Consistent Monte Carlo methods for non-linear applications in light transport

The study of light transport focuses on describing the propagation of light from emitters to sensors through accurately describing the interactions light can undergo with everything in between. Physically-based rendering is the process of applying the laws of light transport to formulate practical algorithms which simulate the flow of light for the purpose of synthesizing images of virtual environments. Unfortunately, there are very few interesting scene configurations which can be computed analytically. Instead, modern solutions predominantly rely on Monte Carlo integration to stochastically estimate the transfer of light since the process is both unbiased and consistent. Meaning, it is expected to give us the correct result, and if we took the average of infinitely many estimates we would obtain the correct answer. However, the ideal properties of Monte Carlo integration are not guaranteed for all problems. In the presence of non-linear perturbations, Monte Carlo integration will almost always result in incorrect solutions. Unfortunately, applications with non-linear perturbations appear throughout light transport from simulating the interactions between light and participating media, to rendering complicated visual phenomena like caustics, to even modern applications in differentiable rendering. In this dissertation we focus on formulating consistent solutions for general non-linear problems within light transport. We achieve this by devising a general recipe outlining for both scientists and practitioners alike how to address and account for the many difficulties which can arise when estimating these difficult problems. This involves discussing various ways of reconstructing non-linear problems into alternative forms which are amenable to unbiased Monte Carlo integration. We then review ways of verifying that these unbiased estimators are efficient enough to be employed in practice. When this is not the case, we review and introduce methods for constructing biased, but still consistent, solutions. Through the application of our recipe, we introduce the first general unbiased rendering algorithm capable of rendering non-classical participating media, the first truly unbiased photon-mapping algorithm for rendering razor sharp caustics, and various consistent techniques for efficiently rendering participating media comprised of unknown black box densities