IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 1

Radiance Caching for Efficient Global IlluminationComputation

Jaroslav Křivánek, Pascal Gautron, Sumanta Pattanaik, and Kadi Bouatouch

Abstract— In this paper we present a ray tracing basedmethod for accelerated global illumination computationin scenes with low-frequency glossy BRDFs. The methodis based on sparse sampling, caching, and interpolatingradiance on glossy surfaces. In particular, we extend theirradiance caching scheme proposed by Ward et al. [1]to cache and interpolate directional incoming radianceinstead of irradiance. The incoming radiance at a pointis represented by a vector of coefficients with respect to ahemispherical or spherical basis. The surfaces suitable forinterpolation are selected automatically according to theroughness of their BRDF. We also propose a novel methodfor computing translational radiance gradient at a point.

Index Terms— Global illumination, ray tracing, hemi-spherical harmonics, spherical harmonics, directional dis-tribution.

I. INTRODUCTION

MONTE Carlo ray tracing is the method of choicefor computing images of complex environmentswith global illumination [2]. Even for the radiositymethod, high quality images are created by final gather-ing, often using Monte Carlo ray tracing [3].

Monte Carlo ray tracing is, however, expensive whenit comes to computing indirect illumination on surfaceswith low-frequency BRDFs (bi-directional reflectancedistribution functions). Too many rays have to be tracedto get a reasonably precise estimate of the outgoing radi-ance at a point. Fortunately, a high degree of coherencein the outgoing radiance field on those surfaces [1], [4]–[6] can be exploited by interpolation [1], [7] to obtain asignificant performance gain.

Our goal is to accelerate Monte Carlo ray tracing-based global illumination computation in the presence ofsurfaces with low-frequency glossy BRDFs. We achieveit through sparse sampling, caching, and interpolatingradiance on those surfaces. In particular, we extend Ward

Manuscript received July XXX, 2004.J. Křivánek is with the University of Central Florida (UCF),

IRISA/INRIA Rennes, and the Czech Technical University. E-mail:jkrivane@irisa.fr

P. Gautron is with IRISA/INRIA Rennes and UCF. E-mail:pgautron@irisa.fr

S. Pattanaik is with UCF. E-mail: sumant@cs.ucf.eduK. Bouatouch is with IRISA/INRIA Rennes. E-mail: kadi@irisa.fr

et al.’s irradiance caching [1], [8] to glossy surfaces. Irra-diance caching is based on the observation that reflectedradiance on diffuse surfaces due to indirect illuminationchanges very slowly with position. However, this appliesto surfaces with arbitrary low-frequency BRDFs. Moti-vated by this observation, we extend Ward et al.’s workto cache and interpolate the directional incident radianceinstead of the irradiance. This allows us to accelerateindirect lighting computation on surfaces with glossyBRDFs. We call the new method radiance caching.

The incoming radiance at a point is represented by avector of coefficients with respect to spherical or hemi-spherical harmonics [9]. Due to the basis orthogonality,the illumination integral evaluation (Eq. 1) reduces to adot product of the interpolated incoming radiance coeffi-cients and the BRDF coefficients. Radiance interpolationis carried out by interpolating the coefficients. We en-hance the interpolation quality by the use of translationalgradients. We propose novel methods for computinggradients that are more general than the method of Wardand Heckbert [8].

Radiance caching shares all the advantages of theWard et al.’s work. Computation is concentrated invisible parts of the scene; no restrictions are imposedon the scene geometry; implementation and integrationwith a ray tracer is easy. Our approach can be directlyused with any BRDF represented by (hemi)sphericalharmonics, including measured BRDFs.

This paper extends the initial description of radiancecaching given in [9] (included with the submission). Themain contributions of this paper are the extension of irra-diance caching to glossy surfaces, an automatic methodfor selecting BRDFs suitable for radiance caching, newmethods for computing translational radiance gradient,and integration of radiance caching in a ray tracer.

The rest of the paper is organized as follows. Sec-tion II summarizes the related work. Section III gives anoverview of how radiance caching works and how it isintegrated in a ray tracer. Section IV details differentaspects of radiance caching. Section V presents theresults. Section VI discusses various topics not coveredin the algorithm description. Section VII concludes thepaper and summarizes our ideas for future work.

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II. RELATED WORK

a) Interpolation: Interpolation can be used inglobal illumination whenever there is a certain level ofsmoothness in the quantity being computed. Radiositymethod uses interpolation in the form of surface dis-cretization e.g. [10]–[12]. In the context of Monte Carloray tracing, approaches have been proposed for screenspace interpolation [5], [13]–[15]. The goal of thesemethods is to display an approximate solution quickly.However, they do not accelerate the computation of thefinal high quality solution, which is the objective of ourwork. Object space interpolation has also been usedfor the purpose of fast previewing [16], [17]. Sparsesampling and interpolation for high quality renderingwas used in [1], [7]. The approach of Bala et al. [7] issuitable only for deterministic ray tracing. Ward et al. [1]use interpolation only for diffuse surfaces. Our approachextends this work to support caching and interpolationof the directional incoming radiance on glossy surfaces.

b) Caching Directional Distributions: Caching di-rectional distributions has been used to extend the ra-diosity method to support glossy surfaces, e.g. [18]–[25]. It has also been used in Monte Carlo ray tracingon diffuse surfaces [6], [26], [27]. Slusallek et al. [6]and Kato [26] use reprojection of radiance samples.Tawara et al. [27] selectively update a radiance samplelist in time to exploit temporal coherence. Storing lightparticles in the scene can also be thought of as cachinga directional distribution [28], [29].

c) Spherical Function Representation: A represen-tation of functions on a (hemi)sphere is necessary forincoming radiance caching. Piecewise constant repre-sentation [6], [24], [26], [27] is simple but prone toaliasing and memory demanding. Unless higher orderwavelets are used, even wavelet representation [21]–[23], [25], [30] does not remove the aliasing problems.Nevertheless, wavelets are a viable alternative to the useof spherical or hemispherical harmonics, especially forhigher frequency BRDFs.

Spherical Harmonics [18], [20], [31]–[37] removethe aliasing problem and are efficient for represent-ing low-frequency functions. However, representation ofsharp functions require many coefficients and ringingmight appear. Hemispherical harmonics [9] are bettersuited for representing functions on a hemisphere. Basisfunctions similar to spherical harmonics are Zernikepolynomials [38], [39] and hemispherical harmonics ofMakhotkin [40]. Unlike for spherical harmonics, therotation procedure is not available for these basis func-tions. We choose hemispherical and spherical harmonicsbecause they are the only basis for which an efficient

rotation procedure is available. They also have good anti-aliasing properties, low storage cost and are easy to use.

d) Illumination Gradient Computation: Arvo [41]computes the irradiance Jacobian due to partiallyoccluded polygonal emitters of constant radiosity.Holzschuch and Sillion [42] handle polygonal emitterswith arbitrary radiosity. Ward and Heckbert [8] computeirradiance gradient using the information from hemi-sphere sampling. Our gradient computation is also basedon hemisphere sampling. We use gradients to improvethe smoothness of the radiance interpolation. One ofthe algorithms we use for gradient computation wasindependently developed by Annen et al. [43].

e) Irradiance Caching: Ward et al. [1] proposeirradiance caching as a means of computing indirectdiffuse inter-reflections in a ray tracer [2]. They samplethe irradiance sparsely over surfaces, cache the resultsand interpolate them. For each ray hitting a surface,the irradiance cache is queried. If irradiance records areavailable, the irradiance is interpolated. Otherwise a newirradiance record is computed by sampling the hemi-sphere and added to the cache. In [8] the interpolationquality is improved by the use of irradiance gradients.

We retain the basic structure of the original algorithm,but each record stores the incoming radiance functionover the hemisphere. This allows us to apply the inter-polation to glossy surfaces.

III. ALGORITHM OVERVIEW

Radiance caching is a part of a ray tracing approachto global illumination. At every ray-surface intersection,the outgoing radiance is evaluated with the illuminationintegral:

L(θo, φo) =∫ 2π

0

∫ π/2

0Li(θi, φi)f(θi, φi, θo, φo) cos θi sin θidθidφi,

(1)

where L is the outgoing radiance, Li is the incomingradiance and f is the BRDF. The integral is split intoparts and each of them is solved by a different technique:

• Direct illumination uses deterministic method forpoint light sources and area sampling for area lightsources [44].

• Perfect specular reflections/refractions are solved bytracing a single deterministic secondary ray.

• Ward’s irradiance caching computes the indirectdiffuse term for purely diffuse surfaces.

• Two different techniques may be used for glossysurfaces.Low-frequency BRDF. Our radiance caching com-

putes the indirect glossy and diffuse terms.

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High-frequency BRDF. Monte Carlo importancesampling computes the indirect glossy term andirradiance caching computes the indirect diffuseterm.

Radiance caching is not used for high-frequency BRDFssince many coefficients would be needed. Moreover,high-frequency BRDFs are well localized and impor-tance sampling provides good accuracy with a fewsecondary rays. The distinction between low- and high-frequency BRDFs is done automatically as describedin Section IV-A. The steps of the rendering algorithmrelated to radiance caching are shown in Figure 1.

// preprocessing - BRDF conversionfor (every surface in the scene) do

if (surface suitable for radiance caching) thenCompute and store the HSH representation of the surface’sBRDF.

end ifend for// rendering with radiance cachingfor (every ray-surface intersection p) do

Retrieve the HSH representation of the BRDF at p.if ( HSH representation of the BRDF not available) then

// high-frequency BRDFImportance sampling computes the indirect glossy term.Irradiance caching computes the indirect diffuse term.

else// low-frequency BRDF; use radiance cachingif ( radiance cache records exist near p ) then

Compute the HSH coefficients of the incident radiance atp by gradient-based interpolation.

elseCompute incident radiance at p by sampling the hemi-sphere above p.Compute HSH coefficients of the incoming radiance.Compute the translational gradient of the coefficients.Store the new radiance record in the radiance cache.

end ifCompute the outgoing radiance at p as the dot product ofthe coefficient vector of the incoming radiance and that ofthe BRDF.

end ifend for

Fig. 1. Outline of the radiance caching algorithm. (HSH stands forhemispherical harmonics.)

The i-th radiance cache record contains:• position pi,• local coordinate frame (ui,vi,ni),• hemispherical harmonics coefficient vector Λi rep-

resenting the incoming radiance,• two derivative vectors ∂Λi∂x and

∂Λi∂y representing the

translational gradient,• harmonic mean distance Ri of objects visible

from pi.Λ denotes a coefficient vector and λml denotes a coeffi-cient, that is Λ = {λml }. Each record stores the incidentradiance function in a view-independent manner so that

it can be reused for different viewpoints. The records arestored in an octree as described by Ward et al. [1].

IV. RADIANCE CACHING DETAILS

A. BRDF Representation

We represent the BRDFs using the method of Kautzet al. [35] which we briefly describe here. We discretizethe hemisphere of outgoing directions. For each dis-crete outgoing direction (θo, φo), we use hemisphericalharmonics to represent the BRDF over the hemisphereof incoming directions. The n-th order representationof a cosine weighted1 BRDF f(θo,φo) for an outgoingdirection (θo, φo) is

f(θo,φo)(θi, φi) ≈n−1∑

l=0

l∑

m=−l

cml (θo, φo)Hml (θi, φi) , (2)

where

cml (θo, φo) =∫ 2π

0

∫ π/2

0f(θo, φo, θi, φi)H

ml (θi, φi) sin θidθidφi.

(3)

Hml are the hemispherical harmonics basis functions [9].We sample the outgoing hemisphere for (θo, φo) usingthe parabolic parametrization [45].

Adaptive BRDF Representation: Hemispherical har-monics representation for all scene BRDFs is computedbefore the rendering starts. For each outgoing direction,the adaptive representation of f(θo,φo) uses the minimumorder n sufficient to avoid exceeding the user specifiedmaximum error, measured as described in [33] (seeFigure 2). If no order n < nmax is sufficient for the spec-ified error, the hemispherical harmonics representationis discarded: radiance caching will not be used for thatBRDF and that outgoing direction. After applying thisprocedure, only low-frequency BRDFs are representedusing the harmonics and radiance caching is used forthem. This constitutes an automatic criterion for discern-ing low- and high-frequency BRDFs in our renderingframework. nmax is user specified; we use nmax = 10 forour examples. The aim is to have nmax such that a BRDFis classified as low-frequency if and only if radiancecaching is more efficient than Monte Carlo importancesampling. While nmax = 10 was a good value for ourscenes, it would not have to be so in other ones. Highernmax allows using radiance caching for higher frequencyBRDFs. The higher frequency BRDFs, the more rayswill be needed to sample the hemisphere for a new recordand the less reuse will be possible for that record.

1All BRDFs are multiplied by the cosine term cos θi beforecomputing the harmonics representation.

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Fig. 2. Adaptive BRDF representation for (a) Phong BRDF withexponent h = 15 and (b) anisotropic Ward BRDF [46] with kd = 0,ks = 1, αx = 0.6, αy = 0.25. The order of the hemisphericalharmonics representation adapts to the BRDF without ever exceedingthe specified maximum representation error (here 5%). The colordisks represent BRDF representation error for different outgoingdirections (θo, φo). Directions are mapped on the disk with theparabolic parametrization. (One can imagine the disks as lookingat the hemisphere from the top.) The graphs represent one scanlinefrom the disk images (i.e. fixed y and varying x component of theoutgoing direction). In the case of the Ward BRDF, radiance cachingis not used for some directions, since the representation error wouldbe too high.

B. Incoming Radiance Computation

Whenever interpolation is not possible at a point p,a new radiance record is computed and stored in thecache. We represent the incoming radiance Li by a vectorof hemispherical harmonics coefficients Λ = {λml }as Li(θ, φ) ≈

∑n−1l=0

∑lm=−l λ

ml H

ml (θ, φ), where n is

the representation order. The coefficients λml for ananalytical Li would be computed with the integral λml =∫ 2π0

∫ π/20 Li(θ, φ)H

ml (θ, φ) sin θdθdφ. Our knowledge of

Li is based only on sampling (ray casting). Hence, wecompute λml by a Monte Carlo quadrature with uniformsampling:

λml =2π

N

N∑

k=1

Li(θk, φk)Hml (θk, φk), (4)

where Li(θk, φk) is the incoming radiance coming fromthe sampled direction (θk, φk) and N is the numberof sampled directions. We use a fixed N but adaptivehemisphere sampling [47], [48] is desirable.

The order n for the incoming radiance representationis equal to the order of the BRDF representation at p.This cuts off high frequencies from the incoming radi-ance. The approach is justified by a low-frequency BRDFacting as a low-pass filter on the incoming radiance[34]. If incoming radiance vectors with different numberof coefficients are interpolated, the shorter vectors arepadded with zeros.

C. Translational Gradient Computation

We want to compute the translational gradient ∇λmlfor each λml computed with Eq. (4). The gradient isused to improve the interpolation smoothness. We didnot succeed in extending the gradient computation ofWard and Heckbert [8] to handle our case since theirmethod tightly couples the cosine probability density andthe cosine weighting used for irradiance computation.

Instead, we have developed two new methods forcomputing translational gradient ∇λml . The first, nu-merical, displaces the center of the hemisphere. Thesecond, analytical, is based on differentiating the termsof Eq. (4). In both cases we compute the gradient∇λml =

[∂λm

l

∂x ,∂λm

l

∂y , 0]

by computing the partial deriva-tives ∂λml /∂x and ∂λ

ml /∂y. The gradient is defined in

the local coordinate frame at the point p. The derivativewith respect to z is not computed since typical displace-ments along z are very small and using it does notvisually improve the interpolation quality. We computethe gradients simultaneously with the computation of thecoefficients λml during the hemisphere sampling.

1) Numerical Gradient Computation: To compute thederivative ∂λml /∂x numerically, we displace the point p,along the local x-axis, by ∆x to p′ (Figure 3). For eachMonte Carlo sample Li(θk, φk) we:

1) Compute the new direction (θ′k, φ′k) at p

′ as(θ′k, φ

′k) =

qk−p′

r′k

. Here qk is the point hit bythe ray from p in direction (θk, φk) and r′k =‖qk − p′‖. We will also denote rk = ‖qk − p‖.See Figure 3 for the various terms used here.

2) Compute the solid angle Ω′k associated with thenew direction (θ′k, φ

′k). The solid angle Ωk associ-

ated with each direction in Equation (4) is uniformand equal to 2π/N . With the displacement of thepoint p, the solid angles no longer remain uniform.The change in solid angle is due to the change indistance rk = ‖qk − p‖ and orientation of thesurface at qk, as seen from the hemisphere centerp or p′. The solid angle before the displacementis Ωk = ∆Ak

cos ξkr2

k

= 2πN , where ξk is the anglebetween the surface normal at qk and the vectorfrom qk to p. The area ∆Ak = 2πN

r2k

cos ξkis the part

of the environment visible through Ωk. It does notchange with the displacement because we assumethat the environment visible from p and p′ isthe same. After the displacement the solid anglesubtended by ∆Ak becomes

Ω′k = ∆Akcos ξ′kr′2k

=2π

N

r2kr′2k

cos ξ′kcos ξk

.

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We now estimate the coefficient λml′ at p′ as

λml′ =

2π

N

N∑

k=1

r2kr′2k

cos ξ′kcos ξk

Li(θk, φk) Hml

(θ′k, φ

′k

),

and finally we compute ∂λml /∂x = (λml

′ − λml )/∆x.The computation of ∂λml /∂y proceeds in a similar

way. This completes the numerical estimation of thetranslational gradient ∇λml at the point of interest.

Fig. 3. Gradient computation by displacing the hemisphere centerfrom p to p′ ((a) before and (b) after the displacement). Thequantities changing with the displacement are (shown in red): sampleray direction (θk, φk), the solid angle Ωk associated with this sample,and the angle ξk between the sample direction and the surface normalat the hit point qk. Neither the hit point qk nor the area ∆Ak visiblethrough Ωk change with the displacement.

2) Analytical Gradient Computation: We rewriteEquation (4) as

λml =N∑

k=1

ΩkLi(θk, φk)Hml (θk, φk), (5)

with Ωk = 2πN for uniform hemisphere sampling. Wehave seen that Ωk does not remain constant with dis-placement of p and therefore it has to be included in thesum and differentiated.

The partial derivative ∂λml /∂x is computed by differ-entiating the terms of the sum in Eq. (5)

∂λml∂x

=N∑

k=1

∂

∂x

(ΩkLi(θk, φk)H

ml (θk, φk)

)=

=N∑

k=1

Li(θk, φk)

(∂Ωk∂x

Hml (θk, φk) + Ωk∂Hml (θk, φk)

∂x

)

(6)The derivative of the basis function is

∂Hml (θk, φk)

∂x=

∂θk∂x

∂Hml (θk, φk)

∂θk+

∂φk∂x

∂Hml (θk, φk)

∂φk,

(7)with [49]

∂θk/∂x =− cos θk cos φk/rk,∂φk/∂x = sin φk/(rk sin θk). (8)

Those derivatives with respect to y would be

∂θk/∂y =− cos θk sin φk/rk,∂φk/∂y =− cos φk/(rk sin θk). (9)

Derivatives ∂Hml /∂θk and ∂Hml /∂φk are given in the

Appendix I.The derivative of the solid angle Ωk is ∂Ωk∂x =

∂∂x∆Ak

cos ξkr2

k

= ∆Ak∂∂x

cos ξkr2

k

. The area ∆Ak = 2πNr2

k

cos ξkis the part of the environment visible through Ωk. Itdoes not change with the displacement. The change ofcos ξk/r

2k with the displacement of p is opposite to its

change with the displacement of qk = (qx, qy, qz), i.e.

∂

∂x

cos ξkr2k

= − ∂∂qx

cos ξkr2k

.

The derivative ∂∂qxcos ξk

r2k

can be computed with the as-sumption that p lies at the origin because only therelative position of p and qk matters (see Figure 4).Since cos ξk = −nk·qkrk and rk = ‖qk‖ =

√q2k + q

2y + q

2z

we have∂

∂qx

cos ξkr2k

=− ∂∂qx

nxqx + nyqy + nzqz(q2x + q

2y + q

2z)

3/2

=−rknx + 3qx cos ξkr4k

. (10)

Here nk = (nx, ny, nz) is the surface normal at qk.Combining this result with ∆Ak = 2πN

r2k

cos ξk, we get

∂Ωk∂x

=2π

N

rknx + 3qx cos ξkr2k cos ξk

. (11)

Plugging Equations (11) and (7) into (6) we get the com-plete formula for ∂λml /∂x. The formulas for ∂λ

ml /∂y

are similar; only Eq. (8) must be replaced by Eq. (9).A similar gradient calculation was also proposed in

[29]. This method disregards the change of Ωk and hencedoes not provide good results. The analytical method wasalso independently developed by Annen et al. [43]. Acode fragment evaluating one term of the sum in Eq. (6)is given in the accompanying material [50].

Fig. 4. Quantities in the computation of ∂∂qx

cos ξkr2

k

.

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3) Discussion: For the derivation of both numericaland analytical methods we assumed:

• The radiance Li(θk, φk) from the point qk incidentat p does not change with the displacement of p.

• Visibility of ∆Ak, the small area around qk, doesnot change with the displacement of p.

Though none of these assumptions is necessarily valid inall scenes, they are reasonable for small displacements.

The numerical and analytical methods are equivalent,their results are indistinguishable. The numerical methodis easier to implement since we do not need to evaluatethe basis function derivatives. The analytical method isnumerically more stable near edges and corners and alsoslightly faster to evaluate.

4) Irradiance Gradient Computation: Note that bothmethods we propose can still be used if Hml is replacedby any other hemispherical function. We also do notrely on uniform hemisphere sampling. Any probabilitydensity p(θ, φ) can be used for sampling. The onlychange is that the solid angle becomes Ωk = 1Np(θk,φk)instead of Ωk = 2πN used for the uniform sampling.

As an example we compute the irradiance gradi-ent ∇E with a cosine-weighted hemisphere sampling.Hml (θ, φ) is replaced by cos θ, the probability densityof sampling in direction (θ, φ) is p(θ, φ) = cos θπ andtherefore Ωk = πN cos θk . The resulting formula for theanalytical method is

∂E

∂x=

N∑

k=1

Li(θk, φk)

(∂Ωk∂x

cos θk −π

N

sin θkcos θk

∂θk∂x

)

with∂Ωk∂x

=π

N cos θk

rknx + 3qx cos ξkr2k cos ξk

.

We implemented this irradiance gradient computationmethod and that of Ward and Heckbert [8] and wecompared them on a sample scene (Figure 5). The resultswere similar for both methods. Ward and Heckbert’smethod gives better results when some surfaces are seenat very sharp grazing angles from the sampling point p.Otherwise our method gives slightly smoother results.

Even though the quality of Ward and Heckbert’smethod is subtly superior to ours, our method providesmany advantages. It works with any sampling distribu-tion and with any function used to weight the radiancesamples. The contribution of radiance samples to thegradient is independent of each other and therefore ourmethod is more easily amenable for parallelization orhardware implementation. We do not assume any strat-ification of the hemisphere. This allows one to use ourgradient computation with different sampling strategies,e.g. quasi Monte Carlo sampling.

Fig. 5. Comparison of irradiance gradient computation. The sceneis a diffuse Cornell box; only first bounce indirect illumination iscomputed. The color-coded images show the difference betweenthe gradient-based interpolation and the reference solution (10,000samples per hemisphere at each pixel). RMS error of the imagesis 0.125 for Ward’s method and 0.131 for our method. The graphshows relative error of the interpolation along a single scanline ascompared to the reference solution. Ward’s method gives better resultswhen there are surfaces seen at very sharp grazing angles from thesampling point. Otherwise our method gives slightly lower error.

D. Radiance Interpolation

If a query to the radiance cache succeeds, the in-coming radiance is interpolated as described in thissection. We use a weighted interpolation scheme similarto the one proposed in [8] for interpolating the coeffi-cient vectors Λi at any required surface point p. Thedifference is that we replace the use of the rotationalgradient by truly rotating the incident radiance function.This aligns the coordinate frame at the position pi of

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the cache record and the frame at p (see Figure 6).The weight wi(p) of record i with respect to p iswi(p) =

(‖p−pi‖

Ri+√

1 − n · ni)−1

, where n is thesurface normal at p, ni is the surface normal at pi, andRi is the harmonic mean distance to objects visible frompi. The coefficient vector of the interpolated radiance iscomputed as a weighted average:

Λ(p) =

∑

S

Ri

(Λi + dx

∂Λi∂x

+ dy∂Λi∂y

)wi(p)

∑

S

wi(p), (12)

where the set S of radiance records used for interpolationat p is defined as S = {i|wi(p) > 1/a} and a is auser-defined desired accuracy. The definition of the set Seffectively represents the criterion used to decide whichradiance cache records can be used for interpolation. If Sis non-empty, interpolation (or extrapolation) is possible.dx and dy are the displacements of p − pi along thex and y axes of the record i’s local coordinate frame.Displacement along z is not taken into account for thegradient-enhanced interpolation, since it is typically verysmall. Ri is the hemispherical harmonics rotation matrix[9] that aligns the coordinate frame at pi with the frameat p.

The interpolation scheme is borrowed from Ward etal. [1]. They derived it from the ‘split sphere’ model thatestimates an upper bound on the magnitude of the changeof irradiance. Although this model does not apply forradiance caching, the results are satisfying. We observethat a lower a has to be used for higher-frequencyBRDFs and interpolation errors are more apparent whensurfaces are viewed from grazing angles. In future workwe want to devise an interpolation scheme suited forradiance caching based on these observations.

Fig. 6. Rotation Ri aligns the coordinate frame at pi with that atp to make interpolation possible.

E. Outgoing Radiance Computation

The incoming radiance obtained by interpolation orhemisphere sampling is integrated against the BRDF tocompute the outgoing radiance. With an orthonormal

basis, the integral evaluation reduces to the dot product[51]:

L(θo, φo) =n−1∑

l=0

l∑

m=−l

λml cml (θo, φo). (13)

λml is an interpolated incoming radiance coefficient andcml (θo, φo) is a BRDF coefficient at p for the outgoingdirection (θo, φo).

V. RESULTS

Figure 7 gives the breakdown of rendering times forthe the three scenes we used to test radiance caching(Cornell Box, Walt Disney Hall, Flamingo). The timingswere measured on a 2.2GHz Pentium 4 with 1 GBRAM running Windows XP. The resulting renderings areshown in figures 8, 9 and 10, and in the accompany-ing video [50]. The maximum hemispherical harmonicsorder for radiance caching was set to n = 10, whichcorresponds to approx. 3.6 kB sized radiance cacherecords.

Fig. 8. Two views of a Cornell Box with glossy back wall renderedusing radiance caching (top) and Monte Carlo importance sampling(bottom).

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Cornell Box1280 × 1280

Disney Hall1440 × 840

Flamingo1280 × 1280

Frame I Frame II Frame I Frame II Frame I Frame II Frame IIIRC filling 28.7 9.2 218.0 169.0 63.5 53.2 63.9RC interpolation 4.6 4.6 17.3 17.0 4.42 11.4 8.7Total 82.9 68.7 357.8 295.6 108.0 101.0 104.0

Fig. 7. Timing breakdown for the test scenes. ‘RC filling’ is the time spent on computing and adding radiance cache records. ‘RCinterpolation’ is the time spent on looking up the existing radiance cache records and interpolating the radiance. ‘Total’ is the total renderingtime. The difference between the total time and the time spent by radiance caching consists of primary ray casting, direct lighting, specularreflections (in Office Space) and irradiance caching (in Disney Hall and Flamingo). All times given in seconds.

We compared the solutions obtained by radiancecaching with those obtained by Monte Carlo importancesampling. Importance sampling uses the surface BRDFas the importance function. The two rendering methodsexhibit artifacts with very different characteristics: high-frequency noise for importance sampling (‘specks’ inimages) and low-frequency error in radiance caching(uneven illumination gradients). It is therefore difficult tocompare rendering times needed to attain the same visualquality. Instead, we have chosen to fix the renderingtime and compare the image quality delivered by thetwo methods.

a) Cornell Box: Figure 8 shows renderings of aCornell Box with a glossy back wall (Phong BRDF,exponent 22), taken from two viewpoints at resolution1280 × 1280. Except for the back wall, all objects areLambertian. Only direct lighting and first bounce indirectglossy lighting for the back wall were computed.

Images in the top row were computed using radiancecaching with the caching accuracy set to a = 0.15 andthe number of rays cast to sample each hemisphere set toN = 6000. The indirect glossy term took 33.3 secondsto compute for the left image; total rendering time was82.9 sec (direct illumination uses 8 samples per pixel tosample the area source). The number of radiance cacherecords was 600. The time spent on the indirect glossyterm computation in the right image was only 13.8 sec,since the records from the left rendering were retainedand only 164 additional record were required.

The indirect glossy term for each of the two bottomimages was computed in 35 seconds using Monte Carloimportance sampling with 12 reflected rays per pixel ona glossy surface. Those rendering methods exhibit highnoise level, whose perception is even amplified in thetemporal domain as shown in the video.

The average time spent on radiance caching for a180 frames long animation with the camera movingbetween the position in the left and right images wasjust 4.9 sec per frame. Most of this time is spent oninterpolation since only a few records are needed foradditional frames. The average frame time with MonteCarlo importance sampling is 35 seconds since this

Fig. 9. Rendering of a simple model of Walt Disney Hall in LosAngeles. The top and the middle images, computed with radiancecaching, show the building from two different viewpoints. The bottomimage was computed with Monte Carlo importance sampling.

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 9

method does not reuse any information from the previousframes. Even though this time is 7 times longer than forradiance caching, the quality is much lower.

b) Walt Disney Hall: Figure 9 shows renderings ofa simple model of Walt Disney Hall in Los Angeles.The curved walls of the real building are covered withbrushed metal tiles, whose BRDF we approximate bya three-lobe Lafortune model [52]. The illumination isdue to the sun (modeled as a directional light), the sky(modeled as a constant blue light) and the surroundingurban environment (modeled as a constant brownishlight). Similarly to the real building (see [50]), wallsreflect the sky or the surroundings depending on theirnormals and the viewpoint (compare the top and themiddle image).

Full global illumination solution with one ray per pixelwas computed at resolution 1440× 840 and then scaleddown. Indirect illumination and the illumination comingfrom the sky and from the surroundings was computedin the same way: with irradiance caching on diffusesurfaces, with radiance caching on glossy surfaces inthe top and the middle image, and with Monte Carlosampling on glossy surfaces in the bottom image.

To capture the high indirect illumination variationson the metal walls, the caching accuracy was set toa = 0.1 leading to as many as 38,000 radiance cacherecords. We used 800 rays for hemisphere sampling. Therendering time was 357.8 s and 295.6 s for the top andthe middle image respectively. Monte Carlo importancesampling in the bottom image used 15 reflected rays oneach visible glossy pixel (fist bounce) and one ray forother bounces. The rendering time was similar to thatfor radiance caching, but the quality of radiance cachingresults is higher.

The BRDF we used for the metallic walls [53] wasrelatively sharp; the Lafortune lobes had exponents of16, 88 and 186. It is impossible to represent sucha BRDF accurately using hemispherical harmonics oforder 10 — the average representation error was over20%. Nonetheless, radiance caching renderings show noserious distortion of the material appearance comparedto Monte Carlo sampling.

c) Glossy Flamingo: Figure 10 shows 3 framesfrom the Flamingo animation. The bird was assignedthe Phong BRDF (exponent 7) and all other surfacesare purely diffuse. The rendering resolution was 1280×1280 pixels. Full global illumination up to 4 bounceswas computed. Irradiance caching was used to computeindirect lighting on diffuse surfaces.

First bounce indirect lighting on glossy surfaces forthe images in the top row was computed using radiancecaching; path tracing was used for deeper recursion

Fig. 11. Time spent on radiance caching for the Flamingo animation(see the accompanying video). Cached records are shared between theframes.

levels. The caching accuracy was set to a = 0.15 and thenumber of rays for hemisphere sampling was N = 1000.Table 7 gives the rendering times for the three imageswhen rendered independently, without retaining radiancecache records from previous renderings. Figure 11 showsthe time spent on radiance caching in a 280 frameslong animation with camera moving between the shownimages (see the accompanying video).

The images in the bottom row use Monte Carlo impor-tance sampling instead of radiance caching. In order tokeep the rendering time the same as for radiance caching,the number of reflected rays per pixel on a glossy surfacewas set to 12, 4, 6 respectively (from left to right).

This scene is particularly challenging for radiancecaching since the glossy surface is curved. On sucha surface, radiance cache records cannot be used forinterpolation at as many pixels as on a flat surface.Moreover, costly alignment is required before each inter-polation. In the left image, the flamingo occupies only asmall part of the screen and therefore one does not takeadvantage of radiance caching’s independence on imageresolution. The quality advantage of radiance cachingover Monte Carlo importance sampling can be seenonly by a very close inspection (see the images in theaccompanying material [50]). However, in the other twoimages the noise introduced by Monte Carlo samplingis more obvious. Notice also that for the animationrendering, the average time for radiance caching is 15seconds per frame. Rendering the same animation usingMonte Carlo importance sampling with only 2 reflectedrays on a glossy pixel leads to 27 seconds per framespent on indirect glossy lighting computation. Notice inthe accompanying video, that the quality obtained byradiance caching is considerably better.

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 10

Fig. 10. Frames from the Flamingo animation rendered using radiance caching (top row) and Monte Carlo importance sampling (bottomrow).

VI. DISCUSSION

a) Rotation Loss: There is a loss of informationwhen radiances are interpolated on a curved surface(Figure 12). A part of the radiance incident at pi shoulddisappear under the surface (marked ‘a’ in Fig. 12) andshould not contribute to the interpolated radiance at p.A part of the radiance actually incident at p is notrepresented by the radiance record at pi (marked ‘b’ inFig. 12) and is missing in the interpolated radiance.

This problem is not due to using a hemisphericalbasis for representing the incoming radiance, but dueto the fact the incident radiance at a surface point isa hemispherical function. Using spherical, instead ofhemispherical, harmonics would not solve this problem.In practice the error introduced by this problem is verysmall because the difference between the normal at pand the normal at any record used for interpolation at pis small. Note also that Ward et al.’s irradiance cachingsuffers from the same problem.

Fig. 12. Information loss when radiances are interpolated on acurved surface.

b) Global vs. Local Coordinates: Incoming radianceat a point can be represented either in the local frameat that point or in the global frame. This influences howthe interpolation at p is performed:

• Incoming radiance in the global framefor (each available record i) do

Update the interpolation sums in (12).end forAlign the result with the local frame at p.Compute the dot product.

• Incoming radiance in the local framefor (each available record i) do

Align the frame at pi with the frame at p.Update the interpolation sums in (12).

end forCompute the dot product.

The final dot product (Eq. 13) is always carried out inthe local frame at p. On curved surfaces, fewer rotationsare needed if the incoming radiance is represented inthe global frame. On the other hand, if the incomingradiance is represented in the local frame, no alignment(even with the BRDF) is needed on flat surfaces. Thelowest number of rotations is obtained if the incomingradiance is represented in the local frame on flat surfacesand in the global frame on curved surfaces. Note thatfull spherical function representation (e.g. using sphericalharmonics) is needed to represent the incoming radiancein the global frame.

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 11

c) Suitability of Hemispherical Harmonics: We use(hemi)spherical harmonics since they are simple, com-putationally efficient (manipulations of vectors of floats),avoid aliasing, and rotation is available for them. Theessential disadvantage is the lack of directional localiza-tion. When we create a new radiance cache record, thefull hemisphere must be sampled, whatever the incomingray direction is. The more directional the BRDF is, themore this approach becomes wasteful. With a basis thatlocalizes in directions, only the required part of thehemisphere would need to be sampled. For this purposeone can use piecewise constant representation [6], [26],[27], but it would presumably introduce aliasing. The useof spherical wavelets [30] is probably a good choice, andis left for further investigation.

d) General limitations: The BRDF frequencies usedin the example scenes are near the limit of what ourtechnique can currently handle. The restriction to low-frequency BRDFs is the main limitation of our technique,because we use hemispherical harmonics. We believethat higher frequency BRDFs can be handled within thepresented framework by using a localized basis suchas wavelets. Additionally, the radiance caching, as agathering technique, cannot solve certain types of lighttransfer phenomena, such as caustics. Those have to besolved by a more adapted technique.

VII. CONCLUSION AND FUTURE WORK

We have presented radiance caching, a method foraccelerating the computation of the indirect illuminationon surfaces with low-frequency glossy BRDFs. Radiancecaching is based on sparse sampling, caching, and inter-polating incoming radiance on those surfaces. Radianceis represented by hemispherical or spherical harmonicsin our approach. The interpolation quality is enhanced bythe use of translational gradients that can be computedwith two novel methods we have presented in this paper.We have also proposed an automatic criterion to decidefor which BRDFs radiance caching is suitable. We haveshown on several examples that radiance caching is moreefficient than pure Monte Carlo sampling at every surfacepoint and delivers images of superior quality.

In the future work we would like to use adaptivehemisphere sampling to compute the incoming radiancecoefficients [29], [48], [54]; use a different representationfor the incoming radiance that localizes better in direc-tions; and devise interpolation criteria better suited forglossy surfaces. In the long term we wish to investigatethe relationship between the frequency content of aBRDF and the suitability of interpolating radiance onsurfaces with that BRDF.

ACKNOWLEDGEMENTSThanks to Vlastimil Havran for letting us use his ray

tracer, Marina Casiraghi and Eric Risser for the WaltDisney Hall model. This work was partially supportedby ATI Research, I-4 Matching fund and Office of NavalResearch.

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APPENDIX IDERIVATIVES OF SPHERICAL AND HEMISPHERICAL

HARMONICSPartial derivatives for spherical harmonics are:

∂Y ml∂θ

(θ, φ) =

−√

2Kml cos(mφ) sin(θ)dP m

l

dx(cos θ) if m > 0

−√

2Kml sin(−mφ) sin(θ)dP

−m

l

dx(cos θ) if m < 0

−K0l sin(θ)dP0

l

dx(cos θ) if m = 0,

∂Y ml∂φ

(θ, φ) =

{0 if m = 0

−mY −ml (θ, φ) otherwise.

The derivative of the associated Legendre polynomials can be foundfrom the recurrence formula:

dP mldx

(x) =

1

x2−1

(xlP ml (x) − (m + l)P

ml−1(x)

)if m < l

−(−1)mx(2m − 1)!!(1 − x2)m

2−1 if m = l,

where x!! is the double factorial (product of all odd integers less thanor equal to x).

The partial derivatives for hemispherical harmonics are:

∂Hml∂θ

(θ, φ) =

−2√

2K̃ml cos(mφ) sin(θ)dP m

l

dx(2 cos θ − 1) if m > 0

−2√

2K̃ml sin(−mφ) sin(θ)dP

−m

l

dx(2 cos θ − 1) if m < 0

−2K̃0l sin(θ)dP0

l

dx(2 cos θ − 1) if m = 0,

∂Hml∂φ

(θ, φ) =

{0 if m = 0

−mH−ml (θ, φ) otherwise.

IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 13

PLACEPHOTOHERE

Jaroslav Křivánek is a Ph.D. student atIRISA / INRIA Rennes and the Czech Tech-nical University in Prague and a visiting re-search associate at the University of CentralFlorida. He earned his Masters in computerscience from the Czech Technical University.His research focuses on real-time and offlinelighting simulation and using visual perceptionfor efficient lighting computation.

PLACEPHOTOHERE

Pascal Gautron is a Ph.D. student at IRISA/ INRIA Rennes, France. He works in collab-oration with the University of Central Floridaand the University of Poitiers. He earned hisMasters in computer science from the Univer-sity of Poitiers. His main research interest ishigh quality real-time rendering using graphicshardware.

PLACEPHOTOHERE

Sumanta Pattanaik is an Associate Professorin the Computer Science Department of Uni-versity of Central Florida. From 1995 to 2001he was a research associate in the Program ofComputer Graphics of Cornell University. Be-fore that he was a post-doctoral researcher inIRISA/INRIA (1993-1995), France. His mainfields of research are: realistic image synthesisand digital imaging. His current research fo-

cuses on: real-time realistic rendering, and the application of visualperception for efficient lighting computation and accurate display. Heis a member of ACM SIGGRAPH, Eurographics and IEEE.

PLACEPHOTOHERE

Kadi Bouatouch is an electronics and auto-matic systems engineer (ENSEM 1974). Hewas awarded a PhD in 1977 and a higherdoctorate on computer science in the fieldof computer graphics in 1989. He researchinterest is: global illumination, lighting sim-ulation and remote rendering for complex en-vironments, real-time high fidelity rendering,parallel radiosity, virtual and augmented reality

and computer vision. He applied his research work to infrared simu-lation and tunnel lighting. He is currently Professor at the universityof Rennes 1 (France) and researcher at IRISA. He is member ofEurographics, ACM and IEEE. He is and was member of the programcommittees of several conferences and workshops, and referee forseveral Computer Graphics journals like: The Visual Computer,IEEE Computer Graphics and Applications, IEEE Transaction onVisualization and Computer Graphics, IEEE transaction on imageprocessing, ACM Transaction on Graphics, etc. He also acted as areferee for many conferences and workshops.