Energy Conserving BSDF Under a Smooth Refractive Interface

This is a part of the blog post series Rendering Layered Materials.

Here I explain how to obtain an energy-conserving BSDF for a given inner BSDF seen and illuminated through an outer smooth refractive interface. The model uses the single-scattering-event simplification, which means that only light paths which bounce from the inner layer exactly once are taken into account and light paths which get reflected multiple times between the smooth interface and the given BSDF are neglected. The model also uses the single-point simplification, which assumes that the point through which the light ray enters the outer material interface is the same as the one through which the light ray exists.

In this part I will first define a few basic radiometric quantities and BSDF and then I will derive the BSDF for a model under a smooth refractive interface. For the sake of clarity, this explanation neglects the reflection contribution of the smooth interface as it is trivial to add to the resulting model.

The radiometry and BSDF explanation is heavily inspired by the legendary Eric Veach’s PhD thesis (it’s more of a book than a PhD dissertation ;-)), which I recommend to read for more detailed explanation and also to everyone who is seriously interested in computer graphics, especially in realistic 3D rendering.

Main idea

One way of obtaining the correct form of a BSDF under a smooth refractive interface its to re-formulate the inner layer BSDF as a function of (non-refracted) incoming and outgoing directions of the whole model $\omega_{i}$ and $\omega_{o}$ rather than as a function of refracted directions $\omega_{i}^{\prime}$ and $\omega_{o}^{\prime}$:

where $f_{s1}$ is the outer smooth interface, $f_{s2}$ is the inner BSDF, $\eta_0$ and $\eta_1$ are the refractive indices of media separated by the outer smooth interface, and $N_g$ is the geometrical normal of the surface.

In other words, we want to express how the layer behaves to the rendered which doesn’t know about the internal mechanics of the model and regards it as a black box.

Radiometric quantities

To handle the problem properly we can grasp it through to the fundamental concepts upon which the model is built. First, we will define a few radiometric quantities needed to define BSDF. There are several ways of doing that, but I will follow the measure-theoretic radiometry way, which defines its quantities as ratios of measures and is explained in more detail for example in the third chapter of the Eric Veach’s dissertation thesis. This approach uses the so-called Radon-Nikodym derivative, which expresses density of one measure with respect to another one. For example the Radon-Nikodym derivative of measure $Q$ with respect to measure $\rho$ is denoted $\frac{\mathrm{d}Q}{\mathrm{d}\rho}$ (similarly to the “normal” derivative).

In the measure-theoretic radiometry case, we usually express the density of electro-magnetic energy measure with respect to some space and/or time measure. The electro-magnetic energy measure directly defines the most basic radiometric quantity, the radiant energy

$$Q \quad \left[J\right]$$

which can be understood as the amount of energy of photons in a given sub-set of space and time. The density of radiant energy per unit of time is radiant flux or radiant power

$$\Phi\left(t\right) = \frac{\mathrm{d}Q\left(t\right)}{\mathrm{d}t} \quad \left[Js^{-1}=W\right]$$

where $t$ is the time parameter and also the time measure. In computer graphics we are very often interested in “amount of light” which arrives at, gets scattered or is emitted from a certain surface point $x$. This can be expressed as density of radiant power per unit of surface area and is called irradiance for incident radiation and radiant exitance or radiosity for scattered and emitted radiation

$$E\left(x\right) = \frac{\mathrm{d}\Phi\left(x\right)}{\mathrm{d}A\left(x\right)} \quad \left[Wm^{-2}\right]$$

where $A$ is the surface area measure. While irradiance measures density of energy incoming to a point of surface from all directions, we often need to measure the amount of light entering or leaving at a point in a single direction. This is what quantity called radiance is for. Formally, it is expressed as the density of radiant power per unit solid angle and unit surface area of a surface perpendicular to the direction. There are several equivalent formulations for radiance, but the most practical for our purposes is

$$L\left(x,\omega\right) = \frac{\mathrm{d}^{2}\Phi\left(x,\omega\right)}{\mathrm{d}A\left(x\right) \mathrm{d}\sigma^{\bot}\left(\omega\right)} \quad \left[Wm^{-2}sr^{-1}\right]$$

Note that this definition uses projected solid angle measure $\sigma^{\bot}$ rather than “normal” solid angle measure $\sigma$ in order to make the quantity independent from relative position of the light direction $\omega$ and the surface normal. Practically, it just adds a simple cosine factor $\cos\left(\theta\right)$ to the solid angle measure $\sigma$, which, in this case, compensates the reduction of area density when direction $\omega$ inclines away from the normal.

Bidirectional scattering density function

Now that we have defined the needed radiometry quantities, we can finally define the bidirectional scattering density function (BSDF), a formal description of the light-scattering properties of a surface point. It expresses how the light outgoing from a point $x$ on a surface in a particular direction $\omega_o$ is dependent on the light incoming to the point from a particular direction $\omega_i$. The surface point $x$ will be omitted from the notation in the following text for clarity.

BSDF geometry.

We denote $L_o\left(\omega_o\right)$ the radiance leaving the point $x$ in the direction $\omega_o$. In general, it is dependent on the radiance incoming to $x$ from all possible directions (from both above or below the surface). Let’s now consider the radiance $L_i\left(\omega_i\right)$ incoming to $x$ through an infinitesimal cone $\mathrm{d}\omega_{i}$ around the direction $\omega_i$ with solid angle size $\mathrm{d}\sigma\left(\omega_{i}\right)$. This generates irradiance denoted $\mathrm{d}E\left(\omega_{i}\right)$ on the surface, which can be computed as

$$\mathrm{d}E\left(\omega_{i}\right) = L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)$$

The incident light can be partially absorbed by the surface and partially scattered in all directions. Let’s denote $\mathrm{d}L_{o}\left(\omega_{o}\right)$ the out-scattered contribution of $\mathrm{d}E\left(\omega_{i}\right)$ to $L_{o}\left(\omega_{o}\right)$. It can be shown experimentally that amount of outgoing radiance $\mathrm{d}L_{o}\left(\omega_{o}\right)$ is proportional to the incoming irradiance $\mathrm{d}E\left(\omega_{i}\right)$ and the BSDF is then defined simply as the ratio between the two:

$$f_{s}\left(\omega_{i}\rightarrow\omega_{o}\right) = \frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{\mathrm{d}E\left(\omega_{i}\right)} = \frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)} \quad \left[sr^{-1}\right]$$

BSDF under a smooth refractive interface

Using the general definition of BSDF, we’ll now derive the formula for a BSDF under a smooth Fresnel surface. For that we will need to prepare 4 prerequisite formulae, which we will then use to build the final BSDF formula.

First, let’s see what happens to the incoming radiance when it passes a smooth interface between medium 0 and 1 with possibly different indices of refraction $L_{i}\left(\omega_{i}^{\prime}\right) = L_{i}\left(\omega_{i}\right) \frac{\eta_1^2}{\eta_0^2} T\left(\theta_i\right)$

where $\frac{\eta_1^2}{\eta_0^2}$ is the radiance scaling due to solid angle compression (see the Veach’s thesis, formula 5.2, p. 141) and $T\left(\theta_i\right)$ is the Fresnel transmission coefficient (how much light is refracted rather than reflected). For our purposes, it will be handy to rewrite the equation a little bit

$$L_{i}\left(\omega_{i}\right) = L_{i}\left(\omega_{i}^{\prime}\right) \frac{\eta_0^2}{\eta_1^2} T\left(\theta_i\right)^{-1}$$

Similarly, we can express what happens to the outgoing radiance

$$L_{o}\left(\omega_{o}\right) = L_{o}\left(\omega_{o}^{\prime}\right) \frac{\eta_0^2}{\eta_1^2} T\left(\theta_o\right)$$

Now, the factor $\frac{\eta_0^2}{\eta_1^2}$ represents the amount of solid angle de-compression. Note that we simplified the notation by using $T\left(\theta_o\right)$ instead of $T\left(-\theta_o^{\prime}\right)$ because the two are equal.

The third piece of our puzzle is how the projected solid angle $\mathrm{d}\sigma^{\bot}$ of a light cone changes when refracted through smooth interface. The relation between the incident and refracted light cone can be found for example in the Veach’s thesis in the equation 5.4 at page 143 and, after a minor rewrite, we will get

$$\mathrm{d}\sigma^{\bot} \left(\omega_i\right) = \frac{\eta_1^2}{\eta_0^2} \mathrm{d}\sigma^{\bot} \left(\omega_t\right)$$

where $\omega_t$ is the refracted direction of $\omega_i$.

The fourth and the last prerequisite is a little technical and I will not go into detail of this one. The point is that the projected solid angle measure of the refracted cone equals the cone of light reaching the inner layer, given that the both layers are parallel to each other:

$$\mathrm{d}\sigma^{\bot} \left({\omega}_i^\prime\right) = \mathrm{d}\sigma^{\bot} \left({\omega}_t\right)$$

Now we finally have all the necessary pieces to build the BSDF for the a layer under a smooth Fresnel surface. Let’s put them into the plain definition of BSDF, simplify it and separate the stand-alone inner layer BSDF to see what compensation factors have to be applied to get the correct formula:

$$\begin{eqnarray*} f_{s}^{\ast}\left(\omega_{i}\rightarrow\omega_{o}\right)&=&\frac{\mathrm{d}L_{o}\left(\omega_{o}\right)}{L_{i}\left(\omega_{i}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}\right)}\\ &=&\frac{\mathrm{d}L_{o}\left(\omega_{o}^{\prime}\right)\frac{\eta_{0}^{2}}{\eta_{1}^{2}}T\left(\theta_{o}\right)}{L_{i}\left(\omega_{i}^{\prime}\right)\frac{\eta_{0}^{2}}{\eta_{1}^{2}}T\left(\theta_{i}\right)^{-1}\frac{\eta_{1}^{2}}{\eta_{0}^{2}}\mathrm{d}\sigma^{\bot}\left(\omega_{i}^{\prime}\right)}\\ &=&\frac{\mathrm{d}L_{o}\left(\omega_{o}^{\prime}\right)}{L_{i}\left(\omega_{i}^{\prime}\right)\mathrm{d}\sigma^{\bot}\left(\omega_{i}^{\prime}\right)}T\left(\theta_{o}\right)T\left(\theta_{i}\right)\frac{\eta_{0}^{2}}{\eta_{1}^{2}}\\ &=&f_{s}\left(\omega_{i}^{\prime}\rightarrow\omega_{o}^{\prime}\right) T\left(\theta_{o}\right) T\left(\theta_{i}\right) \frac{\eta_{0}^{2}}{\eta_{1}^{2}} \end{eqnarray*}$$

Therefore, the only compensation to the inner layer BSDF we need to do is feeding it with refracted directions $\omega_{i}^{\prime}$ and $\omega_{o}^{\prime}$, and multiplying it with the Fresnel transmission coefficients a the solid angle (de-)compression compensation factor $\frac{\eta_{0}^{2}}{\eta_{1}^{2}}$.