Jan Novák

Thomas Müller, Fabrice Rousselle, Jan Novák et al.

We propose neural control variates (NCV) for unbiased variance reduction in parametric Monte Carlo integration. So far, the core challenge of applying the method of control variates has been finding a good approximation of the integrand that is cheap to integrate. We show that a set of neural networks can face that challenge: a normalizing flow that approximates the shape of the integrand and another neural network that infers the solution of the integral equation. We also propose to leverage a neural importance sampler to estimate the difference between the original integrand and the learned control variate. To optimize the resulting parametric estimator, we derive a theoretically optimal, variance-minimizing loss function, and propose an alternative, composite loss for stable online training in practice. When applied to light transport simulation, neural control variates are capable of matching the state-of-the-art performance of other unbiased approaches, while providing means to develop more performant, practical solutions. Specifically, we show that the learned light-field approximation is of sufficient quality for high-order bounces, allowing us to omit the error correction and thereby dramatically reduce the noise at the cost of negligible visible bias.

Thomas Müller, Brian McWilliams, Fabrice Rousselle et al.

We propose to use deep neural networks for generating samples in Monte Carlo integration. Our work is based on non-linear independent components estimation (NICE), which we extend in numerous ways to improve performance and enable its application to integration problems. First, we introduce piecewise-polynomial coupling transforms that greatly increase the modeling power of individual coupling layers. Second, we propose to preprocess the inputs of neural networks using one-blob encoding, which stimulates localization of computation and improves inference. Third, we derive a gradient-descent-based optimization for the KL and the $χ^2$ divergence for the specific application of Monte Carlo integration with unnormalized stochastic estimates of the target distribution. Our approach enables fast and accurate inference and efficient sample generation independently of the dimensionality of the integration domain. We show its benefits on generating natural images and in two applications to light-transport simulation: first, we demonstrate learning of joint path-sampling densities in the primary sample space and importance sampling of multi-dimensional path prefixes thereof. Second, we use our technique to extract conditional directional densities driven by the product of incident illumination and the BSDF in the rendering equation, and we leverage the densities for path guiding. In all applications, our approach yields on-par or higher performance than competing techniques at equal sample count.

Simon Kallweit, Thomas Müller, Brian McWilliams et al.

We present a technique for efficiently synthesizing images of atmospheric clouds using a combination of Monte Carlo integration and neural networks. The intricacies of Lorenz-Mie scattering and the high albedo of cloud-forming aerosols make rendering of clouds---e.g. the characteristic silverlining and the "whiteness" of the inner body---challenging for methods based solely on Monte Carlo integration or diffusion theory. We approach the problem differently. Instead of simulating all light transport during rendering, we pre-learn the spatial and directional distribution of radiant flux from tens of cloud exemplars. To render a new scene, we sample visible points of the cloud and, for each, extract a hierarchical 3D descriptor of the cloud geometry with respect to the shading location and the light source. The descriptor is input to a deep neural network that predicts the radiance function for each shading configuration. We make the key observation that progressively feeding the hierarchical descriptor into the network enhances the network's ability to learn faster and predict with high accuracy while using few coefficients. We also employ a block design with residual connections to further improve performance. A GPU implementation of our method synthesizes images of clouds that are nearly indistinguishable from the reference solution within seconds interactively. Our method thus represents a viable solution for applications such as cloud design and, thanks to its temporal stability, also for high-quality production of animated content.