Multiple-Forward-Scattering in Volume Rendering of Participating Media
Natural volumetric media have phase functions which typically are sharply peaked in the forward scattering direction, with backscatter accounting for only a few percent of the total angular redistribution from a single scattering event. This property has been exploited in the past in the small-angle approximation for radiative transfer, successfully for many engineering and science applications. The small-angle approximation also robustly describes the multiple-forward-scattered behavior of the light field many scattering lengths into the participating medium, including the asymptotic regime, in agreement with experimental measurements and computationally intensive simulations. Physically, the important missing ingredient not found in the small-angle approximation is occasional large angle scatters that reverse the propagation direction of some of the light. This paper introduces a quantitative model of multiple scattering which contains both the multiple-forward-scatter character and a few large-angle scattering events. The model is derived directly from the Green's function representation of radiative transfer, and path integrals are used to construct the appropriate form of the small angle approximation. The model is suitable for media that have internal structure.