Technical SectionImproved stratification for Metropolis light transport☆
Graphical abstract
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 to integration domain . Thus, taking the transformation from primary sample space to path space 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 is sampled uniformly, but the samples are obtained with a non-uniform target density with which integrand needs to be compensated in the quadrature formula: 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
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.
References (26)
- et al.
Arbitrary importance functions for metropolis light transport
Comput Graph Forum
(2010) - et al.
A simple and robust mutation strategy for the Metropolis light transport algorithm
Comput Graph Forum
(2002) - et al.
Equations of state calculations by fast computing machines
J Chem Phys
(1953) - et al.
Metropolis light transport
Proceedings of the SIGGRAPH ’97
(1997) - et al.
On the start-up bias problem of Metropolis sampling
Proceedings of the winter school of computer graphics ’99
(1999) - et al.
Deterministic importance sampling with error diffusion
Comput Graph Forum
(2009) Consequences of stratified sampling in graphics
Proceedings of computer graphics (SIGGRAPH ’96)
(1996)Quasi-monte carlo methods for photorealistic image synthesis
(1998)- et al.
Systematic sampling in image-synthesis
Proceedings of the conference computational science and its applications
(2006) - et al.
Anisotropic Gaussian mutations for metropolis light transport through hessian-hamiltonian dynamics
ACM Trans Graph
(2015)
A variance analysis of the Metropolis light transport algorithm
Comput Graph
Multiplexed metropolis light transport
ACM Trans Graph
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This article was recommended for publication by M Wimmer.