Overview

In this paper published in 2001, the main aim is to enable efficient simulation of effects that BRDF models cannot capture even in highly scattering media that was expensive to simulate with previous methods. It includes an exact solution for calculating single scattering, a diffusion approximation for multiple scattering as well as an image based measurement technique for determining the optical properties of translucent materials.

Computing subsurface scattering consists of not only implementing an accurate light transport algorithm but also ensuring that a suitable local scattering model is implemented. BRDF and BSSRDF have been used to approximate local scattering in a great deal of research. Jensen et al. (2001) describe how in BRDF models, any subsurface scattering is approximated by a Lambertian component which assumes that light scatters at one surface point without taking into account subsurface transport from one point to another. BRDF has been used widely to approximate subsurface scattering due to it being computationally cheap as compared with solving the full radiative transfer equation which is very slow to compute.

Theory

This paper describes a BSSRDF model which is the sum of a diffusion approximation and a single scattering term developed by the authors. Both the diffusion approximation and the single scattering term are derived from BSSRDF and BRDF theory. The elements of theory discussed in this paper are important in understanding the approximation techniques that follow. In particular:

Outgoing Radiance

The outgoing luminance is a calculation of the luminance at the point where light is exiting the material. This is how we relate this to the BSSRDF equation:

Remember the BSSRDF equation is:

fr(x, w, x’, w’) = dLr(x,w)/d Fi(x’,w’)

If we change x and w to xi and wi to show they represent incoming light (radiance) and x’ and w’ to xo and wo to show they represent outgoing light (irradiance) we have:

fr(xi, wi, xo, wo) = dLr(xi,wi)/d Fi(xo,wo)

The outgoing radiance can be defined as:

Lr(xo,wo)

Obviously this value is important as it describes the amount of light emitted from point xo as a result of light entering from point xi and scattering. Mathematically this is calculated by integrating the incident radiance over incoming directions (360 degrees) and the area of the point xi.

Actual light propagation (the process by which light is transmitted through the medium) within the material is described by the radiative transport equation. The equation is included in the paper but for the purposes of this document I will simply outline the important properties that describe it:

• The scattering coefficient: a measure of the extinction of light due to scattering
• The absorption coefficient: a measure of extinction of light due to absorption
• The extinction coefficient: simply the scattering coefficient plus the absorption coefficient
• The phase function: the phase function is a property of the medium and describes the dependence of scattered radiance on the scattering angle. Essentially, it specifies the spherical distribution of the scattered light (Jensen and Buhler 2002, p. 577).

To determine whether the phase function is backward scattering, forward scattering or isotropic you take the mean cosine of the scattering angle.

The reduced intensity describes how the radiance of infinitesimal beam entering a homogenous medium decreases exponentially over distance in the implementation discussed in Case Study 1: Finding Nemo, this is approximated with Renderman's smoothstep function.

Diffusion Approximation

The diffusion approximation is how Jensen et al. (2001) approximate multiple scattering. They describe how multiple scattering events blur light distribution, resulting in light distribution that is relatively uniform. Clearly, relatively isotropic light distribution can help reduce the complexity of diffusion calculations by removing the need to calculate more complex anisotropic distribution.

The process of the diffusion approximation described by Jensen et al. is very complex and as such, I will attempt to describe the method in as simple terms as possible rather than listing all the mathematical calculations involved.

Radiance L(x, w) is approximated by a two-term expansion involving scalar irradiance and vector irradiance. Scalar irradiance is the irradiance flux (raw stream of photons) at point x and is defined as the integral of radiance at point x from direction w (L(x, w)) by the differential solid angle (dw). Vector irradiance represents the direction of irradiance from point x and is defined as the integral of radiance at point x from direction w (L(x, w)) by the direction of radiance (w) by the differential solid angle (dw). Scalar irradiance and vector irradiance are yielded as part of the result of integrating the radiative transport equation over all directions w at point x.

The diffusion approximation is then calculated by substituting the radiance approximation just discussed into the radiative transport equation and then integrating this for all directions w. This equation is then substituted into the result of integral of the radiative transport equation for all directions w at point x.

Sounds confusing? Essentially what they are doing is simplifying the radiative transport equation by assuming subsurface light transport is relatively isotropic. This is achieved by simplifying the calculations for radiance at point x in direction w and substituting the result into the radiative transport equation and then integrating it. This equation is then substituted into the integral of the standard radiative transport equation (which describes the volumetric behaviour of light propagation) to arrive at the classic diffusion equation.

This is not the end of the diffusion approximation however. The paper then discusses how to take into account scattering medium in a finite region of space which involves solving the diffusion equation subject to the appropriate boundary conditions as well as taking into account the fact that a medium may have layers that have different refractive indices (a complex procedure but one that is not necessary for modern day skin rendering).

Single Scattering Term

The single scattering term described in this paper is based on extending the BRDF to a BSSRDF to account for local variations in lighting over the surface.

Final BSSRDF Model