  We consider the problem of multiple scattering on Smith microfacets. This
problem is equivalent to computing volumetric light transport in a homogeneous
slab. Although the symmetry of the slab allows for significant simplification,
fully analytic solutions are scarce and not general enough for most
applications. Standard Monte Carlo simulation, although general, is expensive
and leads to variance that must be dealt with.
  We present the first unbiased, truly position-free path integral for
evaluating the BSDF of a homogeneous slab. We collapse the spatially-1D path
integral of previous works to a position-free form using an analytical
preintegration of collision distances. Evaluation of the resulting path
integral, which now contains only directions, reduces to simple recursive
manipulation of exponential distributions. Applying Monte Carlo to solve the
reduced integration problem leads to lower variance.
  Our new algorithm allows us to render multiple scattering on Smith
microfacets with less variance than prior work, and, in the case of conductors,
to do so without any bias. Additionally, our algorithm can also be used to
accelerate the rendering of BSDFs containing volumetrically scattering layers,
at reduced variance compared to standard Monte Carlo integration.
