Path Tracing Estimators for Refractive Radiative Transfer

Light transport in heterogeneous refractive media is described using the refractive radiative transfer equation (RRTE) that, in addition to light bending due to continuous refraction, also models effects due to volumetric and surface scattering. The light bending effects make this equation significantly more challenging to simulate than its counterpart for homogeneous refractive media, the radiative transfer equation. Existing rendering algorithms are based on photon mapping techniques; these algorithms are efficient but biased, and can introduce significant artifacts in the output images. By contrast, unbiased algorithms such as path tracing, particle tracing, and bidirectional path tracing, are inefficient or even completely intractable for rendering refractive radiative transfer.

The main challenge in applying these unbiased algorithms to heterogeneous refractive media is the difficulty of performing direct connections between two points inside such a medium. Direct connections are needed, e.g., for next event estimation in path tracing and particle tracing, and to connect the source and sensor subpaths in bidirectional path tracing. Connecting two points inside a homogeneous refractive medium is trivial---one only needs to trace the linear segment between the two points. By contrast, connecting two points inside a heterogeneous refractive medium requires finding paths, generally curved, starting and ending at the two points that are solutions of the eikonal equation. This equation accounts for the variable (inversely proportional to refractive index) speed of light inside heterogeneous refractive media. The solutions to the eikonal equation are paths satisfying Fermat's principle, in that they are locally stationary with respect to the time it takes light to traverse them or, equivalently, with respect to optical pathlength.

In this paper, we address this challenge by developing techniques for computing these stationary paths, efficiently and without bias. Our technique is based on the fact that the eikonal equation, and the ray tracing equations derived from it, are differentiable. Therefore, we show that we can use use efficient gradient-based optimization algorithms to search for paths that connect two points, while also satisfying the eikonal equation. Special attention is required to account for the fact that, in heterogeneous refractive media, there may exist more than one stationary paths connecting two points: our optimization-based technique in its simple form would only find one of these paths, resulting in bias, and is not practical for enumerating all stationary paths. To ensure unbiasedness and maintain efficiency, we introduce a Monte Carlo technique that forms an unbiased estimate of the total throughput through multiple stationary paths, through repeated random reinitializations of our gradient-based optimization algorithm. To further improve efficiency, we show how to accelerate direct connections using techniques similar to sphere tracing of signed distance functions, and inside-outside tests based on fast winding numbers.