Multiple Importance Sampling Research Articles

Optimal multiple importance sampling

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.

Photon surfaces for robust, unbiased volumetric density estimation

We generalize photon planes to photon surfaces : a new family of unbiased volumetric density estimators which we combine using multiple importance sampling. To derive our new estimators, we start with the extended path integral which duplicates the vertex at the end of the camera and photon subpaths and couples them using a blurring kernel. To make our formulation unbiased, however, we use a delta kernel to couple these two end points. Unfortunately, sampling the resulting singular integral using Monte Carlo is impossible since the probability of generating a contributing light path by independently sampling the two subpaths is zero. Our key insight is that we can eliminate the delta kernel and make Monte Carlo estimation practical by integrating any three dimensions analytically, and integrating only the remaining dimensions using Monte Carlo. We demonstrate the practicality of this approach by instantiating a collection of estimators which analytically integrate the distance along the camera ray and two arbitrary sampling dimensions along the photon subpath (e.g., distance, direction, surface area). This generalizes photon planes to curved "photon surfaces", including new "photon cone", "photon cylinder", "photon sphere", and multiple new "photon plane" estimators. These estimators allow us to handle light paths not supported by photon planes, including single scattering, and surface-to-media transport. More importantly, since our estimators have complementary strengths due to analytically integrating different dimensions of the path integral, we can combine them using multiple importance sampling. This combination mitigates singularities present in individual estimators, substantially reducing variance while remaining fully unbiased. We demonstrate our improved estimators on a number of scenes containing homogeneous media with highly anisotropic phase functions, accelerating both multiple scattering and single scattering compared to prior techniques.

A null-scattering path integral formulation of light transport

Unbiased rendering of general, heterogeneous participating media currently requires using null-collision approaches for estimating transmittance and generating free-flight distances. A long-standing limitation of these approaches, however, is that the corresponding path pdfs cannot be computed due to the black-box nature of the null-collision rejection sampling process. These techniques therefore cannot be combined with other sampling techniques via multiple importance sampling (MIS), which significantly limits their robustness and generality. Recently, Galtier et al. [2013] showed how to derive these algorithms directly from the radiative transfer equation (RTE). We build off this generalized RTE to derive a path integral formulation of null scattering, which reveals the sampling pdfs and allows us to devise new, express existing, and combine complementary unbiased techniques via MIS. We demonstrate the practicality of our theory by combining, for the first time, several path sampling techniques in spatially and spectrally varying media, generalizing and outperforming the prior state of the art.

Hierarchical russian roulette for vertex connections

While bidirectional path tracing is a well-established light transport algorithm, many samples are required to obtain high-quality results for specular-diffuse-glossy or glossy-diffuse-glossy reflections especially when they are highly glossy. To improve the efficiency for such light path configurations, we propose a hierarchical Russian roulette technique for vertex connections. Our technique accelerates a huge number of Russian roulette operations according to an approximate scattering lobe at an eye-subpath vertex for many cached light-subpath vertices. Our method dramatically reduces the number of random number generations needed for Russian roulette by introducing a hierarchical rejection algorithm which assigns random numbers in a top-down fashion. To efficiently reject light vertices in each hierarchy, we also introduce an efficient approximation of anisotropic scattering lobes used for the probability of Russian roulette. Our technique is easy to integrate into some existing bidirectional path tracing-based algorithms which cache light-subpath vertices (e.g., probabilistic connections, and vertex connection and merging). In addition, unlike existing many-light methods, our method does not restrict multiple importance sampling strategies thanks to the simplicity of Russian roulette. Although the proposed technique does not support perfectly specular surfaces, it significantly improves the efficiency for caustics reflected on extremely glossy surfaces in an unbiased fashion.

Combining Point and Line Samples for Direct Illumination

AbstractWe develop a unified framework for combining point and line samples in direct lighting calculations. While line samples have proven beneficial in a variety of rendering contexts, their application in direct lighting has been limited due to a lack of formulas for evaluating advanced BRDFs along a line and performance tied to the orientation of occluders in the scene. We lift these limitations by elevating line samples to a shared higher‐dimensional space with point samples. Our key insight is to separate the probability distribution functions of line samples and points that lie along a line sample. This simple conceptual change allows us to apply multiple importance sampling (MIS) between points and lines, and lines with each other, in order to leverage their respective strengths. We also show how to improve the convergence rate of MIS between points and lines in an unbiased way using a novel discontinuity‐smoothing balance heuristic. We verify through a set of rendering experiments that our proposed MISing of points and lines, and lines with each other, reduces variance of the direct lighting estimate while supporting an increased range of BSDFs compared to analytic line integration.

Adaptive BRDF‐Oriented Multiple Importance Sampling of Many Lights

AbstractMany‐light rendering is becoming more common and important as rendering goes into the next level of complexity. However, to calculate the illumination under many lights, state of the art algorithms are still far from efficient, due to the separate consideration of light sampling and BRDF sampling. To deal with the inefficiency of many‐light rendering, we present a novel light sampling method named BRDF‐oriented light sampling, which selects lights based on importance values estimated using the BRDF's contributions. Our BRDF‐oriented light sampling method works naturally with MIS, and allows us to dynamically determine the number of samples allocated for different sampling techniques. With our method, we can achieve a significantly faster convergence to the ground truth results, both perceptually and numerically, as compared to previous many‐light rendering algorithms.

Fourier Analysis of Correlated Monte Carlo Importance Sampling

AbstractFourier analysis is gaining popularity in image synthesis as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyse convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling‐based integrators to unify non‐uniform sample distributions [importance sampling (IS)] as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier‐based tools in performing variance analysis for MC integration. At the same time, it reveals that, when combined with correlated sampling, IS can impact convergence rate by introducing or inhibiting discontinuities in the integrand. We demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy which converges slowest and propose several simple approaches to overcome this limitation. We show that smoothing light boundaries (as commonly done in production to reduce variance) can improve (M)IS convergence (at a cost of introducing a small amount of bias) since it removes C0 discontinuities within the integration domain. We also propose practical integrand‐ and sample‐mirroring approaches which cancel the impact of boundary discontinuities on the convergence rate of estimators.

Multiple Importance Sampling for Efficient Symbol Error Rate Estimation

Digital constellations formed by hexagonal or other non-square two-dimensional lattices are often used in advanced digital communication systems. The integrals required to evaluate the symbol error rate (SER) of these constellations in the presence of Gaussian noise are in general difficult to compute in closed-form, and therefore Monte Carlo simulation is typically used to estimate the SER. However, naive Monte Carlo simulation can be very inefficient and requires very long simulation runs, especially at high signal-to-noise ratios. In this letter, we adapt a recently proposed multiple importance sampling (MIS) technique, called ALOE (for `At Least One rare Event'), to this problem. Conditioned to a transmitted symbol, an error (or rare event) occurs when the observation falls in a union of half-spaces or, equivalently, outside a given polytope. The proposal distribution for ALOE samples the system conditionally on an error taking place, which makes it more efficient than other importance sampling techniques ALOE provides unbiased SER estimates with simulation times orders of magnitude shorter than conventional Monte Carlo.

Generalized Multiple Importance Sampling

Importance Sampling methods are broadly used to approximate posterior distributions or some of their moments. In its standard approach, samples are drawn from a single proposal distribution and weighted properly. However, since the performance depends on the mismatch between the targeted and the proposal distributions, several proposal densities are often employed for the generation of samples. Under this Multiple Importance Sampling (MIS) scenario, many works have addressed the selection or adaptation of the proposal distributions, interpreting the sampling and the weighting steps in different ways. In this paper, we establish a general framework for sampling and weighing procedures when more than one proposal are available. The most relevant MIS schemes in the literature are encompassed within the new framework, and, moreover novel valid schemes appear naturally. All the MIS schemes are compared and ranked in terms of the variance of the associated estimators. Finally, we provide illustrative examples which reveal that, even with a good choice of the proposal densities, a careful interpretation of the sampling and weighting procedures can make a significant difference in the performance of the method.

Efficient Energy-Compensated VPLs using Photon Splatting

Efficient light transport simulation is important for many applications in computer graphics. Many-light methods based on virtual point lights (VPLs) especially work well for scenes consist mainly of diffuse materials. Rendered images with VPLs, however, are known to suffer from energy loss due to clamping of the VPL contributions. Removing this clamping, unfortunately, introduces significant variance in images due to the singularities of VPLs. We propose an efficient and practical algorithm which compensates energy loss of VPLs using yet another efficient rendering method -- photon splatting. Recent unified path sampling frameworks show that VPLs are in fact photons. Based on this unified formulation, we propose to reuse VPLs as photons to compensate missing energy of clamped VPLs. This combination is very natural: the singularities of VPLs happen if a shading point is too close to VPLs, but such VPLs can also be seen as neighboring photons that are indeed useful for photon density estimation. We formulate this energy compensation process using multiple importance sampling to combine rendering with VPLs and photon splatting optimally. Our algorithm can be efficiently implemented on GPUs since rendering with VPLs and photon splatting are both suitable for GPUs. The results demonstrate that our method produces more accurate images in the same rendering time than the existing methods. For common use cases of rendering mostly diffuse scenes, our work leads to an efficient, accurate, and practical rendering algorithm based on VPLs.

Manuka

The Manuka rendering architecture has been designed in the spirit of the classic reyes rendering architecture: to enable the creation of visually rich computer generated imagery for visual effects in movie production. Following in the footsteps of reyes over the past 30 years, this means supporting extremely complex geometry, texturing, and shading. In the current generation of renderers, it is essential to support very accurate global illumination as a means to naturally tie together different assets in a picture. This is commonly achieved with Monte Carlo path tracing, using a paradigm often called shade on hit , in which the renderer alternates tracing rays with running shaders on the various ray hits. The shaders take the role of generating the inputs of the local material structure, which is then used by path-sampling logic to evaluate contributions and to inform what further rays to cast through the scene. We propose a shade before hit paradigm instead and minimise I/O strain on the system, leveraging locality of reference by running pattern generation shaders before we execute light transport simulation by path sampling. We describe a full architecture built around this approach, featuring spectral light transport and a flexible implementation of multiple importance sampling ( mis ), resulting in a system able to support a comparable amount of extensibility to what made the reyes rendering architecture successful over many decades.

Multiple importance sampling characterization by weighted mean invariance

In this paper, we examine the linear combination of techniques and multiple importance sampling for Monte Carlo integration from a new perspective of quasi-arithmetic weighted means. The invariance property of these means allows us to define a new family of heuristics. We illustrate our results with several rendering examples, including environment mapping and path tracing.

Optimal Sample Weights for Hemispherical Integral Quadratures

AbstractThis paper proposes optimal quadrature rules over the hemisphere for the shading integral. We leverage recent work regarding the theory of quadrature rules over the sphere in order to derive a new theoretical framework for the general case of hemispherical quadrature error analysis. We then apply our framework to the case of the shading integral. We show that our quadrature error theory can be used to derive optimal sample weights (OSW) which account for both the features of the sampling pattern and the bidirectional reflectance distribution function (BRDF). Our method significantly outperforms familiar Quasi Monte Carlo (QMC) and stochastic Monte Carlo techniques. Our results show that the OSW are very effective in compensating for possible irregularities in the sample distribution. This allows, for example, to significantly exceed the regular convergence rate of stochastic Monte Carlo while keeping the exact same sample sets. Another important benefit of our method is that OSW can be applied whatever the sampling points distribution: the sample distribution need not follow a probability density function, which makes our technique much more flexible than QMC or stochastic Monte Carlo solutions. In particular, our theoretical framework allows to easily combine point sets derived from different sampling strategies (e.g. targeted to diffuse and glossy BRDF). In this context, our rendering results show that our approach overcomes MIS (Multiple Importance Sampling) techniques.

Multiple importance sampling revisited: breaking the bounds

We revisit the multiple importance sampling (MIS) estimator and investigate the bound on the efficiency improvement over balance heuristic estimator with equal count of samples established in Veach’s thesis. We revise the proof for this and come to the conclusion that there is no such bound and henceforth it makes sense to look for new estimators that improve on balance heuristic estimator with equal count of samples. Next, we examine a recently introduced non-balance heuristic MIS estimator that is provably better than balance heuristic with equal count of samples, and we improve it both in variance and efficiency. We then obtain an equally provably better one-sample balance heuristic estimator, and finally, we introduce a heuristic for the count of samples that can be used when the individual techniques are biased. All in all, we present three new sampling strategies to improve on both variance and efficiency on the balance heuristic using non-equal count of samples.Our scheme requires the previous knowledge of several quantities, but those can be obtained in an adaptive way. The results also show that by a careful examination of the variance and properties of the estimators, even better estimators could be discovered in the future. We present examples that support our theoretical findings.

Estimating standard errors for importance sampling estimators with multiple Markov chains

The naive importance sampling estimator based on the samples from a single importance density can be extremely numerically unstable. We consider multiple distributions importance sampling estimators where samples from more than one probability distributions are combined to consistently estimate means with respect to given target distributions. These generalized importance sampling estimators provide more stable estimators than the naive importance sampling estimators. Importance sampling estimators can also be used in the Markov chain Monte Carlo (MCMC) context, that is, where iid samples are replaced with positive Harris Markov chains with invariant importance distributions. If these Markov chains converge to their respective target distributions at a geometric rate, then under two finite moment conditions a central limit theorem (CLT) holds for the importance sampling estimators. In order to calculate valid asymptotic standard errors, it is required to consistently estimate the asymptotic variance in the CLT. Recently Tan and Doss and Hobert (2015) developed an approach based on regenerative simulation for obtaining consistent estimators of the asymptotic variance. It is well-known that in practice it is often difficult to construct a useful minorization condition that is required in Tan and Doss and Hobert ’s (2015) regenerative simulation method. We provide an alternative estimator for these standard errors based on the easy to implement batch means methods. The multi-chain importance sampling estimators depend on Geyer’s (1994) reverse logistic estimator (of ratios of normalizing constants) which has wide applications, in its own right, in both frequentist and Bayesian inference. We also provide batch means estimator for calculating asymptotically valid standard errors of Geyer’s (1994) reverse logistic estimator. We illustrate the method with an application in Bayesian variable selection in linear regression. In particular, the multi-chain importance sampling estimator is used to perform empirical Bayes variable selection and the batch means estimator is used to obtain standard errors in the large p situation where regenerative method is not applicable.

Sample Reuse with Adaptive Multiple Importance Sampling for Policy Search

Sample Reuse with Adaptive Multiple Importance Sampling for Policy Search

An Improved Multiple Importance Sampling Heuristic for Density Estimates in Light Transport Simulations

An Improved Multiple Importance Sampling Heuristic for Density Estimates in Light Transport Simulations

A Derivative-Free Trust-Region Algorithm for the Optimization of Functions Smoothed via Gaussian Convolution Using Adaptive Multiple Importance Sampling

In this paper we consider the optimization of a functional $F$ defined as the convolution of a function $f$ with a Gaussian kernel. We propose this type of objective function for the optimization of the output of complex computational simulations, which often present some form of deterministic noise and need to be smoothed for the results to be meaningful. We introduce a derivative-free algorithm that computes trial points from the minimization of a regression model of the noisy function $f$ over a trust region. The regression model is constructed from function values at sample points that are chosen randomly around iterates and trial points of the algorithm. The weights given to the individual sample points in the regression problem are obtained according to an adaptive multiple importance sampling strategy. This has two advantages. First, it makes it possible to reuse all noisy function values collected over the course of the optimization. Second, the resulting regression model converges to the second-o...

RF法による設計点探索のための全象限初期点設定法の研究（多峰性重点サンプリング・シミュレーションに基づく構造信頼性解析の効率化）

This study describes a method to reliably search and determine design points required for constructing importance sampling probability density functions (p.d.f.s) in the method of estimating the structural failure probability based on multiple importance sampling simulation. In order to search out and determine all of design points without any omission, the initial points of Rackwitz Fiessler (RF) algorithm that performs the design point search are proposed to be set on limit state surfaces within the entire quadrants of the basic random variables space. First, from the standardized normal p.d.f., several samples, referred to as real samples, are generated in the specific quadrants. Using the newly proposed "inter-quadrant relational expression", the coordinates of the real sample points are converted to the coordinates of the sample points called pseudo samples in all the quadrants except those of the real sample points. In this way, real samples or pseudo samples are generated and determined in all quadrants. Next, the coordinates of intersection points of the limit state surface and straight lines extending from the origin and passing through respective real and pseudo sample points are determined. And the intersections determined on the limit state surfaces in all quadrants are proposed to adopt as initial points of RF algorithm. By executing RF algorithm from respective initial points determined, all of design points are searched out completely, and the importance sampling p.d.f.s are constructed by making use of the respective design points determined. Numerical examples for estimating the failure probability of structural systems having multivariable nonlinear limit state functions are shown based on multiple importance sampling simulation using all of the constructed importance sampling p.d.f.s and the proposed method shows accurate estimations of structural failure probabilities.

Screen Space Ambient Occlusion Based Multiple Importance Sampling for Real-Time Rendering

We propose a new approximation technique for accelerating the Global Illumination algorithm for real-time rendering. The proposed approach is based on the Screen-Space Ambient Occlusion (SSAO) method, which approximates the global illumination for large, fully dynamic scenes at interactive frame rates. Current algorithms that are based on the SSAO method suffer from difficulties due to the large number of samples that are required. In this paper, we propose an improvement to the SSAO technique by integrating it with a Multiple Importance Sampling technique that combines a stratified sampling method with an importance sampling method, with the objective of reducing the number of samples. Experimental evaluation demonstrates that our technique can produce high-quality images in real time and is significantly faster than traditional techniques.