##
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.