Padintegralen voor een voorwaartsverstrooiend dipoolmodel

Rendering translucent materials with physically based Monte Carlo methods tends to be computationally expensive due to the long chains of volumetric scattering interactions. In the case of strongly forward scattering media, the problem gets compounded as those materials are typically densely scattering and each scattering interaction becomes highly directional and near-specular. Existing subsurface scattering models try to avoid explicitly sampling the medium by leveraging analytical results, often based on the diffusion approximation with first order similarity theory to implicitly handle scattering anisotropy. We show that these underlying approximations are fundamentally unable to reproduce the behaviour of finite forward scattering media — even in the optically dense limit — as the existence of a boundary surface allows undiffused incident light paths to escape, especially under grazing illumination. These incorrectly handled paths are precisely the ones that are crucial for reproducing the characteristic appearance of forward scattering media. In order to construct a subsurface scattering model that directly assumes forward scattering, we instead turn to the functional integral formalism for radiative transport, which has not yet received wide attention in the computer graphics literature. We provide an exhaustive and fully self-contained derivation of this formalism starting from first principles all the way to our analytical approximation of the infinite medium forward scattering propagator or Green's function, which takes the form of a time-resolved Gaussian weight on the incoming and outgoing positions and directions. Apart from an approximate handling of the path length constraint within the functional integral, the only other approximation at this point is the assumption of strong forward scattering, i.e. the limit of the mean scattering cosine g → 1. In that sense, our model is complementary to the diffusion approximation which assumes isotropic scattering (g → 0). Contrary to the diffusion approximation, no assumptions on path length are made. In particular, the short length limit stays meaningful, producing unscattered Dirac delta-like behaviour as expected. Our final subsurface scattering model uses this analytical infinite-medium monopole approximation within a dipole model to approximate the finite-medium boundary conditions. Here, the BSDF of the boundary is explicitly taken into account. In particular this means handling internal reflections on index-mismatched boundaries which are required to reproduce the coherent small-angle scattering effects in forward scattering media, as well as obtaining correctly saturated colours due to absorption-based colour filtering of long reflected paths within the medium. Overall, our model captures the full behaviour of forward scattering media, ranging from unscattered straight-line propagation to the fully diffusive limit. In particular, we do not require a separately computed unscattered or single-scattering contribution, in contrast to pure diffusion-based models, nor do we require ad hoc regularisations in the optically thin limit of zero scattering. Instead, we have a single singularity by design from the underlying transport laws corresponding to the initial unscattered propagation, which can be accounted for by the extensive analytical importance sampling scheme that we provide. Furthermore, our model has an analytical effective BRDF as a cheap approximation for near-planar geometry under low frequency lighting. Lastly, we identify several avenues for future work and we hope that we can bring a renewed interest into the application of functional integral methods within computer graphics.