Approximate Reflectance Profiles for Efficient Subsurface Scattering

Per H. Christensen and Brent Burley

Abstract: We present three useful parameterizations of a BSSRDF model based on empirical reflectance profiles. The model is very simple, but with the appropriate parameterization it matches brute-force Monte Carlo references better than state-of-the-art physically-based models (quantized diffusion and photon beam diffusion) for many common materials. Each reflectance profile is a sum of two exponentials where the height and width of the exponentials depend on the surface albedo and mean free path length. Our parameterizations allow direct comparison with physically-based diffusion models using the same parameters. The parameterizations are determined for perpendicular illumination, for diffuse surface transmission (where the illumination direction is irrelevant), and for an alternative measure of scattering distance. Our approximations are useful for rendering ray-traced and point-based subsurface scattering.

One-line summary: Simple BSSRDF models for efficient subsurface scattering.

Published as: Pixar Technical Memo #15-04. Pixar, July 2015.

Download the tech memo here: paper.pdf.

SIGGRAPH 2015 talk here: sig15talk.

Update (Sept 2018): In section 6 of the tech memo we wrote: "Unfortunately the cdf is not analytically invertible, ...". But a reader derived the inverse function of the cdf (with help from Mathematica) and kindly shared his result.
The cdf (Eq. 11) is: cdf(r) = 1 - 1/4 e^{-r/d} - 3/4 e^{-r/(3d)} .
The inverse cdf is: invcdf(u) = 3 d ln( (1 + 1/h(u) + h(u)) / (4 (1-u)) ) with h(u) = (9 - 16u + 8u^2 + 4(1-u)\sqrt{5 - 8u + 4u^2})^(1/3) .

Update (Dec 2019): Evgenii Golubev found a (slightly) simpler form for the inverse cdf using the substitution u' = 1-u (which is of course also a valid inverse cdf). See his nice blog post here.

Back to Per's publication page.