Elsevier

Computers & Graphics

Volume 68, November 2017, Pages 11-20
Computers & Graphics

Technical Section
Improved stratification for Metropolis light transport

https://doi.org/10.1016/j.cag.2017.07.032Get rights and content

Highlights

  • Discussion of stratification problems of the Metropolis algorithm, and showing ways for improvement.

  • Discussion of the difficulties of adaptive mutations.

  • Proposal of a simple mutation strategy to improve the stratum of samples and thus reducing the integration error.

  • Analysis and visualization of the method in 2D.

  • The application of the sampling method in 3D global illumination rendering.

Abstract

The approximation error of Monte Carlo methods depends on two factors, how accurately the target density of the samples mimics the integrand and how well the finite number of samples are distributed with the target density. The first factor can be reduced by importance sampling, the second by stratification. The Metropolis algorithm is particularly effective in importance sampling, but is poor concerning stratification if mutation strategies are not carefully designed. This means that although the asymptotic distribution of the samples well mimics most of the factors of the integrand, the empirical distribution of the finite number of samples can be far from the desired distribution. This paper examines this issue and proposes a simple mutation scheme for Primary Sample Space Metropolis Light Transport (PSSMLT), which improves the stratum of samples and thus reduces the integration error. Unlike other approaches, the proposed method does not sacrifice importance sampling for better stratification and uses only the local importance value without requiring derivatives that would be problematic for discontinuous integrands caused by occlusions and texture patterns. The method is simple to implement and to integrate into photo-realistic renderers.

Introduction

In global illumination rendering we need to evaluate high-dimensional integrals in the space of light paths. A light path connects a light source point to the virtual camera via arbitrary number of scattering events where the direction can change. Classical quadrature rules fail in higher dimensions due to the curse of dimensionality, which means that the sample number required for a given accuracy grows exponentially with the dimension. This problem can be avoided by Monte Carlo or quasi-Monte Carlo quadrature, which transforms pseudo-random or quasi-random samples uniformly filling a high-dimensional unit cube U to integration domain P. Thus, taking the transformation from primary sample spaceU to path spaceP also into account, our task is to compute integrals over the high-dimensional unit cube. If importance sampling is involved in path generation, then more important regions are represented by larger volume in primary sample space. We can also say that this primary importance sampling (e.g. BRDF or light source sampling) does one part of the job of optimal importance sampling, which may need further improvement by some secondary importance sampling.

One possibility for secondary importance sampling is the Primary Sample Space Metropolis Light Transport (PSSMLT) method [1], which requires the definition of a tentative transition function for primary sample perturbation and an importance function which determines the target density of generated samples. Based on these functions, the Metropolis method randomly accepts or rejects tentative samples. The resulting estimator is consistent for arbitrary tentative transition functions ensuring ergodicity and for importance functions that are non-zero where the integrand is non-zero. However, the variance of the estimator depends on the selection of these functions, so does the average probability of accepting tentative samples, called the acceptance rate. This paper targets this problem by proposing a simple mutation strategy that simultaneously improves the acceptance rate and makes the samples more stratified in the high-dimensional sample space. Because of this property, we call the new method stratified Metropolis.

The organization of this paper is as follows. In Section 2 we present the targeted challenges of the Metropolis method. Section 3 reviews the related previous work. Section 4 presents our novel solution. Section 5 demonstrates the method with 2D examples and also in global illumination rendering.

Section snippets

Problem statement

Secondary importance sampling means that not even the primary sample space U is sampled uniformly, but the samples are obtained with a non-uniform target densityg(u), with which integrand f(u) needs to be compensated in the quadrature formula: Uf(u)du1Mj=1Mf(uj)g(uj)where M is the number of samples.

To generate samples with a non-uniform target density, the Metropolis–Hastings method [2], [3] explores the sample space with a Markov stochastic process. Having the current sample ui,

Previous work

The challenging problems of defining the importance function and mutation strategies to increase the performance of Metropolis algorithms have been addressed by several papers.

As stratification requires the reduction of the number of rejected samples, our proposed method is related to techniques also aiming at the reduction of rejections [9] and sample correlation [10]. Unlike path tracing type Monte Carlo methods that estimate pixels independently and determine the number of samples in a pixel

Designing mutations

The method presented in this paper is simple and controls the mutation with both the global and the local properties of the integrand. Without the derivatives, the mutation can depend just on the local value of the importance function, while the strength of this dependence is set adaptively. Our goal is to develop a mutation strategy that makes the samples more stratified not only in image space but in the primary sample space and consequently in the multi-dimensional space of the light paths.

Results

In order to analyze the behavior of the proposed method, we first take a two-dimensional example and then consider the global illumination problem.

Conclusions

This paper proposed a new mutation strategy for the Metropolis method, which results in less rejected samples and better stratified accepted ones, and thus reduces the integration error. We observed that improving the acceptance ratio and avoiding too small steps at unimportant regions both require the control of the mutation size by increasing it in less important regions and reducing it when the importance is high. A mutation strategy that is based on the local importance must be asymmetric,

Acknowledgments

Scenes Sphere-thing, Door, and School-corridor are courtesy of Devon Vitkovsky, Giulio Jiang, and Simon Wendsch, respectively. This work has been supported by OTKA K–124124 and VKSZ-14 PET/MRI 7T and SCOPIA projects.

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