Second‐Order Approximation for Variance Reduction in Multiple Importance Sampling

Abstract

Monte Carlo Techniques are widely used in Computer Graphics to generate realistic images. Multiple Importance Sampling reduces the impact of choosing a dedicated strategy by balancing the number of samples between different strategies. However, an automatic choice of the optimal balancing remains a difficult problem. Without any scene characteristics knowledge, the default choice is to select the same number of samples from different strategies and to use them with heuristic techniques (e.g., balance, power or maximum). In this paper, we introduce a second‐order approximation of variance for balance heuristic. Based on this approximation, we introduce an automatic distribution of samples for direct lighting without any prior knowledge of the scene characteristics. We demonstrate that for all our test scenes (with different types of materials, light sources and visibility complexity), our method actually reduces variance in average. We also propose an implementation with low overhead for offline and GPU applications. We hope that this approach will help developing new balancing strategies.

Optimal combination of techniques in multiple importance sampling

“…Our algorithm although heuristic decreases variance against both standard MIS with taking equal count of samples (α1 = α2 = 0.5) and also again the recent paper by Lu et al [2013]. The optimal scheme was considered as taking equal number of samples in pilot stage of sampling, where in pilot stage of sampling we took 20% of all samples.…”

“…As computation cost and contribution of a single integral in the final result vary, they introduced the concept of cost in MIS, approximated the variance and minimized it through a gradient search method and Lagrange multipliers with the cost as constraint. Lu et al [2013], using the fact that balance heuristic MIS is importance sampling with mixture of densities, approximated the variance for the product of two functions representing BRDF and environment map, using a second order Taylor expansion around the value 1/2, which corresponds to equal number of sampling for the two coefficients. From this expansion they obtained optimal coefficients, but the farthest from the 1/2 value for these coefficients the worse the approximation.…”

“…Lu et al [2013], using also the fact that balance heuristic MIS is importance sampling with mixture of densities, approximate the variance for the product of two functions representing BRDF and environment map, using a second order Taylor expansion around α = 1/2. Thus the farthest from this α = 1/2 the worse the approximation.…”

“…Third row -sampling pattern found by our heuristic in Section 4.3. Fourth row -sampling pattern by the method of Lu et al [2013]. Columns: (left) rectangle texture illuminated by City hall, (middle left) rectangle texture illuminated by Grace Cathedral, (middle right) rectangle texture illuminated St'Peters Dome, (right) Phlegmatic dragon illuminated by St'Peters Dome, with rendered images corresponding to Figure 1.…”

“…Lu et al [2013] use rough Taylor second order approximation around the weight 1/2 for the environment map problem, combining the two techniques of sampling BRDF and environment map. Csonka et al [2001] studied MIS, where the cost of sampling was also taken in consideration when minimizing the variance of the combination of stochastic iteration and bidirectional path tracing.…”