Multiple importance sampling (MIS) is a vital component of most rendering algorithms. MIS computes a weighted sum of samples from many different techniques to achieve generalization, that is, to handle a wide range of scene types and lighting effects. A key factor to the performance of MIS is the choice of weighting function. The go-to default - the balance heuristic - performs well in many cases, but prior work has shown that it can yield unsatisfactory results. A number of challenges cause this suboptimal performance, including low-variance techniques, sample correlation, and unknown sampling densities. Prior work has suggested improvements for some of these problems, but a general optimal solution has yet to be found. We propose a general and practical weight correction scheme: We optimize, on-the-fly, a set of correction factors that are multiplied into any baseline MIS heuristic (e.g., balance or power). We demonstrate that this approach yields consistently better equal-time performance on two rendering applications: bidirectional algorithms and resampled importance sampling for direct illumination.

Independently estimating pixel values in Monte Carlo rendering results in a perceptually sub-optimal white-noise distribution of error in image space. Recent works have shown that perceptual fidelity can be improved significantly by distributing pixel error as blue noise instead. Most such works have focused on static images, ignoring the temporal perceptual effects of animation display. We extend prior formulations to simultaneously consider the spatial and temporal domains, and perform an analysis to motivate a perceptually better spatio-temporal error distribution. We then propose a practical error optimization algorithm for spatio-temporal rendering and demonstrate its effectiveness in various configurations.

Monte Carlo rendering makes heavy use of mixture sampling and multiple importance sampling (MIS). Previous work has shown that control variates can be used to make such mixtures more efficient and more robust. However, the existing approaches failed to yield practical applications, chiefly because their underlying theory is based on the unrealistic assumption that a single mixture is optimized for a single integral. This is in stark contrast with rendering reality, where millions of integrals are computed---one per pixel---and each is infinitely recursive. We adapt and extend the theory introduced by previous work to tackle the challenges of real-world rendering applications. We achieve robust mixture sampling and (approximately) optimal MIS weighting for common applications such as light selection, BSDF sampling, and path guiding.

Multiple importance sampling (MIS) is an indispensable tool in light-transport simulation. It enables robust Monte Carlo integration by combining samples from several techniques. However, it is well understood that such a combination is not always more efficient than using a single sampling technique. Thus a major criticism of complex combined estimators, such as bidirectional path tracing, is that they can be significantly less efficient on common scenes than simpler algorithms like forward path tracing. We propose a general method to improve MIS efficiency: By cheaply estimating the efficiencies of various technique and sample-count combinations, we can pick the best one. The key ingredient is a numerically robust and efficient scheme that uses the samples of one MIS combination to compute the efficiency of multiple other combinations. For example, we can run forward path tracing and use its samples to decide which subset of VCM to enable, and at what sampling rates. The sample count for each technique can be controlled per-pixel or globally. Applied to VCM, our approach enables robust rendering of complex scenes with caustics, without compromising efficiency on simpler scenes.

Russian roulette and splitting are widely used techniques to increase the efficiency of Monte Carlo estimators. But, despite their popularity, there is little work on how to best apply them. Most existing approaches rely on simple heuristics based on, e.g., surface albedo and roughness. Their efficiency often hinges on user-controlled parameters. We instead iteratively learn optimal Russian roulette and splitting factors during rendering, using a simple and lightweight data structure. Given perfect estimates of variance and cost, our fixed-point iteration provably converges to the optimal Russian roulette and splitting factors that maximize the rendering efficiency. In our application to unidirectional path tracing, we achieve consistent and significant speed-ups over the state of the art.

Abstract Combining diverse sampling techniques via multiple importance sampling (MIS) is key to achieving robustness in modern Monte Carlo light transport simulation. Many such methods additionally employ correlated path sampling to boost efficiency. Photon mapping, bidirectional path tracing, and path‐reuse algorithms construct sets of paths that share a common prefix. This correlation is ignored by classical MIS heuristics, which can result in poor technique combination and noisy images. We propose a practical and robust solution to that problem. Our idea is to incorporate correlation knowledge into the balance heuristic, based on known path densities that are already required for MIS. This correlation‐aware heuristic can achieve considerably lower error than the balance heuristic, while avoiding computational and memory overhead.

Path guiding is a promising tool to improve the performance of path tracing algorithms. However, not much research has investigated what target densities a guiding method should strive to learn for optimal performance. Instead, most previous work pursues the zero-variance goal: The local decisions are guided under the assumption that all other decisions along the random walk will be sampled perfectly. In practice, however, many decisions are poorly guided, or not guided at all. Furthermore, learned distributions are often marginalized, e.g., by neglecting the BSDF. We present a generic procedure to derive theoretically optimal target densities for local path guiding. These densities account for variance in nested estimators, and marginalize provably well over, e.g., the BSDF. We apply our theory in two state-of-the-art rendering applications: a path guiding solution for unidirectional path tracing [Müller et al. 2017] and a guiding method for light source selection for the many lights problem [Vévoda et al. 2018]. In both cases, we observe significant improvements, especially on glossy surfaces. The implementations for both applications consist of trivial modifications to the original code base, without introducing any additional overhead.

Many existing Monte Carlo methods rely on multiple importance sampling (MIS) to achieve robustness and versatility. Typically, the balance or power heuristics are used, mostly thanks to the seemingly strong guarantees on their variance. We show that these MIS heuristics are oblivious to the effect of certain variance reduction techniques like stratification. This shortcoming is particularly pronounced when unstratified and stratified techniques are combined (e.g., in a bidirectional path tracer). We propose to enhance the balance heuristic by injecting variance estimates of individual techniques, to reduce the variance of the combined estimator in such cases. Our method is simple to implement and introduces little overhead.

Multiple Importance Sampling (MIS) is a key technique for achieving robustness of Monte Carlo estimators in computer graphics and other fields. We derive optimal weighting functions for MIS that provably minimize the variance of an MIS estimator, given a set of sampling techniques. We show that the resulting variance reduction over the balance heuristic can be higher than predicted by the variance bounds derived by Veach and Guibas, who assumed only non-negative weights in their proof. We theoretically analyze the variance of the optimal MIS weights and show the relation to the variance of the balance heuristic. Furthermore, we establish a connection between the new weighting functions and control variates as previously applied to mixture sampling. We apply the new optimal weights to integration problems in light transport and show that they allow for new design considerations when choosing the appropriate sampling techniques for a given integration problem.

Abstract Robust and efficient rendering of complex lighting effects, such as caustics, remains a challenging task. While algorithms like vertex connection and merging can render such effects robustly, their significant overhead over a simple path tracer is not always justified and – as we show in this paper ‐ also not necessary. In current rendering solutions, caustics often require the user to enable a specialized algorithm, usually a photon mapper, and hand‐tune its parameters. But even with carefully chosen parameters, photon mapping may still trace many photons that the path tracer could sample well enough, or, even worse, that are not visible at all. Our goal is robust, yet lightweight, caustics rendering. To that end, we propose a technique to identify and focus computation on the photon paths that offer significant variance reduction over samples from a path tracer. We apply this technique in a rendering solution combining path tracing and photon mapping. The photon emission is automatically guided towards regions where the photons are useful, i.e., provide substantial variance reduction for the currently rendered image. Our method achieves better photon densities with fewer light paths (and thus photons) than emission guiding approaches based on visual importance. In addition, we automatically determine an appropriate number of photons for a given scene, and the algorithm gracefully degenerates to pure path tracing for scenes that do not benefit from photon mapping.