Geometric Sample Reweighting for Monte Carlo Integration
GGeometric Sample Reweighting for Monte Carlo Integration
JERRY JINFENG GUO and ELMAR EISEMANN,
Delft University of Technology
We present a general sample reweighting scheme and its underlying theoryfor the integration of an unknown function with low dimensionality. Ourmethod produces better results than standard weighting schemes for com-mon sampling strategies, while avoiding bias. Our main insight is to link theweight derivation to the function reconstruction process during integration.The implementation of our solution is simple and results in an improvedconvergence behavior. We illustrate its benefit by applying our method tomultiple Monte Carlo rendering problems.CCS Concepts: •
Computing methodologies → Ray tracing;
Additional Key Words and Phrases: Sampling and Reconstruction, MonteCarlo Integration, Sample Reweighting, Rendering
Monte Carlo (MC) techniques form the foundation of realistic imagesynthesis for decades (Cook et al. 1984). The principle is simple: afunction is sampled and the samples are combined to approximateits integral. Standard MC is often referred to as brute-force , as itsimplementation is simple but the variance of the estimation can behigh and convergence slow. One method to usually improve theintegral approximation is to reconstruct the underlying functionfrom the samples and much previous work devoted its attentionto particular cases (e.g., shadows (Egan et al. 2009) or depth offield (Soler et al. 2009)). In this work, we revisit the reconstructionprocess. We derive an easy-to-implement algorithm to computesample weights that generally improves the approximation whencompared to standard weights for general MC integration.Our observation is that standard sample weights are often lessaccurate for lower sampling rates because they do not properlyreflect the integration domain nor the local sample density. Ourweighting scheme considers all samples of a given set and definesweights based on a geometric partitioning of a low-dimensionalintegration domain. It results in a consistent estimator that outper-forms standard weighting schemes. A major contribution of ourwork is the derivation of an unbiased estimator. It builds upon thispartitioning and applies to sets of independent and identically dis-tributed uniform random samples or stratified samples. Specifically,we propose a novel weighting scheme that is easy to implementand builds upon a sound theoretical derivation. It integrates wellinto existing rendering pipelines, can be parallelized in conjunctionwith the unbiased estimator, and we demonstrate its benefit overexisting schemes via several rendering problems.We will first cover prior work and MC integration. We then givethe motivation behind our approach (Sec. 3) and present the core ofour solution (Sec. 4). Numerical performance and applications torendering are presented in Sec. 5.
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MC methods.
Since the 80s (Cook et al. 1984), MC integrationplays a major role in rendering complex effects, such as motion blur,depth of field, and soft shadows. The complete light transport is de-scribed by the rendering equation (Kajiya 1986), which can be solvedusing path tracing as an associated MC solution. Nevertheless, notall samples taken during the evaluation of an integral contributestrongly to the result. One strategy to modify subsequent samplechoices is to rely on previous samples, i.e., a Markov process. Me-tropolis sampling (Veach and Guibas 1997) can handle complex lightpath configurations by extensively exploring contributing pathsonce they are discovered. Multidimensional k-d trees (Guo et al.2018; Hachisuka et al. 2008) can store samples in a global struc-ture, which can then be used as a means to control future sampleplacement.While standard Monte Carlo (MC) methods solve a definite inte-gration I = ∫ Ω f ( x ) d x of a function f over a finite support Ω ⊂ R d by using a random sample set ( { x i ∈ Ω } ) with the resulting esti-mator being ˆ I MC = N (cid:205) Ni = f ( x i ) , importance sampling influencesthe sampling process via a probability distribution function (pdf) p : Ω → R (Veach and Guibas 1995). The resulting unbiased estima-tor is: ˆ I p = N N (cid:213) i = f ( x i ) p ( x i ) , (1)which effectively weighs samples differently. Importance samplingis interesting when having knowledge about the scene. For instance,importance sampling the light source works better in scenes withsmall or point light sources (Debevec 2008; Dutre et al. 2006). Sam-pling according to the BSDF works better with glossy to highlyglossy surfaces(Lafortune et al. 1997; Shirley 1991; Ward 1992). Mul-tiple importance sampling (MIS) combines different such samplingstrategies (Veach and Guibas 1995). Reweighting.
Our solution focuses on the weighting of samples in-terpreted as an improved function reconstruction. Different weightdefinitions have been shown to be beneficial for rendering, e.g.,derived in Sobolev spaces (Marques et al. 2018). However, these pre-vious solutions target hemispherical illumination integrals and arenot generally applicable to other problems. A reweighting schemewas also proposed for addressing firefly artifacts (Zirr et al. 2018) butthe solution is biased and limited to narrow application scenarios.Other specialized reconstruction techniques exist, including solu-tions for soft shadows (Egan et al. 2009), defocus blur (Soler et al.2009), and motion blur (Egan et al. 2011), which lead to significantimprovements. More complex reconstructions for light fields (Lehti-nen et al. 2011) have proven very successful but are biased (thoughconsistent).Our method is independent of the application scenario and unbi-ased. It handles general functions and links the weights to
Voronoi cell volumes. The latter has also been studied in the context ofanti-aliasing problems (Mitchell 1990), for which the 2 dimensional , Vol. 1, No. 1, Article 1. Publication date: January 2016. a r X i v : . [ c s . G R ] A ug :2 • Jerry Jinfeng Guo and Elmar Eisemann voronoi cell volumes bounded with a pixel are directly used assample weights and leads to improved anti-aliasing effects, but thetheory has not been further developed for unbiased solutions, norgeneralized to other contexts. Voronoi cell size has been used asweights for Monte Carlo integration in (Vorechovsk et al. 2016),where two ways of treating boundaries have been proposed. In thiswork, Voronoi cells of given set of samples within a domain areeither bounded and clipped by the domain boundary, or extended byperiodically adding auxiliary samples that extend the domain. Bothapproaches are shown to improve numerical performance of MC in-tegrations. However, as we show in the Sec.4, directly using Voronoicell size as weight results in an biased estimate. Our solution takesadvantage of Voronoi tessellation and remains unbiased. Referring to Eq. 1, the estimator of importance sampling is a sum offunction value f ( x i ) times a weight ∆ ( x i ) , generally: ∆ ( x i ) = Np ( x i ) . (2)Similarly, Riemann integration approximates an integral using func-tion values f ( x i ) times a weight w ( x i ) :ˆ I = N (cid:213) i = w ( x i ) f ( x i ) . (3)The Riemann weights stem from a partitioning of the support Ω into hypervolumes . In 1D, these hypervolumes are intervals. Eachhypervolume contains exactly one sample and its volume definesthe sample’s weight.The weights ∆ ( x i ) are typically easy to compute but cannot beconsidered hypervolumes; they would overlap or introduce gapsand cannot easily be linked to a partitioning of Ω . Only with increas-ing number, due to the stochastic nature of the process, when thesamples densely cover the support Ω , the difference in the weightdefinitions becomes negligible. See Fig. 1 (a), (b) and (c) for an il-lustration. In consequence, especially for low sample counts, theweights do not well reflect an approximation of the function. Our goal is to associate weights to samples that define an improvedfunction reconstruction during the integration. We will first de-fine a consistent solution, inspired by Riemann integration. Thissolution is independent of the sampling pattern and can be appliedon any sample set as a post process to improve the approximation.This reweighting is consistent, but not unbiased for all samplingstrategies. We then propose a modification to obtain an unbiasedestimator for the cases of uniform random samples that are indepen-dent and identically distributed (i.i.d.), and samples with stratification.See Sec section 6 for the possibility to generalize our method forsamples generated with an analytically known pdf.
Riemann integration typically assumes a regular partitioning of thedomain. Using a
Voronoi diagram of the sample points, it is possibleto partition the domain Ω to take sample density into account. AVoronoi diagram is a partition into regions such that the points in Xf(X) X X Fig. 1. Top row: Three integration methods using the same amount offunction evaluations (i.e., samples): (a) Riemann sum through regularbinning (according to right side value) (b) MC integration using uniformrandom samples; (c) MC integration using samples that are distributedaccording to a pdf w.r.t. function value. Notice that in (a) and (b), theassociated bin widths are equal, i.e., . Bin widths in (c) are adjustedaccording to its density determined during sample generation. Notice alsothe overlaps and gaps between sample bins as illustrated in (b) and (c).Bottom row: illustrations of our methods: (d) uniform random samples withour reweighting; (e) samples distributed according to function value withour reweighting; and (f) samples distributed according to function gradientwith our reweighting. Notice the absence of gaps/overlaps and bin widthsbeing adjusted according to sample positions. each region share the same closest sample location. It can be shownthat the Voronoi cell corresponds to the intersection of half spacesdefined by hyperplanes that are equidistant to two sample points.The theory of Voronoi diagrams is beyond the scope of this paperbut more details can be found in (Aurenhammer 1991; De Berg et al.1997).In our case of a D dimensional problem setting, the diagramwill be bounded by the hypercube ( , ) D , the domain from whichsamples are drawn. The volume of each Voronoi cell determinesthe corresponding sample weight and given that the cells are in-tersections of half-spaces, they are convex and their volume canbe easily computed. The resulting estimator of our approach isˆ I CON = (cid:205) Ni = w CON ( x i ) f ( x i ) , where w CON ( x i ) = | V i || Ω | .Implicitly, this construction approximates the integrand via apiecewise-constant representation. Intuitively, to take the mostbenefit from this interpretation, samples should be chosen withrespect to the gradient of the function. Fig. 1 shows an illustrationof this strategy. In principle, even more advanced approximationscould be used, yet it turns out that such weight definitions, whileconsistent, lead to a biased estimate. In the following, we will showthe reasoning behind this and derive an unbiased estimator for i.i.d.uniform sampling and stratified sampling. i.i.d. Uniform Samples. The reason the direct use of the Voronoicells’ volume is biased is due to the samples whose cell shares aboundary with the domain boundary. To illustrate this situation, , Vol. 1, No. 1, Article 1. Publication date: January 2016. eometric Sample Reweighting • 1:3 we will first consider the 1D case with a set X of N (with N > uniform samples X : = { x i ∈ ( , )} before generalizing to D dimensions, and then to stratified sampling. One Dimension.
Let us assume that the one-dimensional sampleset X is sorted from smallest to largest value. We are interestedin the expected extent of each Voronoi cell, for which we need toderive the expected distance between two adjacent samples. Forthis reason, we first determine the expected position of sample x i .From order statics (David and Nagaraja 2004), we know that thedistribution of the i -th i.i.d. sample follows the beta distribution, i.e., p i ( x ) = x i − ( − x ) N − i ∫ t i − ( − t ) N − i d t . The expected position of the ordered i -th sample x i is then: E [ x i ] = ∫ x · x i − ( − x ) N − i ∫ t i − ( − t ) N − i d t d x = N · (cid:18) N − i − (cid:19) · ∫ x · x i − ( − x ) N − i d x = N · (cid:18) N − i − (cid:19) · ∫ x i · N − i (cid:213) k = (cid:18) N − ik (cid:19) N − i − k (− x ) k d x = N · (cid:18) N − i − (cid:19) · ∫ N − i (cid:213) k = (cid:18) N − ik (cid:19) (− ) k x k + i d x = N · (cid:18) N − i − (cid:19) · N − i (cid:213) k = (cid:18) N − ik (cid:19) (− ) k i + k + = N · ( N − ) ! ( N − i ) ! ( i − ) ! · Γ ( i + ) Γ ( N − i + ) Γ ( N + ) = N · ( N − ) ! ( N − i ) ! ( i − ) ! · i ! ( N − i ) ! ( N + ) ! = iN + . Consequently, we have E [| x i − x i + |] = N + for i = N − E [ x − ] = E [ − x N ] = N + . Theexpected weight is then N + for samples x i with i = N −
32 1 ( N + ) for samples x and x N . The latter weights are larger due tothe intervals containing the two boundaries of the domain. Usingthese weights directly, leads to a consistent but biased estimator.To render the estimator unbiased, we introduce a per-samplecorrection coefficient C :ˆ I GR = N (cid:213) i = w GR ( x i ) f ( x i ) , where w GR ( x i ) = | V i | C ( x i ) | Ω | . (4)These factors have to be carefully chosen — for instance, C = x i to equal N f ( x i ) . Following the weightderivation, an unbiased estimator in 1D, we would then define C ( x ) = C ( x N ) = ( N + ) N and C ( x i ) = N + N for all other samples.As most samples still share an identical correction factor, it keepsus close to the interpretation of the Voronoi cell volume. In higherdimensions, the definition is less straightforward. D Dimensions.
To derive the correction coefficient C from Eq.4in D dimensions, we assume a set of N (with N ≥ D , i.e., intu-itively, this results in at least one inner point and two boundarypoints along each dimension) samples in Ω = ( , ) D . We define the boundary order b ( x i ) of a sample as the amount of cell boundaries ofits Voronoi cell that are part of the domain boundary. For instance,in the above one dimensional example, b ( x ) = b ( x N ) =
1, and forall other sample points, we have b ( x i ) = d is defined as: | X d | = card { b ( x i ) = d , ∀ i ∈ [ , N ]} . For such a sample set of N samples,the expected cardinality of samples of order d is E [| X d |] = (cid:0) Dd (cid:1) ( D √ N − ) d D − d . This formula is the d -th term in the bionomial expansionof [ ( D √ N − + ] D . To understand this result, one should recall thatthe expected position of all samples forms a regular grid. Thus, thisgrid will have a resolution of n = D √ N along each axis. Startingwith one axis, we would find n samples with two boundary samplesof order one and all others samples are inner points of order zero.Repeating these samples n times along a new dimension will incre-ment the order of the first repeated set of samples and the last, asthese represent a new boundary along this dimension. For all othersamples, their boundary order remains unchanged. This process canbe done for all D dimensions, thus implying the binomial expansion.To achieve an unbiased estimator, we first compute the expectedVoronoi volume E [ V i ] for a sample x i . For D dimensions, we have D + D . As we are dealing with an i.i.d.uniform distribution, in each dimension, we have n − n + E [ V i ] = (cid:18) (cid:19) b ( x i ) | Ω | n + E [ w GR ( x i )] = / N , thus eachsample should expectedly contribute equally. The following defini-tion of the correction coefficients fulfills this property: C ( x i ) = (cid:18) (cid:19) b ( x i ) Nn + , (5)because E [ w GR ( x i )] = E (cid:20) | V i | C ( x i ) | Ω | (cid:21) = | Ω | E (cid:20) | V i | C ( x i ) (cid:21) = | Ω | (cid:16) (cid:17) b ( x i ) | Ω | n + (cid:16) (cid:17) b ( x i ) Nn + = | Ω | × | Ω | N = N . (cid:4) Stratified Samples.
The extension to stratified sampling is rela-tively straightforward, as each stratum is considered an independentunit. This means that the function is independently integrated ineach stratum and its whole range is a composition of these units. Inconsequence, the boundary observation now applies to the bound-ary of each stratum. For a sample set X of size N generated with S strata, each stratum is expected to contain NS samples. Let n = D √ N and s = D √ S , then the integration problem for a stratum with NS , Vol. 1, No. 1, Article 1. Publication date: January 2016. :4 • Jerry Jinfeng Guo and Elmar Eisemann samples would imply the correction coefficients to be: C ( x i ) = (cid:18) (cid:19) b ( x i ) Nn + s . (6) In this section, we first show the numerical performance (Sec. 5.1)of our scheme and show its application to a few rendering scenarios(Sec. 5.2). In all tests, we compare four different estimators:(1) Standard MC (i.i.d. uniform sampling)(2) Our weighted standard MC (i.i.d. uniform sampling)(3) Stratified MC (stratified sampling)(4) Our weighted stratified MC (stratified sampling)
The numerical performance is tested with two examples: one for a1D MC integration and the other for a 2D MC integration, whichare plotted in Figure 2. The 1D function is given as: f ( x ) = × √− x + . x < x < . −√− x + x − . + .
25 0 . < x < . × ( x − . ) . < x < . . . < x < . − × ( x − . ) . < x < . . × sin ( π · ( x − . )) . < x < . . × sin ( π · ( x − . )) . < x < . . × sin ( π · ( x − . )) . < x < . Lena image (Munson 1996). Thefunctions were chosen to include discontinuities, large-scale varia-tions and small scale changes and led to a representative behaviorof several tests that we have performed. Generally, the MSE dropsas more samples are added (Column 1). Our solutions outperformstandard uniform sampling and even stratified sampling by severalorders of magnitude and converges around 1000, 100 times fasterrespectively in 1D and 100, 10 times faster respectively in 2D.For the case of stratified sampling, we illustrate different amountsof strata for the same sampling count (Column 2). Our weightingscheme makes this parameter less important, as it achieves a betterfunction approximation.We next investigate the impact of distributing samples into batchesfor which we estimate the function integration separately, beforederiving the overall estimate by averaging, which would typicallybe the case for distributed computations. First, we fixate the amountof samples to 100K (Column 3). Notice that the performance ofstandard uniform sampling remains invariant with respect to theamount of samples per batch, as it is already an averaging process.Our solution results in a better approximation for more samplesper batch, as it will approximate the function more faithfully, asexpected. Similarly, stratified sampling also benefits from moresamples per batch, but shows slower convergence.We also investigate the effect of using different batch sizes for uni-form (Column 4) and stratified sampling(Column 5). More batchesthus means a higher overall sample count and all methods improvewith the addition of batches. In all cases, the graphs stop after reach-ing 100K samples. Our solution performs best and the graphs also illustrate the convergence over several batches, due to its unbiased-ness.
We implemented our method in Mitsuba (Jakob 2010), targetingone and two dimensional integration problems, namely motion blur(Sec. 5.2.1), dispersion (Sec. 5.2.2), depth of field (Sec. 5.2.3) andillumination integrals (Sec. 5.2.4). We evaluate MSE and visual ap-pearance, as well as convergence behavior. For all implementations,our reweighting operates at a per-pixel level. We apply our methodon the level of primary samples, thus all applicable local importancesampling techniques are utilized throughout the pipeline.
To simulate motion blur, distribution render-ing samples the time domain: For a pixel ( i , j ) , the luminance L ( i , j ) is given by: L ( i , j ) = ∫ t close t open f ( i , j ) ( t ) d t , with t open and t close being the shutter opening and closing timeand f incorporating the shutter function. Since time is 1 dimen-sional, building a Voronoi partition means sorting and measuringthe distance between samples. We tested our implementation intwo scenes with animation (Fig. 3 and 4). Light dispersion can happen at reflec-tive or refractive dielectric materials, leading to effects such asrainbows, resulting from different wavelengths travelling in differ-ent directions. Spectral sampling simulates multiple wavelengthsin order to capture such effects. To reduce the complexity of theadditional spectral dimension(Bergner et al. 2009), hero wavelengthspectral sampling (Wilkie et al. 2014) can be used as an approxima-tion: ˆ I ( i , j ) = N N (cid:213) k = f ( i , j ) ( λ k ) p ( i , j ) ( λ k ) Our implementation of spectral sampling uses 15-bin wavelengths.Hero wavelength sampling is used with 3 shifted additional wave-length samples (Wilkie et al. 2014). We tested our method with twoscenes configured with dispersive di-electric materials (Fig. 5 and6).As shown in the results, our method brings down colour noisesignificantly and dispersive regions look much smoother at lowsample rate.
A camera with aperture leads to defocusblur/depth of field effects. The aperture is usually modelled as a 2Dshape, e.g., a square, a circle, or a star, which is sampled to determinethe origin of each primary sample ray, which passes through theposition on the focal plane corresponding to the current pixel. Forlens aperture A ⊂ R , we obtain: L ( i , j ) = ∫ A f ( i , j ) ( s ) d s . To determine our weights, we use a 2D Voronoi diagram based onthe aperture samples. We tested a simple glossy sphere illuminatedusing an environment map (Fig. 7). , Vol. 1, No. 1, Article 1. Publication date: January 2016. eometric Sample Reweighting • 1:5
Hi Mauro,I wonder if we can reschedule the meeting tomorrow to next Thursday (23-May)? This week we are busy working on a paper submission this week, tomorrow is the last day and Elmar would be only available in the morning.Next Thursday we would be both available in the afternoon.Let us know what do you think!Cheers,Jerry
Fig. 2. We apply our geometric sample reweighting to one and two dimensional MC integration problems.Fig. 3. Four highly glossy spheres moving in different directions with samples per pixel. In each subfigure: corresponding render, difference with referenceand highlighted regions.Fig. 4. Highly glossy Buddha moving horizontally with samples per pixel. In each subfigure: corresponding render, difference with reference andhighlighted regions. , Vol. 1, No. 1, Article 1. Publication date: January 2016. :6 • Jerry Jinfeng Guo and Elmar Eisemann Fig. 5. Wineglass with dispersive dielectric materials with and samples per pixel. MSE for highlighted four plots are: . , . , . and . respectively. Fig. 6. Torus with dielectric materials with samples per pixel.Fig. 7. Glossy sphere crossing the focal plane of the camera with samples per pixel. Leaving out irrelevant terms, the lu-minance L x at scattering point x with one bounce is given by: L x = L e ( x ) + L direct + L indirect = L e ( x ) + ∫ L f s ( x ) L e ( l → x ) d l + ∫ Ω f s ( x , ω ) L i ( ω ) d ω , where L e denotes light emission and l ∈ L denotes all light sources.In this application, we use light sampling instead of random raysto ensure that the light source is always sampled. Our unbiasedreweighting achieves the best convergence and, as shown in theinsets, also the smoothest results (Fig. 8). Observation.
In all cases, our solution leads to smoother visualresult and less black holes in the falloff regions. From the MSEplots, we can see that standard MC with uniform sampling has theworst performance, while our weighted stratified sampling generally has the best one. Our method improves both uniform sampling andstratified sampling. We can also see that even with uniform samplingas input, our weighted uniform sampling not only improves overthe unweighted version, but also has a performance that is as goodas our weighted stratified sampling. Precompute sample weightsenables a negligible computation overhead.
The reweighting scheme in this paper enables a better approxima-tion than standard MC weights. Our solution is general and doesnot require any prior knowledge about the integrating function.Implicitly, our method approximates this function via a reconstruc-tion from the samples, but does not introduce a bias in the resultingestimator. We showed its practical benefit for various renderingproblems. , Vol. 1, No. 1, Article 1. Publication date: January 2016. eometric Sample Reweighting • 1:7
Fig. 8. Each image uses primary rays per pixel and at each scattering event light samples. Our solution is applied to the samples. While we focus on primary samples that are either i.i.d. uniform orstratified in this work, our method can also handle non-uniform sam-ple sets following a distribution of p ( x ) . The expected position of the i -th sample x i is then N · (cid:0) N − i − (cid:1) · ∫ P − ( x )· x i − ( − x ) N − i d x , where P ( u ) = ∫ p ( u ) d u . Unfortunately, it is necessary to integrate the dis-tribution function. Approximate schemes remain an area of futurework. Similarly, using the method in higher dimensions requiresthe computation of cell volumes in high-dimensional Voronoi dia-grames, which can be costly. One could precompute these weightsbut we left such accelerations as future work. Finally, it is an excit-ing opportunity to exploit the generality of our solution to improveother integration problems. REFERENCES
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