Multiple-Scattering Microfacet BSDFs with the Smith Model

In this paper, we address an important open problem in material modeling. What happens when light scatters multiple times on rough material surfaces? In image synthesis, accurately describing light matter interactions is important for materials to obtain a realistic look. However, the multiple light matter interactions that we can see in this figure are absent from many surface appearance models. Here's an example of this Rendering white glass should be simple, but we can see that the rougher the glass is, the darker its appearance becomes. Even though it should be simple, modeling the appearance of glass that is at the same time rough and white is almost impossible with the current material models. Many material models, such as those rough dielectric plates, are called micro facet materials because the underlying mathematical model assumes that their interfaces are made of microscopic imperfections that we call facets. Those facets are too small to be visible, but the way they are statistically oriented change the way light interacts with the material, causing its rough appearance. Many rendering systems model only the contribution of the first bounce of light. The contribution of multiple bounces is unknown and it is simply set to zero, as if it were neglectable. However, on very rough microsurfaces, the amount of light that scatters multiple times is significant and should not be neglected to avoid energy loss and the noticeable darkening of the material appearance. In summary, modeling rough materials correctly with multiple scattering is a challenging problem. Our multiple scattering model presented in this paper opens up the possibility of modeling rough materials correctly in a practical manner. Beyond fixing the darkening problem, our goal is to derive a physically based model that is able to make accurate predictions compared to reference data. More specifically, we derive the multiple scattering component of a specific kind of microsurface, the Smith microsurface model. Because it is based on simple assumptions and makes accurate predictions for single scattering, it has received widespread industrial adoption and is considered the academic state of the art in computer graphics for modeling many materials. But can we extend this model for multiple scattering and could it be practically incorporated into a classic BSDF plugin? These are the questions we are interested in. Our main insight is to transform this surface scattering problem into a volume scattering problem, which is easier to solve. To achieve that, we show that the Smith microsurface model can be derived as a special case of the microflake theory for volumes. We reformulate the Smith microsurface as a volume with additional constraints to enforce the presence of a sharp interface. This volumetric analogy is very convenient because we know how to compute the light scattering in volumes. It depends on two functions that we derive for this new kind of volume. The first one is the Free path distribution, which tells us how long a ray can travel in a medium before finding an intersection on the micro surface. The equivalent question is what is the height of the next intersection? Once an intersection is found, we need to know in which direction the light is scattering. Again, this is given by a volumetric phase function which depends on both the base material of the surface and the distribution of the micro facets. We derived the phase function for three different surface diffuse, conductive and dielectric and common microfacet distributions such as Beckman and ggx. Now that we know the free path and the phase function of this volumetric model, we know exactly how the light scatters in the medium. From the light propagated in this medium emerges a distribution that has all the expected properties of a classic surface bsdf. It is energy conserving and reciprocal. Furthermore, it is exactly the classic single scattering BSDF based on the Smith microsurface model, but with the addition of higher order scattering. Now that we know that the model is mathematically correct, we are interested in its predictive power. How accurate is this new model? To answer this question, we need some reference data to compare the predictions of the model to. A common way to validate models is to compare their predictions to simulated data obtained by ray tracing triangulated surfaces. Contrary to real world acquisition, the surface used in the simulation has known material and statistics and the collected data are free of noise. There is thus no degrees of freedom left to match the parameters of the model to the simulation. This is why this validation procedure is widely used in the field of optical physics and therefore we chose this to validate our model. We generated random surfaces with known backman statistics and did a ray tracing simulation on them. By comparing the predictions of our multiple scattering model to the results of the ray tracing simulation, we found our BSDF model to accurately predict both the albedo and angular distribution of the accident energy among the scattering orders, and this for a large variety of materials, roughnesses, anisotropy and inclinations. In our supplemental material, we provide an exhaustive set of such validation results. To make the model practical, we implement two procedures, evaluation and importance sampling. Since the BSDF is the expectation of all the paths that can be traced on the microsurface, importance sampling can be done straightforwardly. By generating one path, we construct an unbiased stochastic estimate by tracing one path and evaluating the phase functions at each intersection. With next event estimation, as in classical path tracing, with important sampling and this stochastic evaluation, we have everything required to implement a classic BSDF plugin. Furthermore, our implementation is analytic and does not use per BSDF precomputed Data, which makes RBSDFs usable with textured albedos, roughness and anisotropy in the supplemental materials, we provide a document describing a tutorial implementation for various materials and ready to use plugins for the Mitsuba physically based renderer. Now let's have a look at some results. This image shows a collection of bottles with micro facet materials. The energy loss is significant if multiple scattering is neglected, especially on dielectrics. Without multiple scattering, ROF transmittance appears unnatural, which is hard to compensate for by tuning parameters. With our multiple scattering model, we simulate the expected appearance of rough glass and metals without tuning any parameters. Our model is robust and behaves as expected even with high roughness values. We can see that the model avoids the darkening effects and even produces interesting emerging effects like color saturation. This can be observed on this rough diffuse material since the absorption spectrum of the material is repeatedly multiplied after each bounce on the microsurface. The reflected color appears more saturated after multiple bounces. This emerging effect can also be seen on this gold conductor material. The unsaturated single scattering gold conductor appears strangely dull thanks to our model. The introduction of multiple scattering restores the shiny appearance expected from gold. Note that since our model is parametric and does not depend on any pre computed data, we fully support textured input which is important for creating visually rich images. As an example, this is a dielectric with textured roughness and anisotropy. Thanks for watching.