1. Very rough surface comparison

The ratio of microfacet BRDFs f_μ to the very rough surface MBK BRDF f_vr is

\begin{array}{l} \frac{f_{μ}}{f_{v r}} = (\frac{ρ_{s}}{2 K}) (\frac{2 F G}{Q}) (\frac{{(\cos θ_{i} + \cos θ_{s})}^{2}}{4 \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h}}) {(\frac{σ_{s} \sqrt{2}}{l_{c} σ_{g}})}^{2} \times \\ \exp [- (\frac{\tan^{2} θ_{h}}{2 σ_{g}^{2}} - \frac{l_{c}^{2}}{4 σ_{s}^{2}} \frac{η_{r}^{2}}{{(\cos θ_{i} + \cos θ_{s})}^{2}})] . \end{array}

The exponential term will be analyzed first. The angular dependence is contained in two terms: tan ²θ_h from the microfacet model and $η_{r}^{2} / {(\cos θ_{i} + \cos θ_{s})}^{2}$ from MBK. Although these terms were derived with a completely different physical interpretation (geometric optics versus physical optics), it will be shown here that these terms are equal at all incident and scattered angles, in-plane and out-of-plane. Starting with Eq. (6) and using basic trigonometry identities to rewrite cos θ_h in terms of tan θ_h:

\tan^{2} θ_{h} = \frac{4 \cos^{2} θ_{d} - {(\cos θ_{i} + \cos θ_{s})}^{2}}{{(\cos θ_{i} + \cos θ_{s})}^{2}} .

The denominator already matches the denominator of $η_{r}^{2} / {(\cos θ_{i} + \cos θ_{s})}^{2}$ , so we only need to show that the numerator equals $η_{r}^{2}$ as given by Eq. (10). This is performed by use of the double angle formula for cos(2θ_d) and adding zero:

\begin{array}{l} 4 \cos^{2} θ_{d} - {(\cos θ_{i} + \cos θ_{s})}^{2} = 2 + 2 (2 \cos^{2} θ_{d} - 1) - {(\cos θ_{i} + \cos θ_{s})}^{2} \\ = 2 + 2 \cos (2 θ_{d}) - {(\cos θ_{i} + \cos θ_{s})}^{2} \\ = \sin^{2} θ_{i} + \cos^{2} θ_{i} + \sin^{2} θ_{s} + \cos^{2} θ_{s} + 2 \cos (2 θ_{d}) - {(\cos θ_{i} + \cos θ_{s})}^{2} \\ = \sin^{2} θ_{i} + \sin^{2} θ_{s} + 2 \sin θ_{i} \sin θ_{s} \cos ϕ_{s} = η_{r}^{2}, \end{array}

where in the last line Eq. (5) was used to obtain

η_{r}^{2}

. Thus, we have shown that linear systems direction cosine space for a very rough surface is exactly equal to the geometric microfacet model bisector space:

\tan^{2} θ_{h} = {(\frac{η_{r}}{\cos θ_{i} + \cos θ_{s}})}^{2} .

The angular term in the exponential from the scalar wave optics model, $η_{r}^{2} / {(\cos θ_{i} + \cos θ_{s})}^{2}$ , ranges from 0 to ∞. This is in contrast to the result obtained later in this paper when examining a polished surface. (The angular term in the exponential from the microfacet model, tan²θ_h, always ranges from 0 to ∞.)

From geometric optics σ_g is commonly thought of as the Gaussian width of the probability distribution of microsurface normals. Mathematically, the microfacet surface normals can be calculated from the surface height profile and is related to the normal to the derivative of the surface height profile; however, only the probability distribution of the surface profile is specified. Instead, from Eq. (16), we solve for σ_g for a very rough surface using the remaining terms in the exponential, obtaining

σ_{g, v r} = \frac{σ_{s} \sqrt{2}}{l_{c}} .

Recall σ_s represents the width of the Gaussian in the vertical (height) direction and l_c represents the width of the Gaussian in the horizontal (correlation length) direction. This result is identical to the result obtained in the Hyde polarized BRDF model’s parameterization of the Gaussian distribution given in [18] that was derived using the Method of Moments, suggesting the methodology employed in this comparison is correct. However, Eq. (20) only holds for the very rough surface approximation. A different result for the polished surface analysis is obtained later in the paper in Eq. (25), showing this interpretation of σ_g is only valid for very rough surfaces, and is not true in general. This point will be discussed in more depth after comparing the polished surface approximation to the microfacet model.

Eq. (16) has now simplified to

\frac{f_{μ}}{f_{v r}} = (\frac{ρ_{s}}{2 K}) (\frac{2 F G}{Q}) (\frac{{(\cos θ_{i} + \cos θ_{s})}^{2}}{4 \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h}}) .

Direct comparisons between Q and F vary significantly in magnitude. Stover explains the s polarization, but does not suggest an approximation valid at all angles and indices of refraction for the p polarization (and thus the unpolarized term Q = Q_s + Q_p), although he does present approximations for a few limiting cases [30]. To improve on this understanding, the polarization factor Q was analyzed from the perspective of comparing it to not only Fresnel reflectance, but also to the remaining angular terms from the microfacet model, since the microfacet model uses F instead of Q.

In [13], we discovered that, for in-plane angles, there is an approximate relationship S between the unpolarized Fresnel reflectance and the polarization factor, Q, given as

S = \frac{4 \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h}}{{(\cos θ_{i} + \cos θ_{s})}^{2}} \approx \frac{2 F}{Q},

where the factor of 2 arises from Q = Q_s +Q_p = F_s +F_p = 2F if θ_i =θ_s and ϕ_s = 180° (standard Fresnel reflection). However, in [13], there was not a solid basis for this approximation. In this paper, the rationale for Eq. (22) is now evident, as this relationship explicitly appears in Eq. (21).

In [13], we also only examined the in-plane relationship. In this paper, a more complete analysis is performed by computing the relative difference between 2F/Q and S, including at out-of-plane angles, where the relative difference is defined as

R_{d} = \frac{| 2 F / Q - S |}{2 F / Q} .

Figure 2 was created at θ_i = (15°, 30°, 45°, 60°) for five different indices of refraction. For each angle, the five different indices of refraction were chosen to represent a wide range of materials: high n and high κ: ñ = 4 + 10i; low n and low (but nonzero) κ: ñ = 1.5 + i, low n and zero κ: ñ = 1.4, moderate n and κ: ñ = 1.7+5i, and n < 1: ñ = 0.25+3i. When θ_i = θ_s and ϕ_s = 180° (specular reflection), R_d = 0. In general, even out-of-plane, the error is relatively small if θ_i and θ_s are both small, but increases when either θ_i or θ_s is large. This result suggests that the cross section conversion term σ (θ_i,θ_s) in Eq. (3) and the microfacet distribution normalization 1/cos⁴ θ_h in Eq. (7) both arise from using F instead of Q.

Fig. 2 Surface plot of relative difference R_d for five different indices of refraction: (a-d) ñ = 4+10i, (e-h) ñ = 1.5+i, (i-l) ñ = 1.4, (m-p) ñ = 1.7+5i, (q-t) ñ = 0.25+3i. For each index, four different incident angles are plotted: (a,e,i,m,q) θ_i = 15°; (b,f,j,n,r) θ_i = 30°; (c,g,k,o,s) θ_i = 45°; (d,h,l,p,t) θ_i = 60°. The plots are all symmetric about ϕ_s = 180°. The dotted black line at ϕ_s = 180° on the right side of each plot represents in-plane scatter. In each case, R_d = 0 when θ_i = θ_s and ϕ_s = 180°; R_d is generally small when θ_i and θ_s are small.

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This result also suggests a novel interpretation for G. In geometric optics, G = 1 when the incident and scattered angles are both small–that is, there is no shadowing and masking. This is consistent with the low relative difference observed in Fig. 2 at small incident and scattered angles. However, the relative difference at large angles does not agree with a geometric optics interpretation. Since the BRDF is fundamentally a wave optics problem, better approximations for G may be derived by solving 2FG/Q = S for G instead of from the geometric optics approach detailed in [33]; such a solution is beyond the scope of this paper. Alternatively, Q does have a closed-form solution (although it is substantially more complex than F). For increased accuracy, Q/2 could be used in place of F using this observed relationship.

3.2. Polished surface comparison

Much of the detail for a similar analysis of f_μ/f_p in the polished surface approximation is contained in [13], where the microfacet model was compared to GHS. However, GHS was shown to reduce to MBK in the polished surface approximation [15]. Key points from that analysis are presented here.

The ratio f_μ / f_p results in

\begin{array}{l} \frac{f_{μ}}{f_{p}} = (\frac{ρ_{s} λ^{2}}{8 π^{2} σ_{s}^{2}}) (\frac{2 F G}{Q}) (\frac{1}{4 \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h} {(\cos θ_{i} + \cos θ_{s})}^{2}}) \times \\ (\frac{λ^{2}}{2 π^{2} σ_{g}^{2} l_{c}^{2}}) \exp [- (\frac{\tan^{2} θ_{h}}{2 σ_{g}^{2}} - \frac{π^{2} l_{c}^{2} η_{r}^{2}}{λ^{2}})] . \end{array}

Compared to the very rough surface approximation, the angular dependence of the exponential terms has changed, so the microfacet angular dependence is no longer equal to the wave optics angular dependence. From Eq. (19), instead of the exponential from scalar wave optics having an angular dependence of tan² θ_h, the angular dependence is $η_{r}^{2} = (\tan^{2} θ_{h}) {(\cos θ_{i} + \cos θ_{s})}^{2}$ . Unlike with the very rough surface, the polished surface angular term in the exponential from scalar wave optics, η_r, only ranges from 0 to 4, not from 0 to ∞, but the microfacet model angular term, tan²θ_h, still ranges from 0 to ∞. Additionally, a different relationship for σ_g is observed, given as

σ_{g, p} = \frac{λ}{π l_{c} \sqrt{2}},

which varies with wavelength, unlike in Eq. (20). This shows the microsurface normal interpretation in the microfacet model does not hold for a polished surface, since the microsurface normal distribution should not vary with wavelength.

When making this substitution for σ_g, it is observed that ρ_s remains a unitless quantity, but varies with surface height and wavelength; that is, $ρ_{s} \propto σ_{s}^{2} / λ^{2}$ . Additionally, by applying Eq. (22) to substitute for 2F/Q, it becomes apparent that there remains a (cosθ_i + cosθ_s)⁻⁴ term in the ratio, suggesting the microfacet model would be multiplied by a (cosθ_i + cosθ_s)⁴ term.

Since there is now a wavelength dependence both in the distribution function and in ρ_s, it becomes apparent that a geometric optics model is not suited for this regime. Furthermore, the angular dependence is no longer simply tan²θ_h, but now is η_r. Equation (19) does relate the two expressions, but since the two are not equal, the microfacet model does not appear to be suitable for fitting to polished surfaces, unless corrections for the differences noted here were present.

3.3. Original Beckmann-Kirchhoff modification

In the original BK theory, $F_{b k}^{2}$ was used instead of K, and was described as a geometrical term [27]. Although Krywonos used K in place of $F_{b k}^{2}$ in MBK, $F_{b k}^{2}$ is much simpler to compute and may be desirable in some applications of BRDF if a more complete understanding of its role were developed. That purpose is explored in this section. This geometrical factor $F_{b k}^{2}$ was given as [27]

F_{b k}^{2} = {(\frac{1 + (\cos θ_{i} \cos θ_{s} + \sin θ_{i} \sin θ_{s} \cos ϕ_{s})}{\cos θ_{i} (\cos θ_{i} + \cos θ_{s})})}^{2} .

F_bk is not to be confused with Fresnel reflectance; it represents a geometric term that does not depend on index of refraction. Its connection to the microfacet model was not clear from prior work on MBK in [27, 28]; in this section, we develop an explanation and a minor modification for this term. Rewriting $F_{b k}^{2}$ in microfacet coordinates θ_h and θ_d:

F_{b k}^{2} = \frac{\cos^{2} θ_{d}}{\cos^{2} θ_{i} \cos^{2} θ_{h}} .

Using basic trigonometric manipulations to put S in microfacet coordinates, this term can be rewritten as

\frac{1}{S} = \frac{\cos^{2} θ_{d}}{\cos θ_{i} \cos θ_{s} \cos^{2} θ_{h}} \approx \frac{Q}{2 F} .

These equations suggest the purpose of $F_{b k}^{2}$ is to approximate Q/2 so that when BK theory is multiplied by the Fresnel equation F, it scales approximately correctly. However, there is an issue with $F_{b k}^{2}$ as presented. When θ_i ≠ θ_s, $F_{b k}^{2}$ does not obey Helmholtz reciprocity and differs from this approximation for Q/2F. For this reason, we propose using 1/S rather than the original $F_{b k}^{2}$ term; using 1/S makes the term functionally equivalent to the cross section conversion and microfacet distribution normalization, with the modified angular term that arises from g in MBK.

3.4. BRDF scaling by 1/σ

In [12], experimentally measured BRDF data was presented in direction cosine space, which aligns BRDF data at all incident angles, other than differing in height. We observed that multiplying measured BRDF data by a term inspired by the microfacet cross section conversion, cosθ_i cosθ_s, resulted in an approximate height alignment. With the 1/S term written in micro-facet coordinates above, the scaling in direction cosine space that was observed experimentally in [12] can be explained. When at the specular peak, cosθ_h = 1 (θ_h = 0). Near the specular peak, θ_h is small and thus cosθ_h ≈ 1. Thus, S ≈ cosθ_i cosθ_s as was observed in [12] to be the scaling term that aligned the BRDF magnitude data in direction cosine space.

4. Conclusion

The terms of the MBK BRDF model are now compared to microfacet models, enabling the MBK model to be rewritten in microfacet coordinates (assuming Gaussian statistics) for a very rough surface (g > 800) as

f_{v r} = \frac{K F G}{4 \cos θ_{i} \cos θ_{s}} (\frac{l_{c}^{2}}{2 π σ_{s}^{2} \cos^{4} θ_{h}}) \exp [- {(\frac{l_{c}}{2 σ_{s}})}^{2} \tan^{2} θ_{h}],

and for a polished surface (g < 0.025),

\begin{array}{l} f_{p} = \frac{{(2 π)}^{3} F G {(\cos θ_{i} + \cos θ_{s})}^{4}}{4 \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h}} (\frac{σ_{s}^{2} l_{c}^{2}}{λ^{4}}) \times \\ \exp [- {(\frac{π l_{c}}{λ})}^{2} \tan^{2} θ_{h} {(\cos θ_{i} + \cos θ_{s})}^{2}] . \end{array}

The infinite summation can also be written in microfacet coordinates as

\begin{array}{l} f_{M B K} = \frac{π F G {(\cos θ_{i} + \cos θ_{s})}^{2} K l_{c}^{2}}{2 λ^{2} \cos θ_{i} \cos θ_{s} \cos^{4} θ_{h}} \exp (- g) \times \\ \sum_{m = 1}^{\infty} \frac{g^{m}}{m! m} \exp (- \frac{l_{c}^{2}}{4 m λ^{2}} \tan^{2} θ_{h} {(\cos θ_{i} + \cos θ_{s})}^{2}), \end{array}

where g is defined in Eq. (12). In this form, a direct comparison between the microfacet class of BRDF models given in Eq. (2) is now possible, which is expected to lead to more physical closed-form BRDF models for applications such as remote sensing and computer graphics.

To obtain the necessary relationships that were used to obtain the above form for the MBK model in microfacet coordinates, and to better comprehend what the microfacet model represents, the very rough surface approximation of MBK was compared to the general form of the microfacet model. In the rough surface approximation, the angular dependence of MBK (and thus also of GHS) was shown to be exactly equal to the angular dependence in the microfacet distribution function. The remaining angular terms were compared to the ratio of the standard Fresnel term with the polarization factor Q. This led to a novel approximation for Q, and suggested where certain terms present in the microfacet BRDF models arise. In particular, part of the microfacet distribution function normalization and the cross section conversion term were found to result from using standard unpolarized Fresnel reflectance F instead of the polarization factor Q. The approximation was found to be relatively accurate, except when θ_i or θ_s is large; in the microfacet model, this region is where there would be significant shadowing and masking. Since the BRDF is fundamentally a physical optics problem, this observation suggests that instead of attempting to derive the shadowing and masking term from geometric optics, better results may be obtained by developing a modification to the ratio 2F/Q, particularly at large incident or scattered angles. Alternatively, it is now possible to modify the microfacet model to use Q instead of F using this relationship.

Next, key differences between the polished surface analysis that were detailed in [13] and the rough surface analysis developed in this paper are presented. This resulted in a modification to the angular and wavelength dependencies in the microfacet model for a polished surface, and showed the breakdown of a geometric model for BRDF.

Additionally, the original BK geometric term $F_{b k}^{2}$ was analyzed, developing an explanation for this term relative to both the microfacet and linear systems models. A modification to $F_{b k}^{2}$ in the BK model was also proposed. The physical insight developed in this paper is expected to lead to future development of better closed-form approximations to the BRDF that preserve the relative simplicity of the microfacet model, while increasing the fidelity of the microfacet BRDF approximation.

Acknowledgments

We would like to thank the United States Air Force Research Laboratory, Sensors Directorate, for supporting this research. The views expressed in this paper are those of the authors and do not necessarily reflect the official policy or position of the United States Air Force, the U. S. Department of Defense, or the U. S. government.

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