1. Introduction

Microfacet-based BRDF models were developed to study the scattering behaviour of light off rough surfaces. Since their introduction, they have proven instrumental in predicting reflectance from glossy surfaces of all sorts, and they combine ease of use with a high degree of realism. This has led to widespread use in areas such as remote sensing and material classification [1–3].

In the field of computer graphics, microfacet-based BRDF models greatly aided the development of modern photo-realistic rendering software by allowing highly realistic depictions of solid objects [4–7]. Their versatility can further be enhanced by using them in layered configurations: when they are used in such a way, one can model a wide variety of natural and manufactured surfaces via a highly intuitive process [8, 9]. However, some aspects of these models are not yet as well understood as one might wish. Production renderers usually employ some sort of closed form approximation of the BRDF: these models often result from a given micro-facet distribution and ignore wave properties of light, such as polarisation. Both efficient sampling of these resulting BRDFs, as well as the general shape and properties of the resulting lobes, have been the target of renewed interest.

The rendering community has also long been aware that polarisation is a potentially quite relevant feature of light: and in particular, that the appearance of inter-reflections between specular and glossy surfaces can be considerably influenced by this phenomenon. This of course also applies to multiple reflections between micro-facets: but how much this influences the resulting BRDF of a micro-facet surface does not seem to have been investigated yet. Also, while micro-facet models are potentially able to describe polarising glossy BRDFs, very little work has been conducted in this direction as well. In this paper, we report on our findings in exactly this direction: we built a brute-force micro-facet BRDF simulator that features switchable polarisation support, validate the simulator with state-of-art model from graphics, a library from physics and ellipsometric measurements of real surface samples, and properly assess the impact of including the feature in a system. With this system, we can conclusively show that inclusion of polarisation is needed for highly accurate results of certain surface types – but also that there are other types of surface for which polarisation can apparently be ignored.

The remainder of this paper is organised as follows: first, we review related work in Section 2. In Section 3, we describe the implementation of our brute force simulator. Then, in Section 4, we describe how we validated our system against models from graphics, physics and measured data. In Section 5 we present our findings for several representative surface configurations and conclude with our insights. Finally, we provide an outlook on future work in Section 6.

2. Related work

In the following two subsections, we discuss related work in the areas of micro-facet BRDF models (in particular, those which include multi-bounce effects) and polarisation effects. Previously, inclusion of multi-bounce effects and polarisation were seen as mutually exclusive ways to increase the accuracy and fidelity of BRDF modelling. In this paper, we investigate the combination of both effects, i.e. we essentially aim to perform a virtual analogy to the reflection ellipsometry described in reference [49].

2.1. Models of surface reflectance

In the optics community, a large amount of work has been done on surface reflectance modelling: to just name a few [1, 2, 10–13]. However, BRDF models from optics are usually not directly useful for computer graphics purposes. Modern rendering techniques rely on path tracing based techniques to solve the problem of light transport via a Monte Carlo approach. With this sort of technique, not only is a fast and accurate evaluation of the BRDF model itself needed: it is crucial to also have an efficient sampling scheme for the model. This second point is usually not a concern from the perspective of optics research, so many “BRDFs from optics” lack this crucial component.

Over the last decades, a considerable amount of research on this has also been done within computer graphics. Cook and Torrance [4] introduced the first micro-facet model to computer graphics: this was a simplified version of the Torrance-Sparrow model [10] from optics. It models surface reflectance via single reflections, and accounts for shadowing and masking. The authors assumed multi-bounce reflections could be approximated by a diffuse lobe, but according to recent work [6], this assumption apparently does not hold. Ward and Ashikhmin [14, 15] described anisotropic reflections based on micro-facet theory. Cabral and Westin presented techniques to simulate BRDFs with bump maps and gaussian height fields generated by FFT filtering [16, 17]. He [18] presented a polarising BRDF model based on wave optics, which is capable to represent a variety of materials. Walter [5] simulated light transmission through rough surfaces and introduced the now widely adopted GGX normal distribution to physically-based rendering. Baghre [19] investigated the effectiveness of microfacet BRDF models if non-parametric factor models are used. Raymond presented a BRDF model to simulate scratched materials [20]. Additionally, there are also publications which investigate optical effects in BRDF models, e.g. diffraction [21–23] and interference [24, 25].

Several efforts have also been made to study the geometric properties of the micro-facets that are assumed to be at the root of such BRDF models. For example, Heitz [26,27] provided detailed insights into the masking-shadowing function. Ribardière [7] introduced Student’s t-distribution to this area, which generalised the well-known Beckmann and GGX distributions and provided Smith’s analytical masking-shadowing function. Beyond work on masking-shadowing functions, most of the micro-facet BRDF models in both optics and computer graphics apply masking and shadowing functions to approximate the multi-scattering events on a surface: this can cause an energy loss of up to 20%. Recently, Heitz [6] filled this gap by presenting a multi-scattering micro-facet BSDF model that is based on the Smith model [28].

For multi-layered materials, Harahan and Krueger [29] introduced a layered model that uses linear transport theory, and accounted for multi-scattering by extending the single-scattering method from Cook-Torrance. Dorsey [30] and Gondek [31] provided approaches to model metallic patinas, pearlescent paints and pigmented materials in layered structures. A model that can reproduce a large number of different multi-layer surfaces was presented by Lafortune [32]. Wilkie and Weidlich [8, 33] introduced reflectance properties for diffuse fluorescent surfaces based on a layered micro-facet approach, and then presented a framework to combine several micro-facet models into an unified, expressive BRDF model. Donner [34] presented a layered, heterogeneous spectral reflectance model for human skin with spatially-varying absorption and scattering parameters. Jakob [9] went beyond Stam’s work [35] by offering an accurate and efficient solution with the employment of the radiative transfer equation (RTE) for any layered structure. However, their model only includes multi-scattering effects via approximations for multi-scattering effects in the micro-facet models themselves. There are also layered BRDF models that try to include interference effects [36–39].

2.2. Polarisation rendering

Polarisation is an intrinsic property of transversal electromagnetic waves, and the polarisation state of a light wave can be altered via interactions with matter, such as scattering, reflection, and refraction [40]. As modern realistic rendering techniques are already very computationally demanding without treatment of polarisation, the effects is usually not taken into account, unless specific circumstances demand its inclusion. Numerous publications exist which discuss how to integrate polarisation into modern rendering systems [42–55]. In Section 3.2, we discuss the specific formalisms we used in our simulation.

3. Implementation

In this section, we provide technical details of our simulation. In Section 3.1, we first discuss how we procedurally generate surface meshes which correspond to particular micro-facet distributions: if simulation of a layered surface is desired, several of these micro-facet meshes will be constructed, one for each layer. In Section 3.2, we then briefly review the mathematical formalisms we use to describe polarised light. Finally, the actual simulation is described in Section 3.3: how we shoot a large number of simulated photons at this surface model at a specified incident angle, and trace them until they re-emerge from the surface model, or are considered absorbed.

3.1. Random surface mesh construction

Our random mesh construction is similar to that described in a technical report by Heitz [56]. Using his methods, we are able to procedurally generate isotropic or anisotropic Beckmann heightfields that exhibit Gaussian distributions of their heights and slopes. The heightfields are created from N independent cosine waves (we use the default setting of N = 1000), and then used to generate a corresponding triangle mesh. There are three parameters that control the mesh generation: mesh resolution, height field size and roughness. Generally, for larger height field sizes and rougher surfaces, the mesh resolution should be higher. Figure 1(a) and 1(b) shows two Beckmann height fields with two different roughness parameters(0.1 and 0.4): the height field size we use is 400×400, and the mesh resolutions are 256×256 and 768×768.

 

Fig. 1 Two single layered Beckmann meshes of height field size 400×400, with roughness parameters 0.1(a) and 0.4(b).

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3.2. Formalisms for describing polarised light

Stokes vectors and Mueller matrices are common formalisms for polarised light and optical elements, and in our simulation we use them to describe light rays, and light-surface interactions, respectively. Stokes vectors use four numbers [s0, s1, s2, s3] to describe intensity and polarisation state of light at a given frequency, with s0 representing intensity. Mueller matrices, which are used as attenuation calculus for Stokes vectors, are defined by a 4×4 matrix.

In our simulation, individual micro-facets are assumed to be perfectly smooth, so that the attenuation of light reflecting off them can be described via the Fresnel terms: these yield a complete Mueller matrix for a given incident angle. Assuming there is a light wave travelling towards the interface between two layers, the corresponding indices of refraction for the upper and lower medium are n₀ and n₁ + ik; and θ_i, θ_r and θ_t correspond to the incident, reflection, and refraction angle, respectively.

The incident light wave is decomposed into the two orthogonal polarisation components s and p, denoted by E_s and E_p, respectively. Furthermore, E_rs (E_rp) and E_ts (E_tp) correspond to the reflective and refractive s (p) polarisation components. According to Fresnel’s law, the Fresnel reflective terms, together with their phase shifts, can be expressed as follows:

The Mueller matrix for Fresnel reflection and refraction can then be written as:

For a reflection event, A = F_s + F_p, B = F_s − F_p, $C=-2\sqrt{F_s{F}_p}cos({\theta }_s-{\theta }_p)$ and $S=2\sqrt{F_s{F}_p}sin({\theta }_s-{\theta }_p)$.

Due to the law of energy conservation, the corresponding refractive terms have to be T_s = 1 − F_s and T_p = 1 − F_p for a refraction event: in this case, the retardance also equals zero. The A, B, C, S terms in Mueller matrix for refraction are: A = T_s + T_p, B = T_s − T_p, $C=-2\sqrt{T_s{T}_p}$ and S = 0.

As stated in [47, 48], a polarised light ray must be traced together with its reference frame. As outlined in the cited course notes, operations on polarised light and attenuation elements are only permissible both operands share the same propagation direction, and have the same reference frame. If the operands are co-axial but the reference frames do not match, a rotation must be applied first, in order to bring them both into the same reference frame.

3.3. Virtual ellipsometry simulation

Essentially, our simulation is a specialised form of photon tracing [41]: particles are emitted from an unpolarised light source in the direction of the surface model at a selected angle of incidence, and traced until they leave the simulated surface. Due to facet-facet inter-reflections and the presence of multiple layers, photons can potentially undergo numerous interactions with micro-facets before leaving the simulated surface. If the surface model contains more than one layer, we apply a straightforward form of importance sampling at the inter-layer interfaces: the selection of whether a given path reflects or refracts is randomly chosen based on the Fresnel reflectance and transmittance coefficients for the given angle of incidence.

Figure 2 shows some of the photon propagation scenarios covered by our simulation. The general outline of our technique is as follows in the subsequent enumeration: pseudo code for the algorithm can be found in Code 1 [57], and the flow chart in Fig. 3 gives a graphical representation.

 

Fig. 2 Two example paths for photon tracing on a layered surface model. Note that apart from wave optics effects, we cover all possible interactions of photons with such a surface model - for reasons of clarity, not all possibilities, such as re-propagation of photons back to a lower layer due to multi-scattering within an intermediate layer, are shown here. Interactions with single layer surfaces are a trivial subset of this diagram.

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Fig. 3 Flowchart of our simulation algorithm. The numbers correspond to the algorithm step numbers used in Section 3.3.

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Multiple bounces between layers, as shown on the right of Fig. 2, can go on as long as the photon does not leave the simulated layer stack sideways. Once a photon path leaves the topmost mesh in an upward direction, the photon is splatted and accumulated in a upper hemispherical sensor – which in our case is a polarisation-aware spectral image that is initially totally black in all pixels: the collection of all splatted photons is the final output of our simulator. These spectral images can then be normally tone mapped and displayed, so that output images like Fig. 11 result. Due to a polarisation-aware image file format being used, the polarisation state of each pixel can also be analysed, along the lines shown in [46].

4. Validation of simulator output

Even though the algorithm outlined in the previous section is not particularly complex, we deemed it necessary to validate its output in several ways. Basic single-bounce photon tracing against a single surface was assumed to be unproblematic, but our model has three features which go beyond that. These are the inclusion of layered surfaces, the inclusion of multi-bounce events, and polarisation. We validated each of these in turn: (1) in Section 4.1, the first two features are validated against a state-of-art model from graphics; (2) in Section 4.2, the polarisation calculations are validated against a light scattering library from physics which was published by NIST [58]; (3) in Section 4.3, we conducted ellipsometric measurements on real samples, and compare our results against the obtained data.

4.1. Comparison against published simulation data

One way of checking the single layer performance of our simulation is by comparisons against published single layer simulation results. In a recent paper [6], the authors included a supplement with reference data they used to validate their own multi-scattering BSDF model: consequently, we also used this data to perform the same sort of validation on our own system. We ran our simulation for various incident angles and for different roughness values on single dielectric (η = 1.5) and conductive surfaces. It should be noted that for conductive surfaces, the reference data [6] was generated for a 100% reflective surface, so we also used a similar surface here – but only for this one experiment.

The results of our comparisons are shown in Tables 1 and 2. For each combination of roughness and incident angle, the corresponding reflected (R), refracted (T) and total scattering (TI) intensities (TI = R + T) are listed separately. In both tables, R, T and TI are from Heitz’s data, and R′, T′ and TI′ are from our experiments. The reflected and refracted light intensities are further split into the multi-scattering (left) and the single-scattering (right) simulation results. For example, in table 1, for the experiment with roughness 1.0 and incident angle 1 rad, the multi-scattered reflected and refracted intensities from Heitz’s data are R_m = 0.002 and T_m = 0.053, and the intensities from ours are $R_m^{\prime }=0.003$ and $T_m^{\prime }=0.052$. Both for the dielectric surface in Table 1 and conductive surfaces in Table 2 the reflected and the refracted intensities are well matched by our simulation results. It is also noteworthy that our simulator is energy conservative, i.e. TI′ = 0.999, especially when the roughness parameter rises.

Table 1. Comparisons of our simulation results against previously published simulation data [6] for a single layer dielectric surface (η = 1.5). Here, R, T, TI and R′, T′, TI′ correspond to reflected, transmitted and total scattered intensities from the data of Heitz, and our simulations. The subscripts m and s denote multi-scattered and single-scattered intensities. The worst and best matches are highlighted in italic and boldface, respectively.

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Table 2. Validation comparisons for a single layer conductive surface. The same variable names and labels as in Table 1 are used.

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Both validations are indicative that our surface simulation for reflection, refraction and multi-scattering effects on single layer surfaces is correct, or at least matches previously published results. As we only consider layered surfaces that do not exhibit inter-layer absorption in this paper, this also means that this validation also applies for the simulation of the restricted class of layered surfaces we are focusing on.

4.2. Comparison with SCATMECH

To validate the polarisation capabilities of our simulator, we compared the values of all four Stokes parameters obtained via our simulation to the values obtained with SCATMECH [58], which is a well-known polarised light scattering calculation library published by NIST.

We use the modified polarising micro-facet model provided by SCATMECH to simulate light reflection off a single layered metal surface for the entire hemisphere, as shown in Fig. 4. In that figure, we show representative results for a silver surface at 40° incidence: the refraction index for silver at λ = 400nm is η_Ag = 0.5 + 2.1i. As the slopes of our generated Beckmann surface statistically follow a Gaussian distribution [56], we use a roughness parameter of 0.3 both in our simulation and in SCATMECH, and also use a Gaussian slope distribution there.

 

Fig. 4 Stoke parameters patterns generated by our simulation (top) and by SCATMECH (middle), and their differences plots(bottom). Corresponding Stokes parameter plots have the same colour bars. The differences in Stokes component S3 are due to SCATMECH not considering multi-bounce events.

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As can be seen in Fig. 4, our simulation yields the same polarisation patterns as SCATMECH does. However, the S0, S1 and S2 Stokes parameters obtained via SCATMECH are slightly lower than our results, and S3 computed by SCATMECH is almost zero. But this is an expected deviation, as the analytical model used by SCATMECH does not account for multi-scattering events: and a non-zero S3 parameter can only be generated by multi-scattering events on single layered surface. It is noteworthy that the S2 component from our simulation shows both the negative (blue) and positive (yellow) linear polarisation pattern correctly. These results show that our simulator is capable of computing polarisation patterns that are in line with expectations.

4.3. Comparison against measured data

The previous two verifications demonstrated that our simulator seems to be performing correctly, at least when compared against other computational models. However, we also wanted some sort of verification done against reflectance measurements done on real surface samples, to connect our effort with observable reality.

There is a huge variety of surfaces one could study in such a context, so we had to choose specific samples based on some criteria. We focused on the following properties: (1) easy to obtain, (2) good inter-sample agreement for samples obtained at different times, (3) a reasonably homogeneous surface structure, and (4) a surface micro-structure that can be approximated well by micro-facet models. Based on these criteria, we selected various types of sandpaper as our test samples. These are comparatively rough, but have a crystalline micro-structure which resembles a very rough micro-facet surface. And as sandpaper is subject to well-defined manufacturing tolerances, a reasonable amount of inter-sample agreement can be expected as well.

For mounting, the samples were permanently affixed to a stable, even backing surface. Then, the micro-structure of the samples was examined via a scanning electron microscope (SEM), so that appropriate corresponding surface models could be built. Specifically, we chose to work with 1200 grain sandpaper, as it proved most suitable for representation via a procedural model of rough micro-geometry. We measured this sample with an imaging ellipsometer, and compared simulated ellipsometric measurements with the obtained data: this resulted in reasonable matches being obtained.

4.3.1. Sample preparation

In order to make the samples observable in SEM, they had to be covered with an extremely thin layer of gold. This was done via sputter coating, which evaporates individual gold atoms, and deposits them on the surface, as shown in Fig. 5. This sort of coating is standard practice in electron microscopy, and preserves the micro-structures of samples very well. For us, this necessary coating step had highly welcome side-effect that the samples were now coated in a thin layer of a material that (1) under normal circumstances does not oxidise (i.e. change), (2) has well known reflectance properties (Fresnel reflectance for gold) and (3) is still applied thick enough to completely determine the reflectance characteristics of the micro-geometry of the surface.

 

Fig. 5 Six samples prepared for our experiments, permanently mounted on a stable flat backing surface. A standard gold coating for scanning electron microscopy has been applied. The right sample in the middle row is 1200 grain sandpaper, which was the one we selected for detailed measurements.

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We also worked with a set of corresponding un-coated samples, and also achieved acceptable levels of agreement between simulation and measurements. However, as we had to estimate some object properties of the uncoated samples (in particular, the exact dielectric index of refraction) to achieve a match, we deem the results obtained with the coated samples to be more reliable: the properties of gold are well known, and do not have to be estimated.

4.3.2. Sample micro-structure reconstruction

Due to the high roughness and complex structure of sandpaper, common optics devices, like interferometers, are unable to capture the profile of the surface. The same applies to atomic force microscopes, and 3D reconstruction techniques that are provided by the operating software of the SEM equipment. However, based on the SEM image shown in Fig. 6(a), we were able to build a plausible model of sandpaper micro-structure via procedural modelling. We first generated a large scale “crystal map” made up of a large number of randomly shaped little crystals. These crystals are then placed on a glossy surface to form a surface with a micro-structure that is capable of simulating the multiple bounces between crystals, as shown in Fig. 6(b).

 

Fig. 6 The micro-structure of 1200 grain sandpaper (a) imaged by SEM, and (b) our procedural model. The right image in subfigures (a) and (b) is an enlargement of the image on the left. In subfigure (b), the procedurally modelled surface is generated via a large amount of simulated crystals that are placed on a backing surface.

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4.4. Measurement of reference results

We measured our samples with an Accurion Nanofilm EP3 ellipsometer which operates on a single wavelength in the red region of the spectrum(670nm). The measurement device has a polarisation filter and compensator (e.g. a quarter-wave retarder) on the light source arm, which is positioned at the input angle one wants to measure. On the detector arm, which is positioned at the output angle, it has a camera with a polarisation analyser in front of it. With this device, all possible input states of light can be generated, but due to the absence of a second compensator at the detector arm, not all reflectance states can be detected. And like all conventional ellipsometers, the device can only measure input/output angles that are in the plane of incidence. However, the two arms can be moved independently of each other, so different input and output angles can be measured.

For each pair of input/output angles we systematically generated all canonical input Stokes Vectors (linear polarisation at 0°, 45°, 90° and 135°, and left and right circular light). For each of these, at the detector end, the angle of polarisation analyser was rotated to 0°, 45°, 90° and 135°, respectively. It has to be noted that the raw data obtained with this instrument are relative attenuation values, not absolute measurements. However, two important measurement quantities of reflected light can easily be calculated from this measurement data: the angle of polarisation (AOP) and the degree of linear polarisation (DOLP). The AOP (in degrees) is defined as 180* arctan(|S2/S1|)=(2π), and the DOLP as $\sqrt{(S{1}^2+S2)}/S0$ [59].

For each incident angle, we varied the output angle to record data around the specular reflection area in the incident plane, with the expectation that diffraction effects would not play a significant role in this case [22]. Figure 7, 8 and 9 show the AOP and DOLP comparison results with incident angles of 30°, 45° and 60°, respectively. The pl_angle and co_angle values in each subfigure indicate the polarisation state of incident light, and AOP values are compared for different reflection angles. As can be seen in Fig. 7(a), 8(a) and 9(a), our simulation can match the measured AOP data very well for linearly polarised incident light, and our results are slightly higher for circularly polarised incident light. And in Fig. 7(b), 8(b) and 9(b), our simulated DOLP can capture the trend of the variations of measured DOLP data.

 

Fig. 7 AOP (angle of polarisation) and DOLP (degree of linear polarisation) values from gold-coated 1200 grain sandpaper measurement and simulation at different reflection angles in the incident plane, for an incident angle of 30° (see Section 4.4 for the definition of AOP and DOLP). pl_angle and co_angle show the angle of light source filter and compensator, respectively: the six plots correspond to the canonical inputs mentioned in the text (linear polarisation at 0°, 45°, 90° and 135°, and left and right circular light).

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Fig. 8 AOP and DOLP values for measurement vs. simulation, incident angle 45°. Same experiment as shown in Fig. 7

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Fig. 9 AOP and DOLP values for measurement vs. simulation, incident angle 60°. Same set-up as in Fig. 7.

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5. The influence of polarisation on the shape of the BRDF Lobe

As outlined in Sections 1, 2.1 and 3.2, polarisation can be a relevant feature when light interacts with surfaces. However, it was unclear just how large the effect of including the phenomenon in inter-facet reflection calculations is, and under which circumstances it has a significant impact. As we demonstrate in this section, inclusion of the phenomenon can, for certain types of surface, measurably affect the resulting BRDF shape.

5.1. Choosing the test cases

For layered surfaces, there are three simulation parameters that can affect the importance of including polarisation calculations in brute force estimations of their BRDF:

To investigate the actual impact of these parameters in practice, we proceeded as follows:

In all simulations in this section, we cover the wavelength range from 380nm to 700nm with a uniform sampling interval of 40nm, and use unpolarised light as input. We report the results from both polarising and non-polarising simulations, and provide the difference magnitudes between them. Two kinds of difference magnitude are considered: the maximum difference magnitude (MDM) among all wavelengths per pixel, and their averaged difference magnitude (ADM). For dielectric surfaces, only ADM values are presented, as their refraction indices are usually not strongly wavelength dependent. And for brevity, we only show part of the results in the paper. Complete results, including detailed parameters, rendering results, difference magnitudes plots, as well as simulated BRDF data for each simulation are provided in Dataset 1 [60].

5.2. Dielectric-dielectric two-layer surfaces, incident angles around Brewster’s angle

As mentioned above, for dielectric surfaces, incident angles around Brewster’s angle cause strong reflective polarisation from the first bounce onwards. This makes them prime targets to establish whether the effect plays any role at all: if we cannot see a noticeable difference here, it is unlikely that any other case would yield measurable results.

In simulation 1, the top layer is transparent with an IOR of η = 1.5, and the bottom layer has an IOR of η = 2.4. Both layer interfaces have roughness parameters of α₁ = α₂ = 0.1, which is fairly smooth. Effectively, this configuration simulates a surface like thinly coated diamond. This is not an exceedingly common kind of surface in reality: but recall that we are first trying to establish whether there are any situations at all where the effect actually matters.

Both interfaces in this layered surface have a relative (!) refraction index of ≈ 1.5, with a Brewster angle of ≈ 56°. As both layers are glossy, we choose several angles in the vicinity of this, namely 50°, 56°, 60° and 65°, as incident angles θ. Figure 10 shows the difference magnitudes of simulation 1, we can see that at the grazing angle side of the forward scattering lobe, there are up to 15% ∼ 23% of difference. The reason that 60° and 65° incident angles yield higher differences can be attributed to the fact that, after being refracted at top layer, the incident angle at the bottom layer is closer to Brewster’s angle there. The rendering results from polarising and non-polarising simulations with incident angle θ = 65° are shown in Fig. 11. It can be observed that the non-polarising simulation image on the right has a brighter and larger edge than the one from the polarising simulation on the left. With a maximum delta of 23% and noticeable visual differences, we have now established that there are at least some scenarios where inclusion of polarisation is necessary to achieve reasonable levels of accuracy.

 

Fig. 10 The difference magnitudes(ADM) of the BRDF lobe (from top-down viewing point at the surface) of simulation 1 (top layer: η₁ = 1.5, α₁ = 0.1, bottom layer: η₂ = 2.4, α₂ = 0.1). 15% ∼ 23% of difference can be observed for this double layered dielectric surface. High differences mainly lie at the grazing angle of forward scattering area. Specifically, subfigure (d) corresponds to the difference between Fig. 11(a) and Fig. 11(b).

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Fig. 11 Polarising (left) and non-polarising (right) rendering results of surface in simulation 1 (top layer: η₁ = 1.5, α₁ = 0.1, bottom layer: η₂ = 2.4, α₂ = 0.1) with incident angle θ = 65°, which can be considered as top-down view of the surface illuminated by directional point light source. The right image is larger than the left one, which show mis-estimations of non-polarising simulations. A more direct visual comparison can be seen in Visualization 1. Rendering results of rest simulations are in Dataset 1 [60].

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We also investigated three other dielectric-dielectric surface combinations (top surface first, bottom second): smooth-rough, rough-smooth and rough-rough. These are labeled simulations 2, 3 and 4, and the results are summarised in Table 3. The first column of the table contains the simulation numbers, the second and third columns list the material and roughness parameters of each layer in each simulation. The last column provides the highest ADM values observed in each simulation. From Table 3 and the figures in Dataset 1 [60], it can be observed that simulation 2 yields around 16% maximum difference and still noticeable visual differences, while simulation 3 and 4 have around 7% ∼ 8% differences and weak visual differences. For more direct visual comparisons on simulation 1 and 2, please refer to Visualization 1 and Visualization 2.

Table 3. Simulations summary on difference magnitudes of double layered dielectric surfaces with Brewster incident angles in Section 5.2.

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When comparing the results of simulation 2 with those from simulation 1, one can see that an increasing bottom layer roughness shrinks the lobe area with the highest differences, and decreases the highest difference magnitude. The differences decrease further as the roughness parameter of top layer increases, which can be explained by the facts that (a) less incident light actually hits micro-facet surfaces near Brewster’s angle (fewer and fewer micro-facets are actually aligned with the average surface orientation), and that (b) high roughness surfaces, due to their increasingly de-polarising nature, tend to increasingly average out any differences in upwelling radiation from lower layers.

5.3. General surface simulations

We then tested more general scenarios: we varied the number of surface layers, the layer materials and the roughness parameters. Also, each surface was illuminated from a wider range of incident angles: θ = 10°, 40°, 60° and 80°. With these simulations, we are able to observe how incident angle, surface roughness and material affect the impact of polarisation on layered surfaces. For brevity, we only discuss two cases here (one single layered simulation in Section 5.3.1, and one double layered one Section 5.3.2), and briefly summarise the rest of the general results in Section 5.3.3.

5.3.1. Single layer surfaces

Here, we investigated single micro-facet layers made of two different materials: glass (η = 1.5) and gold, both with an isotropic roughness parameter of α = 0.25 (labeled as simulation 5 and 6).

In both cases there is practically zero difference between polarising and non-polarising computations for incident angle θ = 10°. Differences appear from incident angle θ = 40° onwards, and become larger as the incident angle increases. Which makes sense, as multi-scattering events hardly ever occur for low incident angles, and only happen at higher angles of incidence. Even so, the highest ADM values we observed were around 1% ∼ 2%, which is essentially negligible. As the effects of polarisation are tiny here, we only include the visualisations in Dataset 1 [60]. Overall, our experiments seem to indicate that inclusion of polarisation computations for multi-scattering events yields marginal differences for single layer surfaces. Which is an interesting result insofar, as a lot of “classic” BRDF models in computer graphics are essentially single layer models: and based on these results, not including polarisation for multi-bounce events can (almost) be ruled out as a source of inaccuracy for them.

5.3.2. Double layered surfaces

The scenario we investigated here was a dielectric layer over a metallic base, which corresponds to e.g. thinly painted metal foil (labeled as simulation 7). The top layer is a transparent dielectric with an OR of η = 1.5, and the lower layer is assumed to be made of gold. Both layer interfaces have a roughness parameter of α₁ = α₂ = 0.1.

Compared to the single layer results, the differences are more pronounced, even for an incident angle of θ = 10°. Particularly noteworthy is that comparatively big differences can be observed even if the top layer is smoother than the single layered surface investigated in Section 5.3.1, where almost no difference was seen. Here, energy transport via refraction and reflection compensates the lack of multi-bounce events on the top layer. Figure 12 shows ADM and MDM plots for incident angles of 60° and 80°: 5% ADM and 8% MDM values can be observed for θ = 60°, 7% and 14% for θ = 80°. Typically, the main specular lobe exhibits low difference magnitudes. This can also be seen in other layered simulations, and is due to the dominance of unproblematic single scattering events that contribute to the main lobe.

 

Fig. 12 Difference magnitudes of simulation 7 (top layer: η₁ = 1.5, α₁ = 0.1, bottom layer: gold, α₂ = 0.1) with θ = 60° and θ = 80°. 5% and 7% of AMD values, 8% and 14% of MDM values can be observed at incident angles of θ = 60° and θ = 80°.

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5.3.3. Other simulations

Apart from the single and double layered surface simulations shown in Section 5.3.1 and Section 5.3.2, we also investigated other double and triple layered surfaces with different materials and roughness parameters. The rendering and difference magnitude plots of these simulations are included in Dataset 1 [60], but the results are summarised as cases 8-16 in Table 4, which has the same arrangement as Table 3, except that for conductive surfaces, MDM values are reported.

Table 4. Summary of ADM and MDM differences for all the simulations discussed in Section 5.3.

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Based on the results shown in Table 4, we can make the following observations:

5.4. Summary and conclusion

Based on the previous sections, it can be summarised that inclusion of polarisation in multi-bounce simulations of micro-facet surface reflectance is (a) much more important for multi-layer surfaces than single layer surfaces, (b) is needed for both smooth and rough layers in multi-layer scenarios, and (c) exhibits wavelength dependency for materials with non-uniform IOR.

A more in-depth analysis can be based on the properties of Fresnel reflectance, which is the basis of micro-facet BRDF models that are theoretically susceptible to polarisation issues. A more detailed breakdown of how incident angle, surface material and roughness parameter affect the importance of including polarisation in multi-bounce BRDF simulations is as follows:

Incident angle: generally, certain incident angle ranges generate a higher reflective DOP due to reflection and refraction, which can lead to the next scattering event being considerably mis-estimated by a non-polarising simulation. Susceptible angles usually range from Brewster’s angle of the lower layer (!) to an angle somewhat larger than Brewster’s angle of the top layer. This spread is mainly due to refraction on upper layers: (a) for common dielectric IORs of 1 to 2.5, a larger incident angle introduces slightly stronger polarisation in refracted light, and (b) after refraction, the incident angle at lower layer is usually closer to the Brewster angle of that surface.

Surface material: the differences between polarised and non-polarising simulations are strongly dependent on the basic material class of the micro-facets. Generally, dielectric facets introduce more differences than metallic facets. The potential effect of metals can be quantised: a higher ratio of η/k induces stronger reflective polarisation. For shrinking k, the ratio goes towards infinity (which corresponds to dielectric behaviour, i.e. total reflective polarisation at Brewster’s angle). But any metal with an η/k ratio higher than 2 is already capable of generating highly polarised reflections (DOP ⩾ 0.8) at some incident angles, so that a simplistic assumption of “all rough metals are depolarisers, and need not be treated with polarisation enabled” would be misleading.

Roughness parameter: the influence of the roughness parameter is more complex. Generally speaking, rougher layers should introduce larger differences between polarising and non-polarising simulations, due to the higher number of multi-scattering events that occur on rougher surfaces. However, due to reflectance from single scattering events being dominant in practice, the actual differences turn out to be not so pronounced. Theoretically, the basic material type should have an influence as well: metallic materials generate weak reflective polarisation to begin with, so that a rough metallic layer should always act more or less as a depolariser. Dielectric layers with high roughness could conceivably behave differently, though. But any large difference caused by one rough dielectric layer can only be maintained if transmitted light from this layer does not subsequently interact with another highly rough layer.

Generally speaking, one can base the decision whether a given layered surface needs to be evaluated with polarisation enabled on its material properties: specifically, η/k ratio together with its roughness parameters. Inclusion of polarisation is necessary for layered smooth surfaces with a high η/k ratio value, like dielectric layered smooth surfaces. But it is not necessary for some metallic rough layered surfaces, such as coated aluminium and silver, which possess η/k ratios much less than 1 in the whole range of visible wavelengths.

6. Limitations and future work

For the validation against measured data, we use a procedurally modelled sandpaper-like surface instead of a directly measured geometrical model of real sandpaper structure. The only reason for this was the inability of current measurement equipment to directly yield useable geometry data for such micro-structures. Conceivably, structural differences between the procedurally modelled surface and real sandpaper geometry could be seen as a source of error: but both the visual similarity of the structures, and the qualitative match of the simulation results, seem to indicate that this is not a major concern. However, once technology becomes available that is capable of directly extracting mesh data for micro-structures like sandpaper, the experiment should be repeated with actual surface mesh data, just to make sure.

As future work, we plan to modify an analytical micro-facet model to include the features observed in our simulation datasets. Furthermore, we plan to verify our simulation against existing BRDF models, and to possibly include a simulation of diffraction in our brute force tracer.