The spherical sampling of the incident radiance function entails a high computational cost. Therefore the llumination integral must be evaluated using a limited set of samples. Such a restriction raises the question of how to obtain the most accurate approximation possible with such a limited set of samples. In this thesis, we show that existing Monte Carlo-based approaches can be improved by fully exploiting the information available which is later used for careful samples placement and weighting.The first contribution of this thesis is a strategy for producing high quality Quasi-Monte Carlo (QMC) sampling patterns for spherical integration by resorting to spherical Fibonacci point sets. We show that these patterns, when applied to the rendering integral, are very simple to generate and consistently outperform existing approaches. Furthermore, we introduce theoretical aspects on QMC spherical integration that, to our knowledge, have never been used in the graphics community, such as spherical cap discrepancy and point set spherical energy. These metrics allow assessing the quality of a spherical points set for a QMC estimate of a spherical integral.In the next part of the thesis, we propose a new heoretical framework for computing the Bayesian Monte Carlo quadrature rule. Our contribution includes a novel method of quadrature computation based on spherical Gaussian functions that can be generalized to a broad class of BRDFs (any BRDF which can be approximated sum of one or more spherical Gaussian functions) and potentially to other rendering applications. We account for the BRDF sharpness by using a new computation method for the prior mean function. Lastly, we propose a fast hyperparameters evaluation method that avoids the learning step.Our last contribution is the application of BMC with an adaptive approach for evaluating the illumination integral. The idea is to compute a first BMC estimate (using a first sample set) and, if the quality criterion is not met, directly inject the result as prior knowledge on a new estimate (using another sample set). The new estimate refines the previous estimate using a new set of samples, and the process is repeated until a satisfying result is achieved.
