To appear in the ACM SIGGRAPH conference proceedings

An Empirical BSSRDF Model

Craig Donner∗ Jason Lawrence† Toshiya Hachisuka‡

Henrik Wann Jensen‡ Shree Nayar∗ Ravi Ramamoorthi∗ §

∗Columbia University † University of Virginia ‡ UC San Diego § UC Berkeley

Monte Carlo PathTracing (30 hours)

Diffusion Dipole + Single Scattering (10 min) Our Model + Single Scattering (30 min) Single Scattering Only

Figure 1: Although the appearance of orange juice is dominated by low-order scattering events, it is not accurately predicted by a singlescattering model alone (lower right). Adding the contribution from high-order multiple scattering using the diffusion dipole (left) still failsto capture these effects and produces visible color artifacts. Numerical methods such as Monte Carlo path tracing (upper right) or photonmapping are accurate, but do not provide an explicit model of the BSSRDF and require long rendering times. Our proposed model is compact,efficient to render and can accurately express the complex spatial- and directional-dependent appearance of these types of materials.

Abstract

We present a new model of the BSSRDF based on a large-scalesimulation that captures the appearance of materials that cannotbe accurately represented using existing single scattering modelsor multiple isotropic scattering models. Our model consists of ananalytic function of the 2D hemispherical distribution of exitantlight at a point on the surface and a table of parameter values ofthis function computed at uniformly sampled locations over the re-maining dimensions of the BSSRDF domain. This analytic functionis expressed in elliptic coordinates and has six parameters whichvary smoothly with surface position, incident angle, mean free pathlength and the underlying optical properties of the material (albedo,phase function and the relative index of refraction). Our modelagrees well with measured data, is compact, requring only 100MBto represent the full spatial- and angular-distribution of light withina wide spectrum of materials, and correctly predicts anisotropic anddirectional effects that are not captured by existing models.

1 Introduction

Light propagates into and scatters within all non-metallic materi-als. This subsurface scattering is common in many liquids—suchas orange juice, coffee or milk, and in solids—such as gemstones,leaves, wax, plastics and skin. It gives materials their characteristiccolors, and provides a soft, translucent appearance. Accurate andcompact models of the way light interacts with these materials arenecessary to efficiently render them.

Light scattering in translucent materials is described by the bidi-rectional scattering surface reflectance distribution function S (theBSSRDF [Nicodemus et al. 1977]). The BSSRDF defines the gen-eral transport of light between two points and directions as the ratioof the radiance Lo(~xo,~ωo) exiting at position ~xo in direction ~ωo tothe radiant flux Φi(~xi,~ωi) incident at~xi from direction ~ωi

dLo(~xo,~ωo)dΦi(~xi,~ωi)

= S(~xi,~ωi;~xo,~ωo|σs, σa, g, η), (1)

where S depends on the optical properties of the material—the scat-tering and absorption coefficients σs and σa, the relative index ofrefraction η , and g ∈ [−1 : 1] which parameterizes the anisotropyof the phase function.

1.1 Related Work

Numerical techniques such as Monte Carlo path tracing [Kajiya1986; Jensen et al. 1999] are capable of simulating general BSS-

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To appear in the ACM SIGGRAPH conference proceedings

RDFs. However, these methods are expensive, often requiringhours to days of processing. A related set of techniques focuson simulating participating media (see Cerezo et al. [2005] fora survey), though many have difficulty handling refraction at aninterface. [Jason: add discussion of Premoze here?] Scatteringequations may also be used in this context [Pharr and Hanrahan2000], but these are computationally expensive to evaluate. Photonmapping [Jensen 1996] can render many BSSRDFs but becomesexpensive in both time and space for highly scattering materials.Furthermore, none of these techniques provide an explicit represen-tation of the BSSRDF which is the main goal of this work. Rather,determining the fraction of light that is transported between any pairof points requires a complete simulation.

Existing analytic models of the BSSRDF only apply to two classesof materials. When light scatters exactly once and at a single point,this single scattering has a closed-form analytic solution [Blinn1982; Hanrahan and Krueger 1993]. Such models produce highlydirectional effects since the exitant light is assumed to be scattereddirectly from propagating beams of light. At the other end of thespectrum are highly scattering materials. The BSSRDF in thesecases is often modeled as the superposition of a single scatteringterm and a diffuse term [Jensen et al. 2001]. This diffusion ap-proximation [Stam 1995] is common and can be applied on its ownto materials that have negligible low-order scattering [Jensen andBuhler 2002; Donner and Jensen 2005]. However, this approxi-mation assumes light is scattered isotropically and produces incor-rect predictions when low-order scattering is significant, such as inmaterials like orange juice (see Figure 1). These materials exhibitsignificant absorption of the light as it propagates through the ma-terial so much of the energy is scattered back into the environmentnear the point of incidence. This light exits in areas and directionsoutside of the single scattering regime, but not in the high-ordermultiple scattering regime. The resulting angular distribution ofexitant light is not predicted well by single scattering, diffusion, ortheir sum.

Hybrid methods attempt to combine the simplicity of diffusionwith the accuracy of general numerical techniques. Donner andJensen [2007] introduce a method to model asymmetric diffuse re-flectance by sampling beams of light using diffusion sources seededby single-step photon tracing. This method assumes that the ma-terial is highly scattering and neglects near-source and directionaleffects. Li et al. [2005] couple path tracing with the diffusion ap-proximation for long path lengths, but in materials with moderatescattering or non-trivial absorption the path tracing step becomescomputationally intensive. Furthermore, since these methods relyon numerical simulation they too fail to provide an explicit modelof the BSSRDF.

Measuring the appearance of translucent materials is also a difficulttask. Both Goesele et al. [2004] and Peers et al. [2006] measuredpoint-to-point transport, but ignored the angular distribution of theexitant light. Also, their data is only suitable for reproducing theparticular materials measured. Narasimhan et al. [2006] used di-lution to measure the optical properties of a variety of scatteringmaterials, but relied on numerical techniques for rendering. Insteadof measuring a range of materials, we opt to simulate them insteadand derive an analytic model based on these simulations. In thisregard, our approach is similar to the virtual gonioreflectometrytechnique of Westin et al. [1992] for analyzing the reflectance ofcomplex opaque surfaces. [Jason: add brief discussion of Bouthorshere]

1.2 Our Approach

We propose an analytic model of the BSSRDF that applies to a widerange of materials. Our model is phenomenological and derived

from a large-scale simulation of the subsurface light transport for arange of optical properties and geometric configurations. Althoughour approach was inspired by data-driven reflectance models [Ward1992; Dana et al. 1999; Matusik et al. 2003], our choice to simulatethese effects avoids a difficult acquisition task and provids greatercontrol over the materials we consider. Further, we validate oursimulation and final model using measured data.

Although the full BSSRDF in Equation 1 is a 12D function (4 spa-tial parameters, 4 angular parameters and 4 optical parameters), wemake the common assumption of a spatially uniform, homogeneoussemi-infinite material. This reduces the dimensionality of the BSS-RDF to 8D and makes exploring this space more feasible.1 We usean efficient photon tracing technique and a large cluster of com-puters to reconstruct the 2D hemispherical distribution of exitantlight over a dense sampling of the remaining geometric and opticalvariables (a 6D space). This dataset captures the complete spatialand angular appearance of a wide range of translucent materials.Although this required many months of processing, it only needs tobe created once and is an important contribution of this work.2

Based on a thorough analysis of this simulated data, we propose ananalytic function expressed in elliptic coordinates with six param-eters that accurately captures the features of these hemisphericaldistribution functions. We demonstrate that this function fits thesimulated data well and, in turn, agrees with measured data in-cluding highly scattering materials such as milk and wax whichexhibit clear non-diffuse and anisotropic behaviors. Importantly,the parameters of this function vary smoothly over the 6D space ofremaining geometric and optical parameters. This allows tabulatingand interpolating them away from the samples we considered inour simulation to provide a continuous representation defined overthe full BSSRDF domain. Therefore, our final model consists of atable of parameter settings of this elliptic function (approx. 100MBof space, though in practice only a fraction of the data is neededat one time) that can be used directly for rendering. We presentimages rendered using this model that show complex directionaleffects such as glows around beams that would be impossible torender with existing diffusion-based techniques.

2 Simulating the Space of BSSRDFs

We performed numerical simulations to reconstruct slices of theBSSRDF over a wide range of geometric configurations and mate-rial properties. Similar to Hanrahan and Krueger [1993], we tracephotons into a semi-infinite slab and record how much energy theydeposit at the surface. We discretize the hemisphere of outgoinglight based on the exit trajectory of the photons. This requires sig-nificantly more photons and storage than a standard diffuse trace.

Assumptions and Parameterization: Recall that we assume a ho-mogeneous, semi-infinite material which allows reducing the 12DBSSRDF in Equation 1 to an 8D function. Table 1 summarizes ournotation and the geometry of our setup is illustrated in Figure 2.

Since the scattering and absorption coefficients σs and σa have in-finite range, we choose to parameterize the BSSRDF in terms ofits albedo α = σs/(σs + σa). Note that α ∈ [0..1]. All materi-als with the same albedo have the same exitant response up to adistance scale which is determined by the mean free path length` = 1/(σs + σa). For example, two materials with optical coef-ficients (σs, σa) and (2σs, 2σa) differ only in terms of ` and `/2.Although ` has infinite range, exitant intensity falls-off rapidly with

1 Note that because we assume the material is homogeneous, for any fixedset of optical parameters, the function that must be evaluated during ren-dering is only 3D.

2 The simulated data can be accessed from http://www.whatever

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increasing distance from the source. Therefore, we can consideronly the local region around~xi without missing a significant portionof the BSSRDF. These choices allow replacing the optical proper-ties (σs, σa, g, η) with (α, g, η) and provide a finite range that issuitable for uniform sampling. Just as with isotropic BRDFs, weassume that the angular dependence of the BSSRDF depends ononly 3 variables (θi, θo, φo). Similarly, for a homogeneous BSSRDFthe amount of light exchanged between two surface points~xi and~xodepends only on their relative positions. This allow parameterizingthe spatial dimensions using 2D polar coordinates with the angle θsand radial distance r = ||~xo −~xi||. Note that r is measured in meanfree path lengths. Under the above assumptions, we simplify theBSSRDF as

S(~xi,~ωi;~xo,~ωo|σs, σa, g, η) ≈ S(θi, r, θs, θo, φo|α, g, η). (2)

We next describe a photon tracing algorithm for reconstructingslices of this form of the BSSRDF. In Section 4 we introduce ournew BSSRDF model.

2.1 Simulation Method

Our photon tracing algorithm relies on sampling the probable pathsof light within a material. We discretize the exitant direction ofeach photon at the surface and accumulate its power into bins.However, to accurately resolve the complete hemispherical distri-bution of outgoing light a large number (∼ 1012) of photons arerequired, particularly in the case of strong anisotropic scattering(i.e., |g| > 0.5).

We emit photons along a collimated beam incident on the slab fromdirection −~ωi, such that it makes angle θi with the surface normal~n (see Figure 2). These photons refract into the material and prop-agate a distance d before being scattered or absorbed:

d =−log ξ

σs + σa, (3)

where ξ ∈ [0..1] is a uniformly distributed random number.

Reusing Photon Paths: To reduce the number of photons traced,and thus increase the speed of our simulation, we bias the result.Specifically, at each step each photon takes through the material wecompute the power Φe that a photon propagating in direction ~ωpincident on position~xp would contribute at the surface point~xo if itwere scattered directly there (see Figure 2):

Φe = α p (~q ·~ωp; g) e−(σs+σa) ||~xo−~xp||Ft(~ωo; η) Φp, (4)

!xo

!xi

θi

−!ωi

!xp

θs

!d!ωp

!ωo = (θo,φo)

S(θi;r,θs,!ωo|α,g,η)≈ S(xi,!ωi;xo,!ωo|σs,σa,g,η)

r

Figure 2: Geometric setup for our simulation and model. A colli-mated beam of light arriving at point ~xi from direction ~ωi (shownin red) makes angle θi with the surface normal. A photon from thisbeam that arrives at point~xp propagating in direction~ωp has an ex-itant trajectory~q towards the point xo = (r, θs) on the surface. Thisparticular photon path exits the material in direction ~ωo = (θo, φo).

S BSSRDF~xi Surface location of incident light (source)~ni Surface normal at~xi~ωi Incident direction

θi,φi Incident elevation and azimuthal anglesΦi Incident radiant flux at~xi~xo Surface location of exitant light~no Surface normal at~xor,θs Polar coordinates of xo w.r.t. source~ωo Exitant direction

θo,φo Exitant elevation and azimuthal angles~xp Location of a scattering event inside material~ωp Direction of incoming photon at xpΦp Power of photon arriving at~xpΦe Power of photon leaving~xp arriving at~xo~q Unit vector from~xp to~xos Single scattering plane

σs Scattering coefficientσa Absorption coefficientα Albedop Phase functiong Anisotropy factor in phase function` Mean free path lengthη Relative index of refractionFt Fresnel transmittance

xd ,yd Projection of ~ωo onto unit discν ,µ Elliptic coordinates of (xd ,yd)

~ωpeak Direction of maximum exitant intensitysp Plane of symmetry of exitant distributionH Proposed analytic distribution function

a±,b± Focal points in elliptic coordinate systemks, ke, kc Parameters in analytic distribution function

Table 1: Notation used in this paper.

where Φp is the power of the photon at ~xp, ~q = ~xo−~xp

||~xo−~xp|| is the nor-malized vector from ~xp to ~xo, p is the phase function, and Ft isthe Fresnel transmittance at the surface. If ~q ·~no > sin−1

η , where~no is the surface normal at ~xo and η is the ratio of the indices ofrefraction, then internal reflection occurs and the photon makes nocontribution. Otherwise, Φe is added to the exitant radiance at ~xo.We compute the contribution of each photon path to every exitantpoint we consider during our simulation. This results in a biased butconsistent estimate of the exitant radiance. To normalize the finalradiance values, the power of each photon is scaled by the inverseof the number of photons traced.

Variance Reduction: We use Russian Roulette to determinewhether this photon, having just traveled the distance d, is scat-tered or absorbed. If it is scattered, then we compute its newtrajectory by importance sampling the Henyey-Greenstein phasefunction [Jensen 2001]. Therefore, we do not modify the weight orpower of a photon unless it probabilistically reaches the surface, atwhich point it is internally reflected after having applied the Fresnelterm. We continue to trace each photon until it is absorbed.

2.2 A Database of BSSRDFs

Our photon tracing algorithm allows simulating the full 8D BSS-RDF in Equation 2 to construct a database we will use to developand validate our model. To accurately resolve near-source anddirectional effects, we densely sample the 2D exitant hemisphere(θo, φo) at ∼ 6000 uniformly distributed directions at each exitantlocation (r, θs) on the surface. Since the response of a material typedoes not depend on the the mean free path length `, we fix ` = 1mm

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To appear in the ACM SIGGRAPH conference proceedings

θi = 20◦, r = 1.5mm (∼ 6.6mfps), θs = 0◦ θi = 20◦, r = 0.5mm (∼ 1.8mfps), θs = 90◦

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Dipole response

Candle wax 50% diluted skim milk

Figure 3: Slices of the 2D hemisphere of exitant light at different surface positions and different angles of incidence for wax (left) and50% diluted skim milk (right). A diffusion approximation predicts a flat, diffuse response over the camera angle whereas our simulateddata and model agree well with measurements. We approximated the optical properties of milk as σs = 1.165, σa = 0.0007 (α ∼ 0.99, ` =0.86mm), g = 0.7, η = 1.35 [Joshi et al. 2006], and of wax as σs = 1, σa = 0.5 (α = 0.67, ` = 0.67mm), g = 0, η = 1.4.

in all of our simulations. During rendering we account for the truemean free path length of a specific material by simply applyingthe appropriate scale factor (Section 5). We sample the remainingBSSRDF dimensions as follows:

θi ∈ {0, 15, 20, 30, 45, 60, 70, 80, 88}r ∈ {.01, .05, .1, .2, .4, .6, .8, 1, 2, 4, 8, 10}|

θs ∈ {0, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180}α ∈ {0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99}g ∈ {−0.9,−0.7,−0.5,−0.3, 0, 0.3, 0.5, 0.7, 0.9, 0.95, 0.99}η ∈ {1, 1.1, 1.2, 1.3, 1.4}

Note that r, the distance in mean free paths from the incident beam,increases at an exponential rate to account for the fact that the totalexitant intensity diminishes along a similar trend. The other pat-terns provide a mostly uniform sampling of the remaining dimen-sions. Note that we sample this space more finely near extremelylow and high albedo and for very high forward scattering.

These sampling patterns produce about 850, 000 exitant hemi-spheres, for a total of ∼5× 109 points in this 8D space. Generatingthis amount of data poses a significant challenge since obtainingacceptable noise levels for just one hemispherical slice can requiremany hours of processing time. We performed these simulations on15 Dual-Socket Quad-Core 2.33GHz Intel R© Xeon R© machines, fora total of 120 processing cores. Reconstructing the exitant hemi-spherical distribution for a single set of optical properties requiresabout 67MB of storage–the entire database is roughly 50GB.

2.3 Validation

To verify the accuracy of our simulation and the model we proposein Section 3, we measured the distribution of exitant light for twomaterials: diluted milk and wax. These materials were illuminatedusing a red (635nm) laser dot of diameter ∼1mm at a fixed angleof θi = 20◦ and were imaged using a monochromatic QImagingRetiga 4000R camera with a 60mm lens attached to a motorizedgantry with an angular precision of ∼0.01◦. We moved the cameraalong an arc at a fixed stand-off distance of 1.5m from the sampleand recorded a high-dynamic range image every 5◦ over the rangeθo ∈ [−75◦, 75◦]. This set of images provides 1D angular slicesof the exitant hemispherical distributions for many points on thematerial surface. We repeated this procedure for two arcs: one

parallel to the plane of incident light and one perpendicular to thisplane. The camera was photometrically calibrated and the resultingimages were aligned using standard chart-based calibration tech-niques [Zhang 1999].

Figure 3 compares measured slices of the exitant light to predictionsmade by our simulation (Section 2.1), a standard diffusion approx-imation [Jensen et al. 2001] and our proposed model (we discussthe “Model Fit” curves in Section 4.1). The optical properties weused in these simulations are reported in the caption. Note thatbecause the diffusion approximation predicts a diffuse response,modulated only by Fresnel transmittance, it does not capture theclear asymmetry in the exitant lobe. Our simulation closely fol-lows the measured data. At these distances from the source thereis some visible noise in the simulations as the paths are relativelylong for high-albedo materials. Unfortunately, since the size of thelaser dot is large relative to the mean free path of these materialsit was difficult to obtain accurate measurements very close to thesource. Although these plots show only two configurations of r andθs they are representative of the close agreement we observed overthe entire material surface.

3 Data Analysis

The simulation method described previously produces 2D slices ofthe BSSRDF S(θi, r, θs, θo, φo|α, g, η) over outgoing angles θo andφo. In this section we describe a number of key observations aboutthese functions that will serve as the basis of our model presentedin Section 4. Specifically, we are interested in understanding howthese hemispherical functions vary with respect to the optical prop-erties α, g, and η , along with their dependence on the incident angleof the beam θi and surface position (r, θs).

Figure 5 visualizes several hemispherical slices produced by oursimulation at different parameter settings. The leftmost columnlists the optical properties and geometric configuration. Next tothese values is a 3D visualization of the measured outgoing lightalong with the incident beam (in red) and the path it takes into thematerial after being refracted at the surface (dashed red). The singlescattering plane s is also shown. This plane traces an arc acrossthe exitant hemisphere and is shown as a wedge emanating fromthe refracted beam. Because the overall intensity of outgoing lightis proportional to the distance from the surface location ~xo theseplots are normalized for visualization purposes. These types of vi-

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To appear in the ACM SIGGRAPH conference proceedings

sualizations are helpful in understanding how the shapes of thesedistributions are related to the angular configuration of the setupand reveal important symmetries in the data.

The third column in Figure 5 shows projections of these hemispher-ical functions onto the unit disc. The color scale, shown at right,assigns higher intensity values red and lower values blue–this samescheme is used in the 3D plots as well. We map the hemisphere tothe unit disc using a standard projection

xd =2θo

πcos φo and yd =

2θo

πsin φo. (5)

The variety of shapes present in these 2D distributions indicatesa complex relationship with the remaining parameters of the BSS-RDF. Furthermore, the presence of high-frequency features in thesedistributions means that they would not be suitable for renderingdirectly. Linearly combining these slices in order to interpolateregions of the BSSRDF domain that were not directly simulatedwould produce significant errors. This motivates our goal of pro-viding a compact analytic function that captures the shape of thesedistributions. Interpolating the parameters of this analytic functionprovides a reliable and efficient way of modeling the continuous 8DBSSRDF domain.

3.1 Phenomenological Observations

Based on this collection of simulated data we make the followingkey observations:

Anisotropy: The exitant distribution of light is not diffuse. Weobserve that even highly scattering materials, such as in Figure 5b,exhibit a non-diffuse and anisotropic shape. This follows from thefact that as light is scattered away from the propagating beam oflight it is focused near the single scattering plane. Therefore, themajority of the exitant energy is due to low-order scattering eventsthat occur before photons have lost much of their directionality.

Peak direction and kurtosis: As is true of the examples in Fig-ure 5, we observe that these distributions almost always containa single peaked lobe (exceptions are discussed in Section 7). Wealso observe that the direction of this peak depends on θi. We at-tribute this to the fact that at steeper angles more photons propagatea further distance across the surface than into the surface beforebeing scattered. The shape of this lobe also depends on the surfaceposition (r, θs) since points near s will receive a higher contributionof light. Additionally, we observed that the optical properties af-fect the shape of these exitant lobes. Higher values of α producemore scattering and thus a wider lobe. With larger magnitudes ofg, however, light is less likely to scatter outside the direction ofthe propagating beam and thus produces a tighter, sharper peak.This was especially true for forward scattering materials such asFigure 5c. Finally, since the direction of propagation of the beamdepends on η , it also affects the orientation of the lobe.

Lobe asymmetry: These lobes are typically not rotationally sym-metric about their axes and the degree of asymmetry depends pri-marily on the angle θs. As photons propagate into the material nearthe refracted beam and then scatter away towards the surface, pointson the surface receive the strongest contribution from regions nearthe propagating beam. The distributions at exitant points furtheraway from the propagating beam exhibit greater asymmetries sincethe photons contributing to these locations have traveled a greaterdistance. This asymmetry is particularly visible in Figure 5b.

Lobe shape: The lobe is often aligned with the single scatteringplane s, as in Figure 5a-d, indicating a strong contribution from low-order scattering. However, when the single scattering trajectoryexceeds the critical angle (i.e. when ~q ·~n > sin−1

η), total inter-nal reflection occurs and produces lobes with a wider, less peaked

µ = 0

µ = 0.2

µ = 0.4

µ = 0.6

µ = 0.8

µ = 1

0 0.5−0.5 1−1

−0.5

0.5

−1

1

ν = 0

ν =π2

ν =3π2

ν = π

Figure 4: The elliptic coordinates describe a set of confocal el-lipses. They provide a convenient system for representing our em-pirical BSSRDF model.

shape, but ones that are still not entirely diffuse. This is visible inFigure 5e. Here, the exitant lobe tends to be symmetric about theplane aligned with θs, as shown.

Elliptical to circular isocontours: The black isocontour linesshown in Figure 5 reveal that these lobes have an elliptical shapenear the peak which transitions to a more circular shape furtheraway from the peak. We attribute this to the number of times lightwas scattered by the material before arriving at these different ex-itant directions. Light that exits near s has likely only scatteredtwo or three times giving the exitant distribution a sharp ellipti-cal contour along the projection of the propagating beam onto thehemisphere. Photons that travel longer paths contribute less powerand are likely to exit in more uniform directions, resulting in a morediffuse distribution with circular contours.

4 An Empirical BSSRDF Model

Our goal is to model the 2D distribution of exitant light of the BSS-RDF over the possible range of geometric and optical parameters.The observations made in the previous section indicate that func-tions traditionally used to model BRDFs such as cosine or Gaussianlobes are not applicable in this case. Because these lobes also tendto be sharply peaked generic basis functions such as spherical orzonal harmonics are also unsuitable.

Indeed, the shape, position and size of the dominant lobe of exitantlight has a complex relationship with the underlying optical andgeometric properties. An important observation, however, is thatcross-sectional contours of these lobes change from elliptical tocircular away from the peak direction. This motivates our choice todefine a function for these lobes in elliptic coordinates (µ, ν) [Kornand Korn 2000]. This coordinate system defines a set of confocalellipses that have high eccentricity near the origin and slowly be-come more circular further away (see Figure 4).

The coordinates µ and ν are related to (xd , yd), the projection of ~ωoonto the unit disc given in Equation 5, according to

xd =

{a+cosh µ cos ν, xd ≥ 0a−cosh µ cos ν, xd < 0

yd =

{b+sinh µ sin ν, yd ≥ 0b−sinh µ sin ν, yd < 0

. (6)

Although a+, a−, b+ and b− are typically equal, allowing themto vary independently provides greater control over the shape of

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Simulated Distribution Model Fit

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3D ExitanceProperties

(a)

(b)

(c)

(d)

(e)

s

s

s

s

θo = 165◦

s

s

s

s

Figure 5: Representative 2D slices of S(θi, r, θs, θo, φo|α, g, η) defined over θo and φo. The left column lists the optical properties (α, g, η) andthe geometric setup (θi, r, θs). The second column contains diagrams that show the incident beam in solid red, the propagating beam insidethe material in dashed red, and the distribution of the exitant radiance. The last two columns show false-color plots of these hemisphericaldistributions projected onto the disc according to Equation 5. The “Simulated Distribution” plots were produced with our photon tracingalgorithm and the “Model Fit” plots correspond to our proposed analytic model. To aid comparison, all plots are normalized to unity. Thesingle scattering plane s is shown in the cases where the lobe is symmetric about this plane. Note that all of these distributions except for (e)exhibit this type of symmetry while (e) is symmetric about the plane formed by~xi −~xo and the surface normal~n.

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Path Tracing Our Model Photon Diffusion Diffusion Dipole

Figure 6: A beam of light striking the material surface at a 60◦ angle off the normal. The leftmost image was rendered using path tracing,and shows the expected glow arond the propagating beam. The image rendered using our model closely agrees to the reference path-tracedresult. Both photon diffusion and the diffusion dipole incorrectly predict the angular distribution of energy within this material as well as thespectral distribution of emitted light.

a lobe defined with respect to these coordinates. Note that whenµ = 0, a+ and a− define the distances of the elliptical focal pointsfrom the origin.

Isocontours and Alignment: Elliptic coordinates are well suitedfor capturing the “elliptical-to-circular” we observed in our simula-tions. Considering the projection of these slices onto the disc, weplace the origin of this coordinate system at the peak direction~ωpeak(location of maximum exitant intensity) and define the x-axis to liealong the plane of symmetry sp. This plane is either the single scat-tering plane s or the plane defined by~xi −~xo and the surface normal~n for distributions dominated by internal reflection as previouslydiscussed. Together, we propose the following analytic function ofthese 2D exitant distributions

H(~ωo ;~ωpeak , sp , ks|Γ ) = ks e−keµ − kc χ Ft(~ωo, η) , (7)

where Ft is the Fresnel transmittance in the outgoing direction

~ωo, χ =√

x2d + y2

d is the distance on the unit disc from the ori-gin ~ωpeak, ks is the overall scale of the distribution [Jason: isthis completely determined by the mean free path length? if so,say that and refer to area of paper where this is detailed], andΓ = {a+, a−, b+, b−, ke, kc} is a 6D vector that collects the freeparameters in this function. These control the lobe’s shape, degreeof asymmetry, anisotropy and kurtosis as explained below.

Asymmetry and Anisotropy: The parameters a+, a−, b+, and b−determine the asymmetry of the elliptical contours. When they aresmall the lobe is focused and as they increase the lobe becomesmore diffuse. When they are unequal the lobe is asymmetric andanisotropic.

Lobe Shape: The parameters ke and kc determine the shape of thecross-section of the lobe (i.e., capturing the transition from ellipticalto circular). The parameter ke controlls the falloff along the planeof symmetry sp. Distributions with long, high peaks have lowerke, while tightly focused peaks require larger values. The radiallysymmetric shape is determined by kc. Although these parametersare correlated to one another, especially further from the peak di-rection, maintaining these degrees of freedom provides finer controlover the precise the shape of the lobe near ~ωpeak, where most of thelight is concentrated.

4.1 Fitting the Model to Simulated Data

We fit the six parameters of H to 2D slices produced by our simu-lation for each configuration of the optical and geometry propertiesusing direct analysis. We found that general non-linear optimiza-tion routines were unnecessarily complex and often produced noisyfits which undermines our goal of smoothly interpolating these val-ues over the full BSSRDF domain.

Our fitting procedure is iterative. We first record values of theexitant distribution along a set of isocontour levels, e.g. c1 =10%, c2 = 20%, . . . , c8 = 80%, and c9 = 90% of the peak value.

For simplicity, we only examine points along the xd−axis as thissimplifies the relationship between cartesian and elliptic coordi-nates (since ν = 0 when yd = 0):

µ = cosh−1( xd

a±)

, yd = 0. (8)

At each contour sample, we calculate µ using Equation 8. Thevalues of xd at these samples are readily obtained. More than twosamples results in an overconstrained linear system computed bypartially inverting Equation 7:

− log(

ci H(~ωpeak)Ft(~ωpeak)

)= ke cosh−1

(x±da±

)+ kcx±d (9)

We solve this system to obtain values of ke and kc that best fit thesecontour samples in the least squares sense. We repeat this procedurealong the positive and negative sides of the xd−axis, and retain thesmaller of the two values for both ke and kc.

Given these values for ke and kc it is straightforward to solve fora± and b± from Equation 9. For example, for x±d :

a± =x±d

cosh

x±d kc − log( ciH(~ωpeak)Ft (~ωx)

)

ke

. (10)

The expression for y±d is similar. We solve for these values at eachcontour level, and choose the set of a± and b± which minimizesthe total L2 error over the hemisphere with respect to the originalexitant distribution. We then re-estimate ke and kc using these newvalues of the other parameters and repeat this process ten times oruntil we observe the error changes by less than 1%.

4.2 Final Model

For each slice of the BSSRDF computed in our simulation we storethe best fit parameters of our model H in a large table, a totalof ∼ 100MB.3 Because the two dimensions of ~ωo are the mostdensely sampled in our simulation this is a significant reduction,from ∼6000 points on the hemisphere to the set of inputs to H (12values). Finally, note that in practice only a fraction of this data isused at any one time.

4.3 Model Accuracy

Figure 3 compares the best fits of our model to simulated data, mea-sured data and an approximation of these angular distributions ob-tained with a standard dipole diffusion model. These results showstrong agreement between measured and simulated data and ourproposed model.

The fourth column of Figure 5 compares fits of our model H tosimulated data for several representative exitant distributions. Al-

3 This data can be downloaded, along with the raw simulated data and themeasurements used in Figure 3, from http://www.whatever.

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To appear in the ACM SIGGRAPH conference proceedings

though our model does not provide an exact match it successfullycaptures the basic trends in these distributions over a wide range ofoptical properties and geometric configurations. In particular, notethat H exhibits the characteristic “elliptical-to-circular” pattern.

We also computed the RMS error of our fits over the entire collec-tion of simulated data. These errors have a mean of 3.2× 10−3 andstandard deviation of 9.0 × 10−4. Note that these values have thesame units as the BSSRDF of [m−2sr−1]. The RMS error of thefits shown in Figure 5 from (a) to (e) are 9.3 × 10−6, 1.3 × 10−3,4.2 × 10−6, 3.6 × 10−6, and 4.0 × 10−5, respectively. We also val-idated the process of interpolating the parameters of H to predictexitant distributions that were not directly simulated. We computedthe RMS error between each simulated distribution and that pro-duced by interpolating the parameters of our model using thosevalues at its nearest neighbors – this is akin to a “leave one out”cross validation test. These errors had a mean of 4.9 × 10−4 witha standard deviation of 6.9 × 10−7. We conclude that our model iscapable of fitting individual slices with a high degree of numericalaccuracy and that these parameters can be safely interpolated toreconstruct a continuous representation of the BSSRDF over therange of geometric and optical properties considered in our simula-tions.

As further validation of our model and to compare it to alterna-tive methods, we produced renderings of a beam of light enteringa material with α = (0.07, 0.53, 0.52) and g = 0 (see Figure 6).Although all four rendering methods capture single scattering well,both the diffusion dipole and photon diffusion inaccurately predictthe color and intensity of multiply scattered light. Our method, onthe other hand, captures the correct wavelength-dependence alongwith the directional effects of the internally scattered light.

5 Rendering with the Model

We sample the illumination incident on translucent materials usingstandard techniques (e.g. [Jensen et al. 2001]). To compute a set ofsurface locations with respect to a single shade point, we draw sam-ples from a probability density function proportional to the exitantdiffuse distribution based on the scale factor ks in H. Alternatively,we could use a hierarchical point-based approximation of the irra-diance [Jensen and Buhler 2002] to perform this sampling. Thiswould decrease rendering times without affecting the model itself.

Interpolation: We first reconstruct the BSSRDF that correspondsto the optical properties of the material we are interested in ren-dering. We linearly interpolate the parameters to H based on theclosest neighboring points along the α, g, and η axes. This gives acompact (∼100KB) 3D reconstruction defined over θi, r, and θs.

For a shade point~xo and lit point~xi on the surface we compute r andθs with respect to an orthogonal coordinate system with its x−axisalong X = ~ωi − (~ωi ·~ni)~ni. The surface normal at~xi is~ni which weassume points up and the z-axis of this coordinate system is definedas Z = X ×~ni. This construction leaves θs as the rotation:

θi = cos−1 (~ωi ·~ni)

r = (σs + σa) ||~xo −~xi||

θs = tan−1(

(~xo −~xi) · Z(~xo −~xi) · X

) (11)

To determine the interpolated locations along the r dimension wecompute the distance in terms of mean free paths by scaling by theinverse of the mfp length, i.e. σs + σa.

Handling non-planar geometry Once we have interpolated the pa-rameters of our elliptic function as described above, we next deter-mine the values of θo and φo to use when evaluating H. Because our

Figure 7: A translucent green lozenge with albedo α =(0.16, 0.27, 0.15). Note the global illumination between the sub-surface scattering as predicted by our empirical BSSRDF modeland the ground plane.

model is only valid for semi-infinite materials, care must be takenwhen rendering arbitrary geometry. [Jason: we need to expand onthis point...what type of care, more specific]

We define a separate coordinate system anchored at ~xo using thenormal~no, Xh = ~no × Z and Zh = Xh ×~no, and derive θo, φo as:

θo = cos−1 (~ωo ·~no)

φo = tan−1(

(~ωo − (~ωo ·~no)~no) · Z(~ωo − (~ωo ·~no)~no) · X

).

(12)

Using these values, we locate the nearest 8 points in the 3Ddataset, and interpolate the values of the model parameters(~ωpeak , sp , ks|Γ ). Finally, we calculate the value of the BSSRDFby evaluating H in Equation 7 using these interpolated values.

6 Results

All of the results in this paper were rendered on an Intel R© Xeon R©

2.33GHz Quad-Core processor and those produced with our modelrequired less than one hour of processing time.

Figure 1 compares renderings of orange juice produced using ourmethod to those produced with Monte Carlo path tracing and thediffusion dipole combined with a single scattering term [Jensenet al. 2001]. We used parameters σs = (0.071, 0.1, 0.042), σa =(0.093, 0.16, 1.15), g = 0.9, and η = 1.3, which produces a rela-tively low spectral albedo of α = (0.43, 0.38, 0.035). Since orangejuice is highly forward scattering, simulating single scattering alonedoes not capture its true appearance. However, high-order multiplescattering is also not a dominant effect and the image rendered us-ing the diffusion dipole is too dark. This is because the dipole modelassumes that anisotropic scattering is balanced by many scatteringevents. However, the low albed of this material causes most ofthe light to be absorbed before reaching this regime. Because ourmodel (center) more accurately captures these low order scatteringevents it matches the reference path traced image, but requires sig-nificantly less time to compute.

Figure 7 shows a green lozenge with significant translucency. Thismaterial has a spectral albedo of α = (0.16, 0.27, 0.15) and isforward scattering with g = 0.5. The scene has global illumina-tion; note the bleeding of green light from the lozenge onto the

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To appear in the ACM SIGGRAPH conference proceedings

Clorox Whole Milk Espresso Era Head & Shoulders

Figure 8: Renderings using parameters from [Narasimhan et al. 2006] to show that our model is capable of simulating a diverse range ofmaterials. This image would have required many hours to compute using path tracing and significant storage (many photons) to render usingphoton mapping.

ground plane. With our model, it is possible to efficiently rendera much larger range of translucent materials than previously pos-sible, even in scenes with global illumination. To demonstrate thegenerality of our model we rendered the image shown in Figure 8which combines several of the materials measured by Narasimhanet al. [2006]. With the exception of milk, these materials havesignificant absorption and anisotropic scattering which make themimpossible to render correctly using standard diffusion-based tech-niques. Our model is also able to accurately render highly scatteringmaterials that would present difficulty for diffusion-based methodssuch as milk. The total amount of data required to render this imagewas about 600KB, as each material requires about 100KB.

7 Limitations

Our analytic formula of the exitant distribution of light H did fail tocapture the proper response for some of the materials we observed.In particular, materials with strongly anisotropic backscattering(g � 0) produce two exitant lobes [Jason: add figure showing highback-scattering material]. One is due to low-order scattering andis in the expected direction, but the other is caused by higher-orderscattering. Clearly, our model consists of only a single lobe andcannot handle these cases. However, because most real-world ma-terials are forward scattering we concentrated on modeling the moreprominent lobe and leave these cases for future study.

When light arrives at the surface along grazing angles, internal re-fraction becomes more prevalent and our model is less accurate inthese cases. Also, our assumption of a semi-infinite and flat surfaceoverestimates the exitant light near corners and thin geometric fea-tures. [Jason: say more about this point, including some mention ofthe fact that prior work shares this limitation and the time has cometo study the effect of geometry on these local subsurface scatteringfunctions]

Further research is warranted to determine a simpler relationshipbetween the optical and geometric properties of the BSSRDF andthe parameters in the analytic function H. While some preliminarywork in this direction has indicated that this mapping is complexand non linear, developing a low-dimensional (possibly analytic)relationship would shed further light on this important class of func-tions and increase the efficiency and accuracy of simulating thesetypes of materials.

8 Conclusion

We presented an empirical model of the BSSRDF that is valid overa far wider range of angular configurations and material proper-

ties than existing analytic models. Our model captures both near-source and directional effects including the important contributionof low-order scattering. This model was derived from a large-scalesimulation of the hemispherical distribution of light leaving a ma-terial’s surface over a range of positions from the source, incidentangles and underlying optical properties (scattering and absorptioncoefficients, phase function, and index of refraction). We presentedan analytic function to approximate these hemispherical functionswhich is expressed in elliptic coordinates and has six parameters.We estimated the best-fitting parameters for our simulated data. Be-cause these parameters vary smoothly with respect to the remainingdegrees of freedom they may be interpolated between simulatedlocations to provide a compact yet continuous representation overthe full BSSRDF domain. This allows generating realistic imagesof translucent materials with notoriously difficult optical proper-ties. In particular, many of the results we reported would have beenimpossible to render using diffusion based methods and much lessefficient with more general numerical integration techniques.

9 Acknowledgments

We thank the reviewers for their helpful input along with XXX.Jason Lawrence acknowledges a NSF CAREER award CCF-0747220, NSF grant CCF-0811493 and an NVIDIA Professor Part-nership Award.

References

BLINN, J. F. 1982. Light reflection functions for simulation ofclouds and dusty surfaces. In Computer Graphics (Proceedingsof ACM SIGGRAPH 1982), ACM, vol. 16, 21–29.

CEREZO, E., PEREZ-CAZORLA, F., PUEYO, X., SERON, F., ANDSILLION, F. 2005. A survey on participating media renderingtechniques. The Visual Computer.

DANA, K., GINNEKEN, B., NAYAR, S., AND KOENDERINK, J.1999. Reflectance and texture of real-world surfaces. ACMTransactions on Graphics 18, 1 (January), 1–34.

DONNER, C., AND JENSEN, H. W. 2005. Light diffusion inmulti-layered translucent materials. ACM Trans. Graphic. 24,3, 1032–1039.

DONNER, C., AND JENSEN, H. W. 2007. Rendering translucentmaterials using photon diffusion. In Rendering Techniques, 243–251.

9

To appear in the ACM SIGGRAPH conference proceedings

GOESELE, M., LENSCH, H. P. A., LANG, J., FUCHS, C., ANDPETER SIEDEL, H. 2004. DISCO: aquisition of translucentobjects. ACM Trans. Graphic. 23, 3, 835–844.

HANRAHAN, P., AND KRUEGER, W. 1993. Reflection from lay-ered surfaces due to subsurface scattering. In Proceedings ofACM SIGGRAPH 1993, ACM Press/Addison-Wesley PublishingCo., New York, Computer Graphics Proceedings, 164–174.

JENSEN, H. W., AND BUHLER, J. 2002. A rapid hierarchical ren-dering technique for translucent materials. ACM Trans. Graphic.21, 576–581.

JENSEN, H. W., LEGAKIS, J., AND DORSEY, J. 1999. Renderingof wet materials. In Rendering Techniques, 273–282.

JENSEN, H. W., MARSCHNER, S. R., LEVOY, M., AND HANRA-HAN, P. 2001. A practical model for subsurface light transport.In Proceedings of ACM SIGGRAPH 2001, ACM Press/Addison-Wesley Publishing Co., New York, Computer Graphics Proceed-ings, ACM, 511–518.

JENSEN, H. W. 1996. Global illumination using photon maps. InRendering Techniques, 21–30.

JENSEN, H. W. 2001. Realistic Image Synthesis Using PhotonMapping. AK Peters.

JOSHI, N., DONNER, C., AND JENSEN, H. W. 2006. Noninva-sive measurement of scattering anisotropy in turbid materials bynonnormal incident illumination. Opt. Lett. 31, 936–938.

KAJIYA, J. T. 1986. The rendering equation. In Proceedings ofACM SIGGRAPH , 143–150.

KORN, G. A., AND KORN, T. M. 2000. Mathematical Handbookfor Scientists and Engineers: Definitions, Theorems, and Formu-las for Reference and Review. Courier Dover Publications.

LI, H., PELLACINI, F., AND TORRANCE, K. 2005. A hybridmonte carlo method for accurate and efficient subsurface scatter-ing. In Rendering Techniques, 283–290.

MATUSIK, W., PFISTER, H., BRAND, M., AND MCMILLAN, L.2003. A data-driven reflectance model. ACM Transactions onGraphics (SIGGRAPH 03) 22, 3, 759–769.

NARASIMHAN, S. G., GUPTA, M., DONNER, C., RAMAMOOR-THI, R., NAYAR, S., AND JENSEN, H. W. 2006. Acquiringscattering properties of participating media by dilution. ACMTrans. Graphic. 25, 1003–1012.

NICODEMUS, F. E., RICHMOND, J. C., HSIA, J. J., GINSBERG,I. W., AND LIMPERIS, T. 1977. Geometrical Considerationsand Nomenclature for Reflectance. National Bureau of Stan-dards.

PEERS, P., VOM BERGE, K., MATUSIK, W., RAMAMOORTHI, R.,LAWRENCE, J., RUSINKIEWICZ, S., AND DUTRÉ, P. 2006.A compact factored representation of heterogeneous subsurfacescattering. ACM Trans. Graphic. 25, 3, 746–753.

PHARR, M., AND HANRAHAN, P. 2000. Monte carlo evaluationof non-linear scattering equations for subsurface reflection. InProceedings of ACM SIGGRAPH 2000, ACM Press/Addison-Wesley Publishing Co., New York, Computer Graphics Proceed-ings, 75–84.

STAM, J. 1995. Multiple scattering as a diffusion process. InRendering Techniques, 41–50.

WARD, G. J. 1992. Measuring and modeling anisotropic reflection.In Proceedings of ACM SIGGRAPH , 265–272.

WESTIN, S. H., ARVO, J. R., AND TORRANCE, K. E. 1992.Predicting reflectance functions from complex surfaces. In Com-puter Graphics (ACM SIGGRAPH 1992), ACM Press, NewYork, NY, USA, 255–264.

ZHANG, Z. 1999. A flexible new technique for camera calibration.IEEE Transactions on Pattern Analysis and Machine Intelligence22, 11, 1330–1334.

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