##### Content of review 1, reviewed on October 06, 2013

## Significance Comment

The application of singularity-removing changes of variable to the extended-source integral of diffusion BSSRDFs is quite relevant and useful and will likely appear in future work as well. As such, this will apply to many rendering algorithms and likely see widespread use.

The modified method-of-images kappa term is helpful, but not likely the final answer on compact, approximate searchlight solutions for isotropic scattering.

The diffuse single-scattering term seems like a rather dubious introduction. One of the nice things about Grosjean's modified diffusion theory is that it very accurately decouples from single-scattering. It is important to do this because single-scattering is a highly angular signal that is not well described by diffusion theory. So it seems weird to then go and put it back in a completely diffused form (despite the typical application of these BSSRDFs to volumes with rough interfaces).

## Quality Comment

The details of the method are presented clearly and completely with many nice examples and figures. However, I think the quality of this paper could have been higher if the authors had:

a) considered the reciprocity of their new oblique BSSRDF (or lack thereof) and taken care to measure the angular behaviour of their new formulation: BSSRDFs hold an important place in the level-of-detail chain sitting between explicit structure and the BRDF, and it is quite useful for all three of these to accurately align. It seems unclear to me that graphics papers like this one need images anymore. Carefully presented plots do much more to convince the reader of the accuracy of the proposed transport theory approximations. Showing selected results where a half-space searchlight solution is applied approximately to curved geometry, while pretty, does little to convince the reader of the method's overall robustness. Under uniform illumination a BSSRDF will need to act mostly like a BRDF. The authors tote the angular flexibiliy of their method but provide very little validation of the angular dimension of their solutions.

b) taken more care to clearly position their work relative to previous work. The method is presented as a continuous beam interpretation of previous related techniques. However, the Quantized diffusion method (publon:2802) does exactly this as well. The difference lies in the quadrature rule used to evaluate the beam/track-length integral. The equiangular change of variable used by this paper was also used in 1984 for "neutron-beams" (publon:2898). I mentioned this to one of the authors in July 2012 after the presentation of publon:2900. What is new about applying a beam-integral of a diffusion BSSRDF is that the Green's function has a $\frac{1}{r}$ singularity instead of a $\frac{1}{r^2}$ singularity. This leads to a change of variable involving sinh and not the equiangular sampling. It would have been nice to see this explored or at least mentioned.

c) The label of 'Monte Carlo' seems imappropriate given that the final form (after a change of variable) for the 1D integral in question is to simply do a 5-sample Riemann sum. Positioning the key problem as a one-dimensional integral of a positive smooth function would immediately connect the reader to a relevant body of work on quadrature schemes and other low-order integration methods. We note that a low-order Gaussian quadrature applied after the equiangular change of variables further increases the accuracy of Photon Beam Diffuion for free. The large body of highly related work on track-length estimators could be expanded: most readers in graphics seem unaware of this connection to photon beams. The authors describe Quantized Diffusion with "complicated and error-prone implementation, numerical instability, and high computational complexity". However, after paying a one-time implementation effort to precompute 128 sets of weights and variances (per index of refraction) to describe media of various absorption levels, the use of Quantized diffusion is simply a look-up table and evaluation of Gaussians. Despite needing 45 Gaussians to accurate describe a profile, they describe it everywhere on the surface at any desired resolution. These weights can be computed without numerical difficulty (see publon:2815).

d) The authors equate the 'diffuse mean free path' with the inverse of the reduced transport coefficient when discussing albedo mapping. However, this term is typically used to describe a ficticious mean-free path by looking at the asymptotics of the diffusion Green's function [Jensen and Buhler 2002].

e) Missing related work: the exact BSSRDF for a half-space with smooth Fresnel interfaces has been recently solved exactly for any anisotropic phase function: publon:2901, publon:2902, publon:2903. These solutions are quite complex, but recent versions are becoming much more tractable.

##### Source

© 2013 the Reviewer (CC BY-SA 3.0).