journal of la adaptive irradiance sampling for many-light

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1

Adaptive Irradiance Sampling for Many-LightRendering of Subsurface Scattering

Kosuke Nabata, Kei Iwasaki

Abstract—Rendering a translucent material involves integrating the product of the transmittance-weighted irradiance and the BSSRDFover the surface of it. In previous methods, this spatial integral was computed by creating a dense distribution of discrete points overthe surface or by importance-sampling based on the BSSRDF. Both of these approaches necessitate specifying the number ofsamples, which affects both the quality and the computation time for rendering. An insufficient number of samples leads to noise andartifacts in the rendered image and an excessive number results in a prohibitively long rendering time. In this paper, we propose anerror estimation method for translucent materials in a many-light rendering framework. Our adaptive sampling can automaticallydetermine the number of samples so that the estimated relative error of each pixel intensity is less than a user-specified threshold. Wealso propose an efficient method to generate the sampling points that make large contributions to the pixel intensity taking into accountthe BSSRDF. This enables us to use a simple uniform sampling, instead of costly importance sampling based on the BSSRDF. Theexperimental results show that our method can accurately estimate the error. In addition, in comparison with the previous methods, oursampling method achieves better estimation accuracy in equal-time.

Index Terms—subsurface scattering, BSSRDF, adaptive sampling, many-lights.

F

1 INTRODUCTION

T RANSLUCENT materials such as marble, skin, wax, andso forth, exhibit characteristic soft appearance. This

characteristic appearance comes from subsurface scatteringeffects that the incident light enters the surface of thetranslucent material, scatters multiple times inside it, andexits from different locations on the surface. The bidirec-tional scattering surface reflectance distribution function(BSSRDF) models the subsurface light transport betweenthe point where the light enters and that where the lightexits. Rendering translucent materials using BSSRDF in-volves the costly double integral, namely, the integral ofthe incident radiance over the hemisphere to compute the(transmittance-weighted) irradiance, and the spatial integralover the surface of the translucent material.

Scalable subsurface rendering methods [1], [2], [3] dis-cretize the incident lighting into a huge number of virtualpoint lights (VPLs), and discretize the surface of the translu-cent material into sample points (we refer to these pointsas irradiance points). Then VPLs and irradiance points areclustered by constructing hierarchical structures of VPLsand irradiance points, to compute the double integral ef-ficiently. Though efficient and practical, previous scalablerendering methods can suffer from errors due to clustering.Although these methods cluster VPLs and irradiance pointsso that the error of each cluster is bounded by a user-specifiedthreshold, this does not guarantee that the total error fromall the clusters is bounded by the threshold. As such, carefulparameter tuning and re-renderings are imposed on the userto obtain rendering results with reliable accuracy.

One approach can control the total error from all the

• K. Nabata and K. Iwasaki are with the Department of Systems Engineer-ing, Wakayama University, Wakayama, Japan. K. Iwasaki is also withPrometech CG Research, Tokyo, Japan.E-mail: [email protected]

clusters for scalable subsurface rendering [4]. This method,however, shares the same problem of previous scalableapproaches [1], [2], [3]. That is, the number of irradiancepoints is specified by the user before the rendering, thoughfinding an appropriate number of irradiance points is adifficult task. An inappropriate number of irradiance pointsleads to noise and artifacts in the rendered images (Fig. 1(a))despite the error estimation, since the true value, whichis estimated using all VPLs and irradiance points, wouldbe inaccurate. Using a large number of irradiance pointswould reduce the noise and artifacts, but it also increasesthe computation time (Fig. 1(b)). Importance-sampling theirradiance points based on the BSSRDF compounds thisproblem, since it involves costly ray-triangle intersectionsto generate the irradiance points, which accounts for mostof the computation time.

To address these problems, we propose an adaptivesampling method of the irradiance points for the spatial in-tegration of subsurface scattering. Unlike previous scalableapproaches [1], [2], [3], [4] that prepare the irradiance pointsprior to the spatial integration computation, our methodsamples points on-the-fly so that the relative error of the esti-mated pixel intensity is smaller than a threshold based on aconfidence interval [5]. To do so, our method constructs thehierarchical structures for the triangular meshes of translu-cent materials, instead of pre-sampled irradiance points.Then irradiance points to compute the spatial integral aresampled uniformly from the triangles. This enables us toincrease the number of irradiance points by subdividing thetriangles, which does not enforce on rebuilding the entirehierarchical structure. Moreover, we develop an efficientmethod for partitioning the triangular meshes taking intoaccount the BSSRDF. This makes it possible to generateirradiance points that make large contributions to the pixelintensity, without requiring costly importance sampling.


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( )

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Fig. 1. (a) Using an inappropriate number of irradiance points leads tonoise as shown in the insets in the red frames. To reduce noise, tweakingthe number of irradiance points and re-renderings is often inevitable.Therefore, the total computation time needed to obtain the renderedresults without noticeable noise might become large at all. (b) Using alarge number of irradiance points wastes computation time in generatingthe irradiance points. (c) Our method automatically determines the num-ber of irradiance points. Our method can yield rendering results withouttrial-and-error processes and re-renderings.

Thus our method achieves a significant speedup (up to12x) compared to the previous methods [1], [4] in equal-quality rendering. While we introduce our sampling methodin a many-light rendering framework, we also discuss thepossibility of adapting our sampling method to path tracingin Sec. 5.4.

2 RELATED WORK

In this section, we mainly focus on reviewing the previouswork relevant to ours: many-light rendering and a hierarchi-cal approach for the dipole model [6]. Although our methodwould work for other BSSRDF models if the upper boundof the BSSRDF between the points where the incident lightenters and those where the scattered light exits can be easilyobtained, incorporating sophisticated BSSRDF models (e.g.,[7], [8], [9], [10], [11]) into our method is something plannedfor future work.

2.1 Many-light rendering

Many-light rendering methods approximate the computa-tion of complex illumination by the computation of direct il-lumination from a set of virtual point lights (VPLs) [12], [13],[14]. Since many-light rendering often uses a huge numberof VPLs to render realistic images, scalable approaches thatcluster similar VPLs are widely used. Lightcuts [15] andits variants [1], [16], [17] are a scalable solution to many-light methods using a hierarchical representation of VPLs.These methods cluster VPLs so that the error in each clusteris bounded by a user-specified error threshold. Anotherscalable approach is formulating many-light rendering asa matrix sampling problem [18], [19], [20], [21]. Althoughthese methods can significantly accelerate many-light ren-dering, they do not estimate the total error from all the clusters,which can emerge as noise and artifacts in the renderedimages. As such, to obtain rendered images with reliableaccuracy, these methods impose parameter tunings and re-renderings on the user. To address this problem, Nabata etal. proposed an error estimation framework for many-lightrendering [22]. Although this method can be used to control

the total error, it does not handle subsurface scattering oflight.

Instead of clustering, a vast set of sensor/light pairs hasbeen stochastically evaluated in the context of bidirectionalpath tracing [23], [24]. These methods, however, do not dealwith subsurface rendering.

2.2 Spatial integration for subsurface rendeirngPrevious methods for computing the spatial integral can becategorized into two groups: preparing a dense distributionof irradiance points over the entire surface of a translucentmaterial before rendering (referred to as point cloud methods),and sampling the irradiance points based on the BSSRDF(referred to as importance sampling methods).

2.2.1 Point cloud methodsPoint cloud methods densely distribute the irradiancepoints across the surface of a translucent material before ren-dering, and reuse the irradiance for all the pixels. Jensen andBuhler proposed a hierarchical two-pass approach to rendertranslucent materials [25]. Arbree et al. proposed a single-pass approach using VPLs [2]. This method clusters thetriples of VPLs, points to be shaded, and irradiance points.Wu et al. formulated calculation of the radiance for subsur-face scattering as a matrix sampling problem [3]. Recently,Milaenen et al. proposed a dual hierarchy method combinedwith frequency analysis of the BSSRDF for efficient compu-tation of the integral [26]. Although the point cloud methodscan amortize the cost to compute the irradiance by reusingit, these methods require additional memory to store theirradiance points, which can be prohibitive especially foroptically dense translucent materials (i.e., large extinctioncoefficient), since the point cloud methods often use themean free-path (i.e., the reciprocal of the extinction coeffi-cient) of the translucent material as the maximum distancebetween the irradiance points. In addition, when only asmall fraction of the surface of a translucent material isrendered, densely distributing the irradiance points acrossthe entire surface can be inefficient.

2.2.2 Importance sampling methodsImportance-sampling of the irradiance points based on theBSSRDF addresses the problems of the point cloud methods.By exploiting the characteristics of the BSSRDF that decaysexponentially, importance sampling methods [4], [16] gen-erate the irradiance points in the proximity of a point to beshaded. Walter et al. incorporate the importance samplingof the BSSRDF into the VPL-based methods (i.e., Light-cuts) [16]. Although this method can render the translucentmaterials efficiently by retaining the merits of Lightcuts,this method also inherits the limitation of Lightcuts (i.e.,the total error from all the clusters is not bounded). Sakaiet al. extended the error estimation method [22] to rendertranslucent materials [4]. Although the total error from allthe clusters can be controlled using this method, it inheritsthe problems of importance sampling methods. That is, aninappropriate number of importance-sampled points leadsto noise and artifacts in the rendered images despite estimat-ing the errors, and the computational cost for importance-sampling of the irradiance points is high since it involvesthe costly ray-triangle intersections.


Our sampling method retains the merits of the impor-tance sampling methods over the point cloud methods (i.e.,avoiding wasteful sampling and the additional memory),but bypasses the costly ray-traingle intersections to samplethe irradiance points. More importantly, both point cloudmethods and importance sampling methods lack the abilityto refine the irradiance points so as to improve the imagequality. Since both methods represent the irradiance pointswith a hierarchical structure, refining the irradiance pointsinvolves rebuilding the hierarchical structure. Our methodadaptively and automatically samples the irradiance pointsso that the estimated relative error is less than the user-specified threshold.

3 BACKGROUND

3.1 Subsurface Rendering with a Diffuse BSSRDFThe BSSRDF is used to model subsurface scattering of lightin rendering translucent materials. The outgoing radiance Lat point xo in direction ωo is calculated by integrating theproduct of the BSSRDF, S, and the incident radiance, L, atpoint xi over the surface A of the translucent material andthe hemisphere Ω as:

L(xo,ωo) =∫A

∫ΩL(xi,ωi)S(xi,ωi,xo,ωo)(ni · ωi)dωidA(xi),

where ni is the normal at point xi. The BSSRDF S can berepresented by the sum of two terms, a single scatteringterm S1 and a multiple scattering term Sd. The multiplescattering term Sd is efficiently calculated from the diffuseBSSRDF Rd [6]. Our method mainly focuses on calculatingthe outgoing radiance Ld due to the diffuse BSSRDF. Ld isgiven by:

Ld(xo,ωo) =

Ft(ωo)

π

∫ARd(ri)

(∫ΩL(xi,ωi)Ft(ωi)(ni · ωi)dωi

)dA(xi),

where Ft is the Fresnel transmittance and ri = ‖xi − xo‖.Hereinafter, the point xo used in computing the outgoingradiance Ld is referred to as the shading point. Since theoutgoing direction ωo is uniquely determined by the shad-ing point xo and the camera position, we drop ωo fromLd(xo,ωo) and the Fresnel transmittance Ft(ωo) is replacedwith Ft(xo) for the notation clarity.

3.2 Many-light Methods for Subsurface RenderingMany-light methods represent the incident radianceL(xi,ωi) at the irradiance point xi using virtual point lights(VPLs). By using a set of VPLs L, the outgoing radiance Ld

is calculated by means of:

Ld(xo) ≈Ft(xo)

π

∫ARd(ri)

∑y∈L

I(y)H(y,xi)

dA(xi),

where I(y) is the intensity of the VPL y. H(y,xi), whichencompasses the geometry term G, the visibility functionV , and the Fresnel transmittance Ft at the irradiance pointxi, is given by:

H(y,xi) = G(y,xi)V (y,xi)Ft (ωyxi) ,

where ωyxi is the unit direction from the irradiance point xito the VPL y as ωyxi = (y − xi)/‖y − xi‖.

The pixel intensity for subsurface scattering of light iscalculated by integrating the product of the importancefunction W and Ld over the pixel footprint P .To calculatethe integral over P , the pixel footprint P is discretized intoa set of points, S, using supersampling. The pixel intensityI with supersampling is calculated as:

I =∑xo∈S

∫Aw(xo)Rd(ri)

∑y∈L

I(y)H(xi,y)

dA(xi), (1)

where w(xo) is the weighting function that encompassesthe importance function W and the Fresnel transmittanceFt and is given by:

w(xo) =W (xo)Ft(xo)

πp(xo)|S|, (2)

where p is the sampling pdf of xo.

3.3 Error Estimation using a Confidence Interval

To make an estimate of some quantity (e.g., pixel intensityI) using sampling, it is beneficial to assess the accuracy ofthe estimate, i.e., the error |I − I|, where I and I are theestimator and the true value, respectively. The confidenceinterval

[I −∆I, I + ∆I

]contains the true value I with a

certain probability α as:

Pr(|I − I| ≤ ∆I) = α, (3)

where α is referred to as the confidence level, and ∆I iscalculated by:

∆I = tα

√s2

n,

where tα is the α quantile of the Student t-distribution t(x)that satisfies

∫ tα−tα t(x)dx = α, s2 is the sample variance of

the estimator I , and n is the number of samples. Note that∆I can be calculated by using only the sample varianceand tα without evaluating the true value I . By specifyingthe confidence level α and the relative error threshold ε, theerror |I − I| is bounded by ∆I < εI with probability α.

In the context of scalable many-light rendering, thesampling domain is partitioned into subdomains (clusters).By partitioning the sampling domain into J clusters, ∆I isrewritten as [22]:

∆I = tα

√√√√ N

N − J

J∑j=1

s2j

nj, (4)

where s2j and nj are the sample variance and the number

of samples of the j-th cluster. N is the total number ofsamples N =

∑Ji=1 nj , nj is allocated to j-th cluster based

on Neymann allocation, which is the optimal method forallocating samples to each cluster with a fixed total numberof samples N .


Fig. 2. Overview of our method. (a) shows the triplet (CL,CI ,CS). Unlike previous scalable approaches that cluster irradiance points, our methodclusters the triangular mesh for a translucent material. (b) shows the estimate of the pixel intensity I1 by sampling a VPL, irradiance point, andshading point from the triplet. To sample irradiance points from CI , we first sample a triangle and then an irradiance point is sampled from it.(c) shows the subdivision of the triplet (CL,CI ,CS) into two triplets (CL,CI

l ,CS) and (CL,CI

r ,CS) by subdividing IC CI into CIl and CI

r . (d)when IC that contains one triangle is selected to be subdivided, our method subdivides the triangle into two triangles, and two ICs that contain eachsubdivided triangle are created. This enables us to generate irradiance points adaptively in the proximity of shading points, without the reconstructionof BVH (just replacing the leaf node with an inner node with two child nodes).

4 OUR APPROACH

4.1 Algorithm Overview

Our goal is to estimate the intensity I of each pixel dueto subsurface scattering of light by adaptively sampling theirradiance points so that the relative error of the estimateI is less than a user specified threshold ε. To this end, ourmethod builds upon the previous clustering based approachproposed by Arbree et al. [2]. Our method estimates thepixel intensity in Eq. (1) by partitioning the integral domainA, the set of VPLs L, and that of the shading points S intoclusters. Each cluster is labeled either as an IC (irradiancecluster) CI , a LC (light cluster) CL, or a SC (shading cluster) CS ,respectively. The combination of three clusters (CL,CI ,CS)is referred to as triplet. The pixel intensity I is estimated bysampling irradiance points, VPLs, and shading points fromeach cluster of the triplets.

The technical novelties of our approach are twofold.Firstly, our method adaptively subdivides the integral do-main A of the surface using triangular meshes instead ofusing pre-sampled points, as in the previous methods [2],[4]. This enables us to generate irradiance points on-the-flyand determine the number of the irradiance points automat-ically, freeing the user from specifying this. Secondly, weintroduce an efficient method to partition the entire domain(L × A × S in Eq. (1)) into the triplets taking into accountthe BSSRDF. In contrast to the previous methods [4], inwhich a cluster with the maximum extent is selected forsubdivision, our method selects a cluster so as to reducethe variance of the estimate I . This enables us to sampleirradiance points with large contributions without excessivesubdivision of the surface or requiring costly importancesampling, resulting in a significant speedup.

Fig. 2 presents an overview of our method. ICs arerepresented by a bounding volume hierarchy (BVH) oftriangles. The irradiance points are directly sampled fromthe triangles. The LCs and SCs are represented by binarytrees similar to Multidimensional Lightcuts [1]. The binarytree for the LC and the BVHs for translucent materials areprepared and shared for all the pixels. For each pixel, wefirst generate shading points on the pixel footprint usingsupersampling and cluster shading points using a binarytree.

We start from the triplet comprising each root node(Fig. 2(a)). The estimate of the pixel intensity I is calculatedby sampling VPLs, irradiance points, and shading pointsfrom the triplet. The sample variance is also calculatedto estimate the error ∆I (Sec. 4.2.2). Then the followingprocesses are repeated:

1) Select the triplet (CLj ,CIj ,CSj ) with the largest errorbound ej (Sec. 4.2.3).

2) Select one cluster out of the three clusters CLj ,CIj ,CSj(Sec. 4.3) and subdivide it. Then the triplet is subdi-vided into two triplets (Fig. 2(c)).

3) For the subdivided two triplets, the estimates of thepixel intensities as well as the sample variance arecalculated.

4) Update the estimated pixel intensity I and the error∆I using the sample variance.

5) Terminate if the estimated error ∆I is less than εI .Return to Step 1, otherwise.

4.2 Error estimation for the pixel intensity I

The pixel intensity I in Eq. (1) is calculated by partitioningthe entire domain into each cluster. The pixel intensity Ijdue to the j-th triplet (CLj ,CIj ,CSj ) is calculated by simplyreplacing the entire domain in Eq. (1) with each cluster as:

Ij =∑

xo∈CSj

w(xo)

∫Aj

Rd(ri)

∑y∈CLj

I(y)H(y,xi)

dA(xi),

where Aj is the sum of the areas of the triangles in IC CIj .To estimate Ij , our method samples irradiance points,

VPLs, and shading points as:

Ij =1

nj

nj∑k=1

w(xo,k)Rd(‖xi,k − xo,k‖)I(yk)H(yk,xi,k)

pI(xi,k)pS(xo,k)pL(yk),

(5)

where xi,k, xo,k and yk are the k-th samples of irradiancepoints, shading points, and VPLs, respectively. VPLs andshading points are sampled proportional to their intensities


Fig. 3. Computation of ∆L, ∆I , and ∆S . (a) our method selects one cluster from three clusters CL,CI , and CS so that the subdivision of selectedcluster can reduce the variance of RdF most. To do so, our method uses the range of Rd/d

2 as the criterion (i.e., we consider the cluster withthe largest range of Rd/d

2 as the largest variance). (b) computation of ∆L. The minimum/maximum distances between CL and CI (fixed) aredc − rI − rL and dc − rI + rL. Since CI and CS are fixed in computing ∆L, the mimimum/maximum distances between CI and CS are fixed asrc − rI − rS . ∆I and ∆S can be computed in the similar way as shown in (c) and (d).

or the weighting function values by using the followingprobability mass functions:

pL(yk) =I(yk)∑

y∈CLjI(y)

=I(yk)

Ij,

pS(xo,k) =w(xo,k)∑

xo∈CSjw(xo)

=w(xo,k)

Wj,

where Wj and Ij are the sums of the weighting functionvalues and the intensities over the clusters CSj and CLj ,respectively. Substituting these functions into Eq. (5) leadsto the following equation:

Ij =WjIjnj

nj∑k=1

Rd(‖xi,k − xo,k‖)H(yk,xi,k)

pI(xi,k). (6)

4.2.1 Sampling irradiance pointsOur method samples nj irradiance points uniformly fromthe cluster CIj . If the cluster CIj contains Tj trianglest1, . . . , tTj with areas a1, . . . , aTj, we first sample atriangle in proportion to its area, then an irradiance point isuniformly sampled from the triangle. That is, the probabilitydensity function (pdf) of the sample irradiance points is:

pI(xi,k) =a∑Tjl=1 al

· 1

a=

1∑Tjl=1 al

=1

Aj, (7)

where Aj is the sum of the areas of the triangles in CIj .

4.2.2 Estimate of I and ∆IBy substituting the pdf pI(xi,k) into Eq. (6), the pixel inten-sity Ij due to the j-th triplet is calculated using:

Ij =AjWjIjnj

nj∑k=1

Rd(‖xi,k − xo,k‖)H(yk,xi,k). (8)

The estimate of the pixel intensity I is calculated by sum-ming Ij over all the triplets as I =

∑Jj=1 Ij where J is

the number of all triplets. Following a previous method forestimating errors [22], our method uses nj = 2 samples(thus the total number of samples N =

∑Jj=1 nj = 2J). The

estimate of the error ∆I = |I − I| in Eq. (4) is simplified to:

∆I = tα

√√√√ 2J

2J − J

J∑j=1

s2j

2= tα

√√√√ J∑j=1

s2j , (9)

where tα is the α quantile of the t-distribution and s2j is the

sample variance of Ij for the j-th triplet as:

s2j =

A2jW

2j I

2j

2(Rd(r1)H(y1,xi,1)−Rd(r2)H(y2,xi,2))2,

where r1 = ‖xi,1−xo,1‖ and r2 = ‖xi,2−xo,2‖, respectively.

4.2.3 Triplet selection using the error boundTo reduce the error ∆I , our method selects the triplet withthe maximum error bound and subdivides it. Although thestandard deviation of each triplet can be used as the errorbound, the computational cost of the standard deviation ishigh since it involves costly evaluations of RdH in Eq. (8)over all VPLs, irradiance points, and shading points in thetriplet. Instead, our method calculates the error bound ejfor the j-th triplet as follows (the derivation is shown inAppendix A.):

ej =AjWjIj2√nj

RudGuFut ,

where Rud , Gu, and Fut are the upper bounds of the diffuseBSSRDF Rd, the geometry term G, and the Fresnel termFt in the triplet, respectively. The details for computing theupper bounds are shown in Appendix A.

4.3 BSSRDF-aware Cluster SelectionOnce the triplet (CLj ,CIj ,CSj ) with the maximum errorbound is chosen, our method selects one cluster to subdividethe triplet. Previous scalable approaches [1], [4] basically usethe diagonal lengths of the bounding boxes of the clustersas the criterion for selecting the cluster. This, however, doesnot work in our method, since we use uniform samplingover the surface of the translucent material for generatingirradiance points, and irradiance points with large contribu-tions to the shading points are rarely sampled until all theclusters are sufficiently subdivided.

Instead, we propose a BSSRDF-aware cluster selectionmethod. As shown in Eq. (8), the variance of Ij stems fromthe diffuse BSSRDF Rd between CIj and CSj , and H betweenCLj and CIj , since AjWjIj is constant in the triplet. In theproduct RdH , the diffuse BSSRDF Rd decreases exponen-tially with respect to the distance r between the shadingpoint and the irradiance point, and the geometry term Gdecreases in inverse proportion to the square of the distance


d between the VPL and the irradiance point. As such, ourBSSRDF-aware cluster selection uses Rd/d2 as the criterion.That is, our method selects the cluster with the maximumrange of Rd/d2 from CLj ,CIj , and CSj . Specifically, we definethe range of Rd/d2 for each cluster c ∈ L, I, S when theother two clusters are fixed, ∆c to be:

∆c =Rd(rmin)

d2min

− Rd(rmax)

d2max

, (10)

where rmax and rmin are the maximum and minimumdistances between the irradiance points in CIj and theshading points in CSj . dmax and dmin are the maximumand minimum distances between the VPLs in CLj and theirradiance points in CIj , respectively.

To simplify the computation of the maximum and min-imum distances between two clusters, our method uses thebounding sphere of each cluster in calculating ∆ as shownin Fig. 3.

∆L =Rd(rmin)

d2min

− Rd(rc − rI − rS)

(dc − rI + rL)2, (11)

∆I =Rd(rmin)

d2min

− Rd(rc − rS + rI)

(dc − rL + rI)2, (12)

∆S =Rd(rmin)

d2min

− Rd(rc − rI + rS)

(dc − rL − rI)2, (13)

where dc is the distance between the centers of the boundingspheres of CLi and CIi , and rc is that of CIi and CSi . rL,rI , and rS are the radii of the bounding spheres of CLj ,CIj , and CSj , respectively. Since ∆ has a common upperbound Rd(rmin)/d2

min, our method only calculates eachlower bound Rd(rmax)/d2

max and selects the cluster withthe minimum lower bound to be subdivided.

LC and SC are subdivided by traversing the correspond-ing binary trees. That is, the cluster corresponding to aninner node is replaced with two clusters corresponding tothe child nodes in the binary tree. IC is subdivided in thesame way by traversing the BVH for triangular meshesof the translucent material. When IC containing a singletriangle is selected to subdivide, our method subdivides thetriangle into two triangles by halving the longest edge andgenerates two ICs containing each subdivided triangle.

5 RESULTS AND DISCUSSION

In this section, we compare our method with Multidimen-sional Lightcuts (MDLC) [1] that is combined with impor-tance sampling of the BSSRDF [16] to handle subsurfacerendering, and the error estimation method for Lightcuts [4](we refer to this as EELC) in terms of the estimation accuracyand the computational efficiency. The estimation accuracy ismeasured by the ratio of pixels whose relative errors areless than ε. To measure the estimation accuracy, our methoduses only the pixels that cover the translucent materials. Inall images, the relative error threshold ε was set to 2%, andthe confidence level α for Eq. (9) was set to 95%. That is,the estimation is accurate when the ratio is close to α (i.e.,95% in our experiments). The reference image is renderedby our method with the relative error threshold parameterε = 0.1%. We also measured the root mean square error(RMSE) to assess the image quality.

5.1 Experimental setup

We tested our method on four scenes. The BSSRDF param-eters are appropriately scaled from the measured parame-ters [6]. The first scene contains a single Buddha model withSkin material parameter. The second scene contains twoBuddha models with Skim Milk and Marble parameters.The third scene has a human head model with texture.The BSSRDF parameters for the human head model aremanually selected. In the Salle de bain scene, the walls, thewashstand, and the bathtub are made of marble. The bound-ary surfaces of the translucent objects except for the humanhead model are specular materials. The image resolution forthe Salle de bain scene is 720×480, and that for other scenesis 512× 512. All images are rendered on a PC with an IntelCore i9-7980XE CPU.

At each pixel, 16 rays are traced for supersampling andthe intersection points are gathered as shading points. Ifa ray intersects a specular material, the intersection pointis gathered as the shading point, and the reflected ray isrecursively traced.

Both MDLC [1] and EELC [4] necessitate the user tospecify the number of irradiance points. Unless otherwisestated, the number of irradiance points per shading point,4096, is chosen for EELC in all scenes so that the estimationaccuracy of EELC comes close to that of our method. Thesame number is used in MDLC. For equal-time comparisonsbetween EELC and our method, the numbers of irradiancepoints used in EELC are manually selected for each sceneafter several trial-and-error processes. The relative errorthreshold ε is set to 2% for both MDLC and EELC, and theconfidence level α is set to 95% for EELC for comparisons.

5.2 Comparison to MDLC, EELC, and Path Tracing

Fig. 4 shows comparisons between MDLC (second column),EELC (third and fourth columns), our method (fifth col-umn), and the reference (sixth column). As shown in Fig. 4and Table 1, our estimation accuracy is very close to α inthese scenes, indicating that our method can control theerror accurately. The images rendered by our method areindistinguishable from the reference images as shown inFig. 4. As shown in the second column (MDLC) in Fig. 4,the estimation accuracy with MDLC is low compared to ourmethod and noticeable noise can be seen even for a largenumber of irradiance points, since MDLC fails to bound thetotal error from all the cluster triplets.

In an equal-time comparison to EELC (the third columnin Fig. 4), the estimation accuracy of our method is up to32.6 percentage points better than that of EELC in all scenes.Although EELC is capable of controlling the total error fromthe true value of the pixel intensity I , the true value itselfis not accurate when a small number of irradiance pointsis used. Therefore, noticeable noise can be seen especiallyin the Two Buddhas scene and the Head scene. The Salle debain scene, which contains vast areas of translucent material(e.g., the walls), is a challenging scene for our method sincesampling irradiance points with large contributions fromvast areas requires a large number of partitions, resultingin prohibitively long computation times. Nevertheless, theestimation accuracy of our method (94.8%) is better than


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Fig. 4. Comparison between Multidimensional Lightcuts (MDLC) [1], Error Estimation for Lightcuts (EELC) [4], and our method. From left to right,the reference image (first), MDLC (second), EELC (third and fourth, where the render time and the estimation accuracy are set to be equal to thoseof our method by adjusting the number of the irradiance points in EELC), our method (fifth), and the reference (sixth). The rightmost images arethe visualiation of the relative errors with false color. The percentage shown under each image indicates the estimation accuracy, which is the ratioof pixels whose relative errors are less than ε. The RMSEs and the render times are also shown. To compare the render times, we set EELC ofequal-quality (with 4096 irradiance points per shading point) to be the baseline method (underlined). The numbers in the parenthesis show thespeedups of each method compared to the baseline. The intensities of the insets are scaled for better clarity. Perceptible noise can be seen inimages rendered by MDLC and EELC in equal-time, while our method does not provide perceptible noise thanks to our error estimation method. Inequal-euqlity comparison, our method achieves up to 12.4x speedup.


TABLE 1Statistics of the numbers of VPLs and the triangles used for four scenes, the render times, the estimation accuracy, and RMSEs for each method.Numbers with # indicate the number of the irradiance points per shading point used for the importance sampling in MDLC [1] and EELC [4]. Thebold numbers indicate the best performance (highest estimation accuracy, lowest RMSEs and render times). In all scenes, our method achieves

high estimation accuracy very close to the confidence level α = 95% compared to MDLC and EELC in equal-time. Moreover, our method achievesa significant speedup compared to MDLC and EELC in equal-quality.

Scene Num. of Render times / estimation accuracy / RMSE (×10−3)VPL tri. our method MDLC EELC equal-time EELC equal-quality

(#4096) (#4096)Buddha 0.998M 1.09M 455s 94.8% 3.99 2203s 40.7% 16.5 456s (#296) 81.6% 7.64 2442s 93.2% 6.79

Two Buddhas 1.037M 2.17M 1229s 94.8% 5.64 9158s 34.0% 22.7 1229s (#449) 62.2% 12.0 9658s 90.5% 7.36Head 0.309M 0.28M 180s 94.9% 5.85 2203s 71.5% 17.3 180s (#323) 73.2% 14.5 2225s 92.5% 12.0

Salle de bain 1.018M 1.22M 2398s 94.8% 2.92 3051s 52.7% 9.53 2403s (#3029) 92.1% 3.61 3184s 92.7% 3.53

TABLE 2Computation times for the importance sampling in EELC [4] with 4096

samples per shading point. Since the importance sampling ofBSSRDF [27] involves costly ray-triangle intersections, the computation

times dominate about 90% of the render times in all scenes.

Scene time for render time occupancy (%)importance-sampling

Buddha 2155s 2442s 88.2%Two Buddhas 8985s 9658s 93.0%

Head 2145s 2225s 96.4%Salle de bain 2897s 3184s 91.0%

that of EELC (92.1%) in equal time, due to our BSSRDF-aware cluster selection method discussed in Sec. 5.3.

In an equal-quality comparison with EELC, 4096 irra-diance points per shading point are required for EELCto achieve a high estimation accuracy, but it also incursextensive computation times for importance-sampling. Asshown in the fifth column of Fig. 4, our method achieves 1.3xto 12.4x speedups compared to EELC with equal-quality.Since importance sampling of the BSSRDF [16], [27] involvescostly ray-triangle intersections, generating a large numberof irradiance points to obtain accurate results can cause abottleneck to the rendering. Table 2 shows the computationtimes for importance sampling of the irradiance points usedin EELC (please note that the computational time for theimportance sampling in EELC is same as that in MDLC sincethe importance sampling process is common in EELC andMDLC). As shown in Table 2, the computation times forthe importance sampling account for about 90% in all fourscenes. On the other hand, our adaptive sampling accountsfor 9.17% (Buddha), 8.32% (Two Buddhas), 19.3% (Head),and 25.2% (Salle de bain) of each render time, respectively.Since our method samples the irradiance points uniformlyon the triangles with a simple pdf, our method efficientlygenerates a large number of irradiance points.

The rightmost images in Fig. 4 visualize the relativeerrors in false color for each method (from left to right, topto bottom, MDLC, EELC in equal time, EELC with equal-quality, and our method). As shown in these images, ourmethod consistently controls the relative errors to be lessthan the threshold ε.

Table 1 lists the statistics for each scene (i.e., the numbersof VPLs and triangles), the estimation accuracy, the rootmean square error (RMSE), and the computation time foreach method. The number of irradiance points used forequal-time comparison with EELC in each scene is alsolisted. As shown in Table 1, our method yields both the best

estimation accuracy and the smallest RMSEs compared toMDLC and EELC.

Fig. 5. Equal-time comparison between our method (left, 180s) and pathtracing (right, 178s). As shown in the bottom insets, the noticeable noisecan be seen in path tracing, while our method does not yield perceptiblenoise due to our error estimation.

Fig. 5 shows an equal-time comparison for the Headscene between our method and path tracing based on BSS-RDF importance sampling [27]. Although path tracing isgood at handling such a scene (i.e., diffuse BSSRDF materialand a large light source represented by an environmentmap), the rendering results by path tracing produce notice-able noise as shown in the insets at the bottom Fig. 5. Onthe other hand, our method does not yield noticeable noisesince our method can accurately control the error.

Fig. 6 shows convergence plots of the RMSE for MDLC,EELC, and our method by changing the relative errorthreshold parameter ε. In Fig. 6, we include the convergenceplots for MDLC and EELC using uniform sampling ofthe irradiance points, which are explicitly referred to asMDLC(US) and EELC(US) to distinguish them from thoseusing importance sampling. Both MDLC(US) and EELC(US)with 4096 irradiance points per shading point are fast butprovide large errors. Increasing the number of irradiancepoints per shading point up to 65536 (i.e., at least 1M irra-diance points per pixel) does not perform well compared toMDLC and EELC. For MDLC and EELC, since the compu-tation time for importance sampling dominates the rendertime, the change in the relative error threshold parameterε has little impact on the render time when ε ≥ 0.5%. Asshown in Fig. 6, the RMSEs using our method are smaller


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Fig. 6. Convergence plots of RMSEs for our method (dark blue solid line), EELC with importance sampling (sky blue solid line) and uniform sampling(light green/light purple solid lines), and MDLC with importance sampling (dark orange dashed line) and uniform sampling (light gray/light orangedashed lines) on the log-log scale. RMSEs and the render times are measured by changing the relative error threshold ε. The numbers after ”US”(i.e., 4096 and 65536) indicate the number of the irradiance points per shading point used in uniform sampling. Both MDLC(US) and EELC(US) donot perform well compared to MDLC, EELC, and our method. While the error reduction rates in EELC and MDLC begin to deteriorate when tighterror thresholds (ε < 0.5% for EELC and ε < 0.05% for MDLC) are used, our method reduces RMSEs steadily thanks to our adaptive sampling.

than those for EELC and MDLC in all scenes except for theSalle de bain scene. In the Salle de bain scene, RMSEs ofour method with ε = 1%, EELC with ε = 0.5%, and MDLCwith ε = 0.05% are similar (0.001476, 0.00144, and 0.001582,respectively), but the render times for EELC with ε = 0.5%(4007s) and MDLC with ε = 0.05% (3659s) are smaller thanthat of our emthod with ε = 1% (6622s). However, the errorreduction rates in EELC and MDLC drop abruptly whenε < 0.5% for EELC and ε < 0.05% for MDLC as shown inFig. 6. On the other hand, our method yields stable errorreduction and better convergence compared to EELC. Wealso discuss the similar behavior of the estimation accuracyin Sec. 5.4.3.

5.3 Cluster Selection ComparisonWe investigate the effect of our BSSRDF-aware cluster se-lection method on the rendering performance. We compareour selection method with the previous method that se-lects the cluster with the maximum extent (i.e., with thelargest diagonal length of the bounding box) from threeclusters of a triplet. The previous cluster selection methodis incorporated into our uniform sampling of the trianglesand the irradiance points. We refer to the previous clusterselection method used in MDLC and EELC as Max-Extent.Fig. 7 visualizes the number of triplets J used to control therelative error to less than the threshold ε = 2%. The toprow visualizes the number of triplets partitioned by usingour BSSRDF-aware cluster selection method. In all scenes,the shadow regions (e.g., under the jaw and around thefoot of the Buddha model, the corner of the walls, and theshelf under the window in the Salle de bain scene) havemany triplets that satisfy the termination criterion ∆I < εI ,since the tolerance εI is small in the shadow regions. Thisproperty is common to MDLC and EELC, and we leave thesolution to this for future work. The bottom row in Fig. 7visualizes the number of triplets partitioned by using Max-Extent. Compared to the top row in Fig. 7, the numberof triplets in our method is much smaller (1.79x to 7.65x)than that for Max-Extent to satisfy the termination criterion∆I < εI .

Table 3 shows comparisons between the render timesfor our method and Max-Extent. The estimation accuracyand RMSEs for Max-Extent are also shown in Table 3.Although Max-Extent can achieve high estimation accuracyand RMSEs similar to ours, the render times also increase

TABLE 3Statistics of the render times, the average numbers of the triplets, the

estimation accuracy, and RMSEs for the previous cluster selectionmethod (Max-Extent). The numbers in the parenthesis show the ratio ofthe render times and the average numbers of the triplets of Max-Extent

with respect to our method.

Scene Max-Extentrender time #triplets accuracy RMSE

Buddha 3325s (7.31x) 141k (7.65x) 94.8% 0.00314Two Buddhas 4760s (3.87x) 93.1k (3.84x) 94.9% 0.00510

Head 330s (1.83x) 15.7k (1.79x) 94.8% 0.00519Salle de bain 20175s (8.41x) 359k (6.67x) 94.9% 0.00294

by up to 8.4 times. This indicates that Max-Extent requiresexcessive subdivisions of the triplets to generate irradi-ance points with large contributions. Although our methodsamples the triangles and the irradiance points uniformly,our BSSRDF-aware cluster selection method can efficientlysample irradiance points with large contributions. The num-bers in parentheses show the improvements in render timesand the numbers of the triplets for our method. Since thecomputation cost of our method is linear with respect to thenumber of triplets J , this number directly affects the rendertime as shown in Table 3. As expected, the improvementsin the numbers of triplets using our method are similar tothose for the render times (both numbers are shown in theparentheses in Table 3).

5.4 Discussion5.4.1 Application to Path TracingOur method has the potential to be integrated into a MonteCarlo path tracing framework. While we partition the inte-gral domain into the triplets for the error estimation, parti-tioning the integral domain (i.e., stratified sampling) can beused for sampling light paths. Let us explain how to samplea light path x = x0x1x2x3 with length three by using ourmethod, where x0 and x3 are on the camera and a lightsource (not restricted to VPL), and x1 (corresponding tothe shading point) and x2 (corresponding to the irradiancepoint) are on the surfaces of a translucent material. Lightpaths with length k ≥ 3 can be sampled by continuing totrace the path from x2 in the similar way as path tracing.

We first sample a shading point x11, an irradiance point

x12, and a point on the light source x1

3 from the root triplet(CL1 ,CI1,CS1 ). This corresponds to sampling a light path


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Fig. 7. Visualization of the number of the triplets in false color using our BSSRDF-aware cluster selection method (top) and the previous selectionmethod, Max-Extent, (bottom). The average numbers of the triplets for both selection methods are listed. The numbers in the parenthesis show theimprovement of our method with respect to the previous method.

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a cluster to be subdivided using our BSSRDF-aware clusterselection method. If, for instance, IC CL1 is selected as shownin Fig. 8 (right), and the root triplet is subdivided into twotriplets (CL1 ,CI2,CS1 ) and (CL1 ,CI3,CS1 ), a new light path x2

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from CI2 instead of CI1. By repeating these processes, ourmethod can sample the light paths in the path tracingframework.

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5.4.2 ScalabilityWe experimented on the scalability of our method withrespect to the VPL count. Fig. 9 shows the relationshipbetween the numbers of VPLs and the render times bydoubling (or halving) the numbers of VPLs listed in Table 1.By increasing the number of VPLs eight times, the increasesin render time for each scene are only 13% (Buddha),

17% (Two Buddhas), 15% (Head), and 21% (Salle de bain),respectively. As shown in Fig. 9, our method scales wellwith respect to the VPL count. In contrast, our methodscales linearly with respect to the number of irradiancepoints, while MDLC and EELC scales sublinearly due to thehierarchical structure of the (pre-sampled) irradiance points.However, as shown in Table 2, a computational bottleneckresides in importance sampling of the irradiance points, andthe cost for importance sampling scales linearly with thenumber of irradiance points.

5.4.3 Robustness to various relative error thresholdsFig. 10 shows graphs of the estimation accuracy with differ-ent relative threshold parameter (ε) values. We comparedthe estimation accuracy between our method and EELC(with 4096 irradiance points per shading point) by changingε from 10% to 0.5%. As shown in Fig. 10, the estimationaccuracy of EELC for ε ≥ 2% is close to 95%, while this fallssharply to 70% in all the scenes when ε = 0.5% is specified.This is because the maximum number of irradiance pointsis fixed for EELC, and additional irradiance points arerequired to achieve a tight error threshold. On the otherhand, the estimation accuracy of our method is consistentlyvery close to 95% for different ε, indicating that our adaptivesampling of the irradiance points is robust in achieving highestimation accuracy against various relative error thresholdparameter values.

5.4.4 Mesh qualitySince our method clusters the triangular mesh, we inves-tigated the effect of the quality of the triangular meshon the rendering performance in the Salle de bain scene.The left wall in the Salle de bain scene consists of onlytwo triangles, which indicates that our method works wellfor low-resolution triangular meshes. We subdivided thetwo triangles of each wall 128 times, then we applied afiner triangular mesh to our error estimation method. Therender time was 2430s, and we could not see any significant


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Fig. 9. Plots of the render times and the number of VPLs on the log-log scale show the scalability of our method. The number above eachplot shows the increase in the render time when the number of VPLs isincreased by eight times. This indicates that our method scales well tothe VPL count.

difference to that (2398s) obtained with the original lowtriangular mesh. We also measured the render times for theBuddha scene by subdividing the triangular mesh of theBuddha model (1.09M triangles). The render time increasesby 13.0% and 22.8% when the original triangular meshis refined to 2.92M (2.68×) and 3.99M (3.66×) triangles,respectively. The computational overhead is sublinear withrespect to the number of triangles.

5.4.5 Image resolutionFinally, we applied our method to higher resolution images.We rendered the Head scene with a resolution of 10242 withsupersampling (4 rays per pixel). The estimation accuracyand RMSE are 94.9%, and 0.0062, respectively, which aresimilar to those with a resolution of 5122 (see Table 1).The render time for the Head scene for 10242 is 600s. Thisindicates that our error estimation framework works wellregardless of the image resolution.

6 CONCLUSIONS

We have proposed an adaptive sampling method of theirradiance points for subsurface rendering with the diffuseBSSRDF model so that the relative error is less than theuser specified threshold. We proposed a BSSRDF-awarecluster selection method to efficiently partition the triplet.We showed that our method can eliminate the need forspecifying the number of the irradiance points, and ourmethod can render images indistinguishable from the refer-ence solution without a tedious trial-and-error process andre-renderings. Besides, thanks to the use of simple uniformsampling combined with the BSSRDF-aware cluster selec-tion, our method achieved significant speedups comparedto the previous methods that require costly importancesampling.

Although our method lifts the limitation of the previousscalable subsurface rendering methods, our method still hasthe following limitations. Our method necessitates speci-fying the numbers of shading points. A small number ofshading points leads to aliasing and artifacts in the renderedimages. However, in our method, we can identify that the

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Fig. 10. Comparison of the estimation accuracy betwen our method(solid line) and EELC (dashed line) for various relative error threshold (ε)values. We measured the estimation accuracy for ε = 10%, 5%, 2%, 1%,and 0.5% with the same confidence level α = 95%. The estimationaccuracy for EELC decreases for tight error thresholds (e.g., it falls downto around 70% in all the scenes when ε = 0.5%), while that of ourmethod is consistently close to 95%.

cause of such artifacts comes from the lack of shadingpoints, efforts to tweak parameters are smaller than those inprevious methods. Since our method is based on the many-light rendering framework, our method inherits the disad-vantages of many-light rendering, such as the singularitiesdue to the geometry term. However, since our method canhandle a large number of VPLs efficiently as shown inTable. 1 and Fig. 9, those singularities can be mitigated.

In future work, we would like to investigate whethermore sophisticated BSSRDF models can be applied to ourerror estimation framework. Moreover, we would like toextend our method to heterogeneous translucent materials.

ACKNOWLEDGEMENTS

We thank the anonymous reviewers for their constructivesuggestions and comments. This work was partially sup-ported by JSPS KAKENHI Grant Numbers 15H05924 and18H03348.

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Kosuke Nabata received his B.S., M.S., andPh.D degrees from Wakayama University in2013, 2015, and 2019, respectively.

Kei Iwasaki received his B.S., M.S., and Ph.D.degrees from the University of Tokyo, in 1999,2001, and 2004, respectively. He is currently anassociate professor at Wakayama University.

journal of la adaptive irradiance sampling for many-light

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