Extended path integral formulation for volumetric transport

We propose an extension of the path integral formulation amenable to the expression of volumetric light transport with photon beam estimates. Our main contribution is a generalization of Hachisuka et al.'s extended path space formulation [Hachisuka et al. 2012] to light transport in participating media. Our formulation supports various point- and beam-based volumetric density estimators, unifying them with path integration in the spirit of the work by Křivánek et al. [2014]. One unique and useful property of our formulation is that it recasts beam-based density estimation as Monte Carlo path vertex sampling in a higher-dimensional space, rather than beam merging in a lower-dimensional space, which enables a practical algorithm for beam estimators with 3D-blur kernels. We thus establish a complementary theoretical foundation for the development of rendering algorithms using points, beams, and paths in participating media.