A BRDF Database Employing the Beard-Maxwell Reflection Model

Harold B. Westlunda Gary W. Meyerb

Department of Computer and Information ScienceUniversity of Oregon · Eugene, Oregon · U.S.A.

AbstractThe Beard-Maxwell reflection model is presented as

a new local reflection model for use in realistic imagesynthesis. The model is important because there is apublic domain database of surface reflection parameters,the Nonconventional Exploitation Factors Data System(NEFDS), that utilizes a modified form of the Beard-Maxwell model. Additional surface reflection parametersfor the database can be determined because a measure-ment protocol, using existing radiometric instruments,has been specified. The Beard-Maxwell model is alsoof historical significance because it predates many com-puter graphics reflection models and because it includesseveral features that are incorporated into existing localreflection models. The NEFDS is described and a spe-cial shader is developed for use with NEFDS. The shadermakes use of the alias method for determining randomvariates from discrete probability distributions. Realis-tic images are synthesized from the existing database andfrom samples that were characterized using the measure-ment protocol.

Key words: BRDF, local illumination

1 Introduction

Computer graphics has made great progress in the simu-lation of object appearance. Reflection models have beendeveloped that characterize both the spectral and the spa-tial distributions of light reflected from an object’s sur-face. Image synthesis systems have been constructed thatuse Monte Carlo techniques to evaluate these reflectionmodels and simulate the objects in the context of glob-ally illuminated environments. With sufficient process-ing time, these programs are capable of producing in-dividual photorealistic images. Recent developments inthe area of real-time shading have made it possible toutilize sophisticated reflection models in interactive pro-grams [18, 20]. These advances in real-time rendering are

aCurrent address: Radical Entertainment, Vancouver, BC, Canada.hwestlund@radical.ca

bCurrent address: Department of Computer Science and En-gineering, University of Minnesota, Minneapolis, MN, U.S.A.meyer@cs.umn.edu

likely to bring renewed attention to the subject of surfacereflection modeling.

Unfortunately, the accurate synthesis of object appear-ance is currently limited by the small amount of read-ily available surface reflectance data. Many sophisti-cated computer graphics reflection models have been pro-posed. The best of these models characterize both thespectral and the spatial distributions of reflected light andare therefore appropriate for modeling object appearance.These models have a sound theoretical foundation andoften contain measurable surface reflection parameters.Regrettably, measurement protocols and actual measureddata are seldom available for the models. This situationis in contrast to object shape information where polygondata is now commercially available.

This paper presents the Beard-Maxwell [30] reflec-tion model which has not been discussed in the com-puter graphics literature even though a large database ofmeasurements exists for the model. The model is of his-torical importance because it predates many of the cur-rent computer graphics reflection models and foreshad-ows several of the features that have been incorporatedinto these models. The model has practical significancebecause the public domain Nonconventional ExploitationFactors Data System (NEFDS) [1] utilizes a modified ver-sion of the Beard-Maxwell model and contains over 400materials for which the parameters of the modified Beard-Maxwell model have been measured. Furthermore, theNEFDS can be extended because a measurement protocolexists, using standard radiometric instruments, to acquiremodel parameters for additional materials.

The paper also describes a rendering system that hasbeen developed for use with NEFDS. The Radiance soft-ware package [2] has been modified so that it can be em-ployed to make pictures with data taken from NEFDS.Since the Beard-Maxwell model is not an invertible func-tion, this necessitated the use of a probability mass func-tion (the discrete counterpart of the probability den-sity function) and the alias method for generating ran-dom variates. The research discussed in this paper alsoincludes an attempt to add additional material to thedatabase. The measurements that were necessary to ac-

complish this are described, and the quality of the ap-proximation provided by Beard-Maxwell and NEFDS isevaluated.

This paper is divided into four major sections. In thenext section, a brief review is done of existing computergraphics reflection models. The following section in-troduces the complete Beard-Maxwell model and showshow it has been modified for use in the NEFDS. Anoverview of the NEFDS is given in the penultimate sec-tion and example renderings that were made using thedata are discussed in the last section. These pictures weremade using a modified version of a public domain render-ing program and data from both NEFDS and from newmeasurements made using the NEFDS measurement pro-tocol. The paper concludes with a summary of the workand suggestions for further research.

2 Background

2.1 BRDFIn order to create a realistic image of an object, the lightreflection properties of the object’s surface must be spec-ified. The most common means of quantifying surfacereflection of light is by utilizing the bi-directional re-flectance distribution function (BRDF). The BRDF, ρ, isdefined as the ratio of differential reflected radiance todifferential incident irradiance:

ρ(Θi; Θr;λ) =dLr(Θi; Θr;λ)

dEi(Θi;λ)

=dLr(Θi; Θr;λ)

dLi(Θi;λ) cos θidωi(1)

where the subscripts i and r denote incident and reflectedrespectively, Θ = (θ, φ) is the direction of light propa-gation, λ is the wavelength of light, L is radiance, E isirradiance, and dω is an element of solid angle [31]. Thissurface reflection geometry is shown in Figure 1.

2.2 Physics Based BRDFsOver the past century, a wide variety of analytical reflec-tion models have been created to represent the physics oflight reflection off surfaces. They have been developed inthe fields of physics and engineering accompanying thedevelopment of radio transmission, radar, heat transfertheory, remote sensing, astronomy, and many other areasof research. A few relevant models are presented here.

In 1963, Beckman and Spizzichino [6] developed amodel of reflection based upon wave optics. Their re-flection model was derived through the use of the Kirch-hoff approximation of the Helmholtz boundary condi-tions. Also in 1963, Hapke [15] developed a reflectionmodel which predicted the light reflection off the lunarsurface using scattering theory. He was able to account

Figure 1: Light reflection geometry

for both forward and backward scatter through his selec-tion of the scattering function.

Torrance and Sparrow [37], in 1967, created an an-alytical reflection model based on a surface composedof small, randomly distributed, mirror-like facets. Thesemicro-facets were distributed using a Gaussian based dis-tribution, which was in correspondence to the measuredsurface reflection. Their model was novel in its inclusionof a term for representing the shadowing and masking ofneighboring facets which helped to predict the measuredoff-specular peak.

2.3 Physics Based BRDFs in Computer GraphicsThe work of Torrance and Sparrow was first introduced tocomputer graphics by Blinn [7] who extended it by utiliz-ing the facet distribution proposed by Trowbridge and Re-itz [38]. This distribution is built on the assumption thatthe facets are ellipsoids of revolution. Another adapta-tion of Torrance and Sparrow’s reflection model to com-puter graphics was offered by Cook and Torrance [10].They incorporated the Fresnel term into the model to cap-ture the wavelength dependency of first surface reflection.Recently Ashikhmin et al. [5] have shown how to pro-duce BRDFs from very general micro-facet distributionsby using a model with a simple shadowing term.

Blinn [8] utilized the work of Hapke to develop a re-flection model for use in the simulation of the rings ofSaturn. He utilized several functions to represent thephase function of light off the scattering particles, includ-ing the Henyey-Greenstein [19] function and a functionof Rayleigh scattering. Blinn also discussed the exten-sion of the model to account for shadowing and multiplescattering.

He et al. [16] utilized the work of Beckman andSpizzichino in addition to the work of Torrance andSparrow to generate a model which accounted for manyphysics derived attributes. Their model utilized polariza-

tion of light, directional Fresnel effects and a more com-plex distribution of micro-facets. It was the first model toprovided a unified means of predicting scatter from sur-faces with autocorrelation distances much smaller, on thesame order, or much greater than the wavelength of in-cident light. Beyond He et al., Stam [36] is one of thefew to have attempted the development of more generalphysics based BRDFs in computer graphics.

2.4 Empirical BRDFs in Computer Graphics

As pointed out by Ward [29] the very complexity thatgives these physics based BRDFs validity also hamperstheir general use in computer graphics. In Ward’s paperhe presented a reflection model using an elliptical Gaus-sian based formula which is ideally suited for fitting tomeasured data. Ward demonstrated its validity throughmeasurement of several surface BRDFs which were fit tohis reflection model.

Lafortune et al. [26] presented a BRDF model builtupon sums of cosine lobes specifically constructed tofit measured BRDF data. The number, orientation, andshape of the cosine lobes are variable, allowing for a con-vincing fit to a wide variety of BRDF distributions.

Dana et al. [11] fit measured data to both the Oren-Nayer model [32] and the Koenderink et al. decomposi-tion [22]. Oren and Nayer built a model of diffuse reflec-tion based on direct and indirect reflections off diffusemicro-facets. Koenderink et al. decomposed the BRDFinto a variable-order vector of coefficients, fit to data.

2.5 BRDF Storage Representations in ComputerGraphics

Fitting measured data to a BRDF model is avoided if thedata rather than the model parameters is stored. Gondeket al. [14] utilized this technique with an adaptive subdi-vision scheme over the sampled reflection hemisphere.Schroder and Sweldens [35] as well as Lalonde andFournier [27] offer an efficient method of storing and uti-lizing BRDFs using spherical wavelets. Cabral et al. [9]and Westin et al. [47] offered the same using sphericalharmonics.

3 A New Reflection Model

The modified Beard-Maxwell model presented in this pa-per incorporates the best attributes of physical and empir-ical BRDFs. It is a physics based model capturing subtleBRDF characteristics required in realistic image synthe-sis. Its empirical measurements are themselves built uponthe physical model, providing a means to set the requiredparameters accurately with relatively little effort. Sinceonly a minimal set of defining parameters are used, themodel requires minimal storage space.

3.1 Beard-Maxwell Reflection ModelThe Beard-Maxwell model presented by Maxwell etal. [30] was originally used to describe the reflectionproperties of rough, painted surfaces displaying Fresneleffects and later was applied to a wider variety of sur-faces. Their model is built on the assumption that thematerial surface is a three dimensional terrain of micro-facets of varying orientation. In this model, reflectedlight is the result of only two physical occurrences. Lightis reflected off one of the micro-facets (first surface re-flectance) and light is scattered out of the surface afterhaving first entered the sub-surface medium (volumetricreflectance). The Beard-Maxwell reflectance model thustakes the form

ρ(Θi; Θr) = ρfs(Θi; Θr) + ρvol(Θi; Θr) (2)

where ρfs and ρvol are the first surface and volumetric re-flectance functions respectively. For the sake of nota-tional clarity the wavelength of this wavelength depen-dent reflectance function is not listed as an explicit pa-rameter. We will take this same liberty to hide other func-tional parameters which are tangential to current discus-sions.

First surface reflection causes light to be reflected inthe specular direction (i.e., mirror reflection) off each in-dividual micro-facet as determined by the micro-facet’snormal rather than the macro-surface normal. There-fore the distribution of the first surface reflectance is de-termined by the distribution of the micro-facet normalswhich in turn is driven by the density function Ξ(Θ), therelative density of micro-facet normals (per steradian) indirection Θ. Maxwell et al. calculated the first surfacereflectance to be

ρfs(Θi,Θr) =R(β)Ξ(H)

4 cos θi cos θrSO (3)

where H is the half angle vector, β is the bistatic angle(i.e., the angle between either the incident or reflecteddirection and the half angle vector), R(β) is the Fresnelreflectance and SO is a shadowing and obscuration1 term.Figure 2 is a diagram of the geometry used by the Beard-Maxwell model.

The shadowing and obscuration term in (3), SO, ac-counts for the height distribution of the micro-facets.Shadowing and obscuration are due to intersections be-tween the other surface facets and the incident and re-flected light rays respectively. Torrance and Sparrow [37]accounted for the light ray intersections with a purely an-alytical function derived from theory. Blinn [7] and Cook

1Obscuration is often termed masking in the literature, but we willuse obscuration here to follow the original discussion in both Maxwellet al. [30] and the NEFDS documentation [4].

Figure 2: Beard-Maxwell BRDF model geometry

and Torrance [10], who were the first to introduce shad-owing and obscuration in computer graphics, also used apurely analytical function. Beard and Maxwell chose tocreate a function with free parameters set using empiri-cal data. This function, which has a maximum value ofone for both specular reflection (θH = 0) and pure back-scatter (β = 0), is defined as

SO =1 +

θHΩ e−2β/τ

1 +θHΩ

(1

1 + φnΩθiΩ

)(4)

where the free parameters Ω and τ modify the falloff ofSO in the forward-scattered and back-scattered directionsrespectively, and φn is a parameter computed from geom-etry which affects the rate of change of SO [30]. The pro-cess of gathering empirical data used to set the equivalentfree parameters in the modified Beard-Maxwell model isdiscussed below in Section 3.2.

Rather than attempting to measure Ξ of (3) directly,Maxwell et al. replaced it with the measured zero-bistatic(β = 0) first surface reflectance at the half angle,ρfs(H, H). SO = 1 when β = 0, so (3) can be rewrittenas

Ξ(H) =4ρfs(H, H) cos2 θH

R(0). (5)

This is simplified further by their assumption that the sur-face is isotropic. In that case to generate reflectance val-ues for all incident and reflected directions, ρfs(H, H)needs only be sampled at the angles 0 ≤ θH ≤ π

2 andφ = 0.

Light reflected from the first surface is assumed tomaintain its original polarization (i.e., incident and re-flected light are of like polarization) while any light re-flected through volumetric scattering is assumed to betotally depolarized. In this way, separation of the firstsurface reflectance from the volumetric reflectance caneasily be performed by using a polarized illuminant and

then measuring the like and cross polarized light. Thefirst surface reflectance can be expressed in measurablequantities by using the four combinations of polarization,ρ⊥⊥, ρ⊥‖, ρ‖⊥, and ρ‖‖2:

ρfs = (ρ⊥⊥ − ρ⊥‖) + (ρ‖‖ − ρ‖⊥). (6)

Similarly, the volumetric reflectance can be expressed inmeasurable quantities:

ρvol = 2ρ⊥‖ + 2ρ‖⊥ (7)

The volumetric reflectance of (2), ρvol is representedby either Lambertian reflectance, ρD, or directional dif-fuse reflectance, ρddif:

ρvol =

ρD

ρddif =2ρvf(β)g(θN )

cos θi+cos θr

(8)

where ρv is the base directional diffuse reflectance whenboth θi and θr are zero and where f(β) and g(θN ) areempirically determined functions that characterize differ-ential subsurface scattering and substrate reflection re-spectively. The directional diffuse component accountsfor any subsurface directional scattering and is selectedonly if there is any directional reflection unaccountedfor by the first surface reflectance. The Lambertian re-flectance is selected if the first surface reflectance ac-counts for all directional reflectance. The directional dif-fuse component of (8) is based on a model of a subsurfaceof finite thickness built on a reflective base. Maxwell etal. found that in practice surfaces were well representedby using a constant valued scattering function and bymodeling the surface as having infinite thickness. Withthese simplifications, f and g can be both set to unity andthe volumetric reflectance becomes:

ρvol =

ρDρddif = 2ρv

cos θi+cos θr

(9)

3.2 Modification to Beard-MaxwellThe NEF uses a modified form of the Beard-Maxwell re-flection model to characterize the reflection properties ofsurfaces in the NEF database. The NEF Beard-Maxwell(NEF-BM) model includes changes in an attempt to moreaccurately represent surfaces other than paint, the originaltarget of the Beard-Maxwell model. Additionally, fixedradiometric measurements are introduced which can be

2We follow the same notation as in Maxwell et al. to denote polar-ized reflectance by ραiαr , where the orientations of αi and αr indicatethe polarization of the incident and reflected light respectively. The po-larization state of the incident light is either parallel (‖) or perpendicu-lar (⊥) to the plane of incidence. The polarization state of the reflectedlight is described in the same way, but with respect to the reflectanceplane.

used to determine appropriate values for the model’s freeparameters.

In the Beard-Maxwell model, only one volumet-ric component was utilized, either the Lambertian re-flectance or the directional diffuse reflectance. In theNEF-BM model both are used. The volumetric Lamber-tian reflectance is referred to as simply diffuse reflectanceand is attributed to multiple first surface reflections. Thevolumetric directional diffuse reflectance is itself referredto as volumetric reflectance, ρvol, and is attributed to allthe reflectance due to subsurface light interaction.

The NEF-BM model uses a simplified form of (4) forits shadowing and obscuration function:

SO =1 +

θNΩ e−2β/τ

1 +θNΩ

.

The model also uses the simplified form of directionalsubsurface scattering from (9). So the complete NEF-BMBRDF can be written

ρ(Θi; Θr) =R(β)

R(0)

ρfs(θH , 0; θH , 0) cos2 θHcos θi cos θr(

1 +θHΩ e−2β/τ

1 +θHΩ

)+ ρD +

2ρvcos θi + cos θr

(10)

As in the BM model, the NEF-BM performs the zero-bistatic BRDF measurements at all four combinations ofpolarization using (6) and (7) to generate the first surfacereflection parameters as well as the diffuse and volumet-ric coefficients. Additionally the NEF-BM utilizes twoother series of radiometric measurements.

The first additional set of measurements is a seriesof specular BRDF measurements for parallel polarizedsource and receiver over 10 deg ≤ θi ≤ 80 deg withφi = 0 deg and φr = 180 deg. For each sampled incidentdirection, the receiver direction is varied about the mirrordirection in the plane of incidence by θi − 5 deg ≤ θr ≤θi + 5 deg. Data obtained from these measurements pro-vides a means of calculating the complex index of refrac-tion components, n and k which are required for comput-ing the Fresnel reflection coefficient, R. Determinationof n and k is performed by utilizing Brewster’s angle,the angle of minimum reflectance of parallel-polarizedlight [17, 4].

The second series of measurements uses a fixed sourcedirection and varying receiver direction, in the plane ofincidence, over −90 deg ≤ θr ≤ 90 deg. These mea-surements offer means to select the shadowing and ob-scuration parameters, τ and Ω, in order to reduce the er-ror away from the forward-scattered and back-scattereddirections. The values of τ and Omega are set using a

least square fit of the modeled BRDF, with shadowingand obscuration, to this measured data. This second se-ries of measurements also provides a good verification ofthe previously selected model parameters.

We now introduce the dependency of the BRDF on thewavelength of light which has hitherto been consideredconstant. The naive method of characterizing spectral de-pendency would be to perform all required measurementsat a sufficiently dense sampling of wavelengths. Howeverthis would undermine one of the primary motivations ofthe NEF-BM model which is to represent the BRDF us-ing relatively few measurements.

The solution was found in the observation that formany surfaces, the BRDF at one wavelength is similarin shape, but not necessarily magnitude, to the BRDF ofthe surface at nearby wavelengths. To utilize this infor-mation, the measurements detailed above are only per-formed at key reference wavelengths and intermediatevalues are linearly interpolated. The interpolated valuesare then scaled by directional hemispherical reflectance(DHR) [31] values, ρd(λ), measured at a dense samplingof wavelengths. The full spectral dependent BRDF thenis:

ρ(Θ;λ) = ρd(λ)[

ρ(Θ;λj)∫ ∫ρ(ΘDHR;λj) cos θidθidφi

(λk − λλk − λj

)+

ρ(Θ;λk)∫ ∫ρ(ΘDHR;λk) cos θidθidφi

(λ− λjλk − λj

)](11)

where j and k are the indices of the reference wave-lengths bounding λ, Θ is the arbitrary geometry (Θi; Θr),and ΘDHR is the geometry of DHR measurement(Θi; ΘDHR).

This method offers the additional benefit of increasedaccuracy. DHR measured values are typically more ac-curate than the corresponding integrated values of mea-sured BRDFs. Scaling the BRDFs to ensure their inte-grated values match the DHR values corrects propagatedmeasurement errors.

3.3 Limitations of the ModelThese measurements generate a reflection model whichfaithfully represents the actual reflection behavior ofmany target surfaces. However, a model is an approxi-mation; there are limitations to the types of BRDFs it canaccurately represent. Some surface BRDFs simply can-not be accurately approximated with the NEF-BM modelusing any combination of parameter values. For a modelwhich was initially constructed to represent simple paintreflection, it covers a broad range quite accurately, butcare must be used in applying it to new BRDFs.

Another limitation of the NEF-BM model is its inac-curacy at grazing angles. The model was created primar-ily for remote sensing. Because remote sensing involvesinterpreting imaged data from aircraft or satellites, accu-racy at grazing angles was not a large concern. For usein computer graphics, however, grazing angles are to beexpected in every scene.

The problem lies in the fact that the NEF-BM BRDFat grazing incidence, Θgi, integrated over the reflectedhemisphere, results in values greater than one:

∫

Ωr

ρ(Θgi; Θr)dΩr =

∫

Ωr

ρfs(Θgi; Θr)dΩr

+

∫

Ωr

ρDdΩr +

∫

Ωr

2ρvcos θgi + cos θr

dΩr

= 1 + πρD + 4πρv

Consequently, light incident at grazing angles will resultin more energy reflected than was incident. In practicethis has not been a problem mainly because of the fore-shortening of incident energy at grazing angles.

4 NEFDS

The NEFDS allows those working in the area of re-mote sensing to perform complex radiometric calcula-tions while taking into consideration the properties of thetarget material, the material background, the measuringsensors, and the atmospheric environment. At the heartof this system, and of particular interest to researchersin the field of computer graphics, is the BRDF database.This database contains the BRDFs of hundreds of sur-faces from a dozen different categories. In this section,we will present a discussion of the BRDFs available andoffer an overview of the relevant portions of the NEFDS.

4.1 Available Material TypesThere are currently over 400 materials available in theNEFDS. These materials correspond to a wide variety ofobjects ranging from dirt to tree canopies. The materi-als fall into 12 different categories: asphalt, brick, cam-ouflage, composite, concrete, fabric, water, metal, paint,rubber, soil and wood.

Although the number of materials is large and the va-riety is wide, the selection is limited by two key condi-tions. One requirement for the inclusion of a material inthe database is that the BRDF of the material must bewell represented by the NEF-BM BRDF model. For ex-ample, the BRDF should not be characterized by extremeanisotropy since the NEF-BM BRDF model only workswith isotropic data. (There are in fact some materials in-cluded in the NEFDS which are anisotropic. To representanisotropic material using the NEF-BM model, the mate-rial’s BRDF is averaged to isotropy.)

Figure 3: Spectral reflectance function for material 0221,red dirt

The second condition is a result of the data’s origin. Asmentioned before, the main use of the NEFDS has been inthe area of remote sensing. Because of this, the materialsin the database were selected on the basis of applicabilityto that field of work. They include objects which wouldbe viewed from a remote sensor (e.g., a satellite). Thisstill offers a potential wealth of BRDF data for use incomputer graphics but is currently constrained to this onearea. This however does not restrict future measurementand the inclusion of other materials having more directapplicability to computer graphics.

4.2 Operational Modes

NEFDS can be accessed either through the XWindowsinterface, NefMenu, or by using command line control.A discussion of the full control offered by NEFDS is be-yond the scope of this paper (see the NEF Users Guidefor detailed discussion [1]). Instead a brief overview willbe given.

The model parameter data files used to specify theNEF-BM models are located in the material and groupsfiles [3]. These data files can be viewed interactivelythrough NefMenu, providing fully titled and organizedfields. NefMenu also allows a convenient means ofbrowsing the materials, organized within intuitive groups.

Most interactive programs in NEFDS are availableboth through the XWindows interface and by commandline control. For example, the spectral distribution ofhemispherical reflectance can be graphed either throughNefMenu or through the program NefPlot. Such a plot ofmaterial number 0221, red dirt, is shown in Figure 3.

The NEFDS may be queried for BRDF values of mate-rials or material groups at ranges of wavelengths for anygiven geometry. The XWindows control offers interactivecontrol over these parameter settings allowing the userto quickly determine key reflection attributes. The com-mand line counterpart, BRDF allows BRDF values to beobtained through a batch process. This is the technique

utilized for the BRDF sampling used to generate BRDFtables for rendering discussed in the next section.

5 Rendering data

In this section, BRDF data obtained from NEFDS wasused to render synthetic images. NEFDS was queried andthe discretely sampled BRDFs were stored in data files.These files were then used as lookup tables to determineBRDF data during the execution of the rendering pro-gram. The program was selected based on the physicallyaccurate global illumination solution that it produces. Inaddition, the program was extended to efficiently executeMonte Carlo integration over the sampled BRDFs.

5.1 The Alias MethodThe inverse method is used in some rendering systems togenerate random variates of the probability density func-tion (PDF) corresponding to the BRDF. In general analyt-ically generating random variates of arbitrary PDFs is notpossible. However, there are techniques available to cre-ate the random variates of an arbitrary probability massfunction (PMF), the discretized counterpart to the PDF.Gentle [13] and Knuth [21] both offer a nice overviewand discussion of the many methods available. For thisarticle, the alias method proposed by Walker [40, 41, 42]was selected because it generates random variates in con-stant time.

Many variations of Walker’s original model have beenproposed offering advantages in one way or another [12,23, 24, 25, 33, 39]. The modification by Vose [39] waschosen because it allows for initialization in O(n) timeversus the original paper’s time of O(n log n). Vose alsopresents several optimizations for memory and executiontime although they are not used here.

The alias method is most easily viewed as a form ofthe rejection method in which the rejected values are re-cycled into usable data. It requires an initial setup whichneed only be performed once. After the setup is complete,random variates are created by transforming variates of auniform distribution.

Consider the random variate X which can take on anyof the t values,

X = x1, x2, ..., xt (12)

with corresponding probability

P = p1, p2, ..., pt. (13)

Ensure that the probabilities form a valid PMF by requir-ing that

∑tc=1 pc = 1.

In the setup process, the alias method creates two newlists, the rejection list and the alias list. The rejection listis a list of probabilities,

R = r1, r2, ..., rt, (14)

whose elements ri determine whether or not an alias is tobe used. Each probability ri in the rejection list forms itsown PMF together with its complement, 1− ri. The aliaslist,

A = a1, a2, ..., at, (15)

consists of t indices which each may take on integer val-ues 1 through t, representing the index of the alias. Thesetup process is performed only once.

After the setup has been completed, selection of therandom variate X is performed by choosing two uni-formly distributed variates, u ∼ U(0, 1) and i ∼Ud(1, t), where Ud(1, t) is a discrete uniform distribu-tion over the integers 1 through t. X is given the valuexi if u ≤ ri (u was not rejected). X is given the valuexai

if u > ri (u was rejected so the alias was used). Thetime required to generate this random variate is equal tothe total time required to generate u, perform a compar-ison, and then lookup the final value. Assuming thesethree actions can be performed in constant time, so canthe generating step.

Consider a BRDF discretely sampled at s reflected di-rections Θr,1,Θr,2, ...,Θr,s, and for each reflected di-rection, t incident directions, Θi,1,Θi,2, ...,Θi,t, for atotal of st samples. The samples each have a representa-tive solid angle ∆ωi,n and a representative projected solidangle ∆Ωi,n = (Θi,n ·N)∆ωi,n corresponding to Θi,n,where 1 ≤ n ≤ t.

The continuous PDF, hΘr,m , associated with the re-flected direction Θr,m then becomes the PMF, g′Θr,m , by

g′Θr,m(Θi,n) =

∫

∆Ωi,n

hΘr,m(Θi)dΩi

≈ hΘr,m(Θi,n)∆Ωi,n

where 1 ≤ m ≤ s and 1 ≤ n ≤ t. The approximationis valid for sufficiently small ∆Ωi,n and will be used ingenerating the random variates in the following section.However, it does require a renormalization to ensure thatthe sum of each PMF is 1. When renormalized and ap-plied to the BRDF, the PMF becomes

gΘr,m(Θi,n) =hΘr,m(Θi,n)∆Ωi,n∑tc=1 hΘr,m(Θi,c)∆Ωi,c

=ρ(Θi,n; Θr,m)∆Ωi,n∑tc=1 ρ(Θi,c; Θr,m)∆Ωi,c

(16)

We are now ready to define the lists from (12) and (13)required to generate random variates of the discretelysampled BRDF. For generating random variates of theincident directions corresponding to reflected directionΘr,m use

X = Θi,1,Θi,2, . . . ,Θi,t

and

P = gΘr,m(Θi,1), gΘr,m(Θi,2), . . . , gΘr,m(Θi,t)

5.2 The Rendering ProgramThe Radiance Lighting Simulation and Rendering Sys-tem (Radiance) [28, 44] was used to generate the syn-thetic images in this work. Radiance is a suite of pro-grams built around an advanced distributed raytracer de-signed for realistic image synthesis. It was selected be-cause it is a physics based rendering system designedto accurately model the light behavior of a scene us-ing physical units [44]. Using such a system reinforcesthe validity of the results obtained by this physics basedBRDF. Additionally, the source code to Radiance is pub-licly available [2] and the program is currently in wideuse, aiding future work.

Radiance is a distributed ray tracer which utilizesMonte Carlo importance sampling. The direction ofthe ray is stochastically selected so that the distributionmatches the corresponding distribution function associ-ated with the reflection model. This requires an inver-sion of the distribution function to map a uniformly ran-dom number to the desired distribution. Ward’s imple-mentation uses the relatively straightforward implemen-tation of the Gaussian inversion to achieve this. That is,Ward’s reflection model is based on the Gaussian func-tion which provides easy inversion and thus a straightfor-ward method of Monte Carlo integration. This method,however, is not compatible with the BRDFs generated byNEF-BM which are not of a Gaussian nature.

A new shader,3 called iBRDF was built for Radianceusing the alias method described in the preceding sec-tion. This provides a means for discretized versions ofarbitrary isotropic BRDFs to be used in the creation ofsynthetic images.

Uniform sampling is used to discretize the BRDF. TheBRDF is separated into M, N, and R divisions of the re-flected θ, incident θ, and incident φ respectively. Sincethe BRDF is isotropic we define the coordinate systemso that φr = 0. For most of the BRDFs generated us-ing NEFDS, uniform sampling was found to be sufficient.Good results were obtained by sampling at 40 values of θiand θr between 0 and π/2 and at 80 values of φr between0 and π 4. For some BRDFs containing greater variationin the reflectance lobe, a larger number of samples wasrequired.

The iBRDF data is accessible via two different func-tions. The first is a direct lookup of the BRDF values

3Radiance uses the term material type rather than shader.4Another consequence of isotropic reflection is the need to only

sample half the hemisphere. The other half (i.e., π < φr < 2π) can bedetermined using the laws of reciprocity.

given incident and reflected directions. This lookup re-quires only constant time since the sample grid consist offixed divisions of θi and φi. For this function, tri-linearinterpolation is used to interpolate between the three sam-pled axes. The second function creates random variatesdistributed as gΘi in (16) using the alias method. Inter-polation is performed using a stochastic version of linearinterpolation for both the selection of the sampled φr andthe selection of the incident direction within the selected∆ωi.

Radiance separates the BRDF into two components,diffuse and specular. The diffuse component refers onlyto pure Lambertian while specular refers to the remainderof the BRDF. The advantage of this separation is that thediffusely reflected radiance is slowly varying across thesurface. Radiance utilizes this by caching the diffuselyreflected radiance values and interpolating between thosecached results whenever possible [46, 45]. Addition-ally, since the diffuse reflectance is pure Lambertian, thecached values are valid for all reflected angles. The abovetwo functions provide the required BRDF information forspecular reflectance. The diffuse reflectance is easily ob-tained by finding the minimum reflectance value of theBRDF for each Θr and using this as the diffuse portionof the BRDF, ρd. The specular portion of the BRDF usedin the alias method is then the difference

ρspecular(Θi; Θr) = ρ(Θi; Θr)− ρd(Θr).

Radiance requires that all spectra be reduced to anRGB tristimulus equivalent, and it performs separate il-lumination calculations for each of the R, G, and Bcomponents. This limits the circumstances under whichthe color calculations will be exact to those where lightreaching the viewing plane has struck only a single col-ored object (or that object and other spectrally neutral sur-faces in the scene). Three separate iBRDF data sets areproduced, one for each of the three axes of the RGB colorspace used in Radiance.

5.3 Rendering NEFDS materialSampled BRDFs (at λ = 550nm) of several NEFDSmaterials were used to generate images in the modifiedrendering program. Figure 4 shows three vases mod-eled with the NEFDS materials Bare Construction Lum-ber, Gloss Paint on Metal, and Scored Aluminum. Fig-ure 5 is rendered using the same materials and addition-ally the NEFDS materials Cement, and Weathered Con-crete along with the addition of a texture map.

The variance in the reflected light over these five sam-pled BRDFs is shown even clearer in viewing the BRDFsdirectly. Figures 6 and 7 show the BRDFs of cementand lumber respectively. Notice the significant differencein geometry that can be characterized by the NEF-BM

Figure 4: Vases with NEF materials (left to right) BareConstruction Lumber, Gloss Paint on Metal, and ScoredAluminum. Image from [43].

model. This strength (the ability to capture a wide vari-ety of BRDF distributions) coupled with the systematicmethod of measurement make for a very powerful ren-dering tool.

5.4 Fitting New Models to NEF-BMNot only does NEFDS offer hundreds of pre-existing ma-terials, it also allows new materials to be added. The mea-sured BRDF values of two metallic paint samples wereused to create two new modeled BRDFs using the NEFBeard-Maxwell model. The two samples, termed fineand coarse, were respectively dominated by small andlarge metallic flakes. The paint with a greater numberof fine flakes had a larger diffuse component due to moreedge scattering. This is easily captured by the NEF-BMmodel as can be seen in the rendered image of Figure 8.Measurements were performed at the National Instituteof Standards and Technology (NIST) using the referencespectrogoniophotometer, the Spectral Tri-function Auto-mated Reference Reflectometer (STARR) [34].

The first measurement5 discussed here is the near zero-bistatic BRDF measurement. This measurement is usedto derive the distribution of micro-facets, which are re-sponsible for the first surface reflection in the Beard-Maxwell model. Measurement of the near zero-bistaticBRDF is performed in a single plane, from the surfacenormal to grazing angles. Source and detector are held ata fixed six degree separation (near zero) in this plane of

5Measurements were performed in the four polarization states,ρ⊥⊥, ρ⊥‖, ρ‖⊥, and ρ‖‖. Unless noted otherwise, the measured val-ues listed are the calculated unpolarized values, computed from thesemeasured polarized values.

Figure 5: Cubes with textured NEF materials (top row)Cement, Gloss Paint on Metal, (bottom row) Bare Con-struction Lumber, Scored Aluminum, and WeatheredConcrete. Image from [43].

Figure 6: BRDF of Cement

measurement.Figure 9 shows the measured versus modeled BRDF

values at the near zero-bistatic angles for the fine andcoarse metallic paint samples. As can be seen there isa very good fit to the measured data at the points outsidethe specular peak. The crossover point at about 18 de-grees is matched for both measured and modeled data.

In addition to the near zero-bistatic measurements, thespecular BRDF measurements were performed using par-allel polarized source and receiver over 10 deg ≤ θi ≤80 deg with φi = 0 deg and φr = 180 deg. As men-tioned earlier, for each sampled incident direction, the re-ceiver direction was varied about the mirror direction inthe plane of incidence by θi − 5 deg ≤ θr ≤ θi + 5 deg.This provided the means by which the complex index ofrefraction was determined, used in determining the Fres-nel reflection. Lastly, the shadowing and obscuration

Figure 7: BRDF of Bare Construction Lumber

Figure 8: Coarse and fine metallic paint on vases

0.01

0.1

1

10

0 10 20 30 40 50 60

BR

DF

(1/

sr)

Exitant theta (deg.)

Modeled 100CMeasured 100C

Modeled 100FMeasured 100F

Figure 9: Comparison of measured and modeled six de-gree bistatic angle BRDF values

Table 1: Comparison of measured versus modeled out-of-plane BRDF values for coarse metallic flake sample.

θi θr φr Coarse metallic flake BRDF(degs.) (degs.) (degs.) Measured (1/sr) Modeled (1/sr)

-30 45 45 0.269 0.255-30 45 90 0.0635 0.0698-30 45 135 0.0389 0.0405-45 45 45 0.207 0.223-45 45 90 0.0469 0.0499-45 45 135 0.0331 0.0332-60 45 45 0.150 0.173-60 45 90 0.0401 0.0469-60 45 135 0.0325 0.0316

Table 2: Comparison of measured versus modeled out-of-plane BRDF values for fine metallic flake sample.

θi θr φr Fine metallic flake BRDF(degs.) (degs.) (degs.) Measured (1/sr) Modeled (1/sr)

-30 45 45 0.254 0.321-30 45 90 0.1023 0.1001-30 45 135 0.0627 0.0622-45 45 45 0.210 0.295-45 45 90 0.0727 0.0769-45 45 135 0.0479 0.0454-60 45 45 0.179 0.240-60 45 90 0.0581 0.0641-60 45 135 0.0386 0.0377

terms, τ and Ω, were determined with the fixed source,varying receiver measurements.

The model parameters for the NEF Beard-Maxwell re-flection model are selected on the basis of in-plane BRDFmeasurements only. In order to verify that the NEFBeard-Maxwell model is in fact an appropriate model torepresent the BRDF of the surface, validation of out-of-plane BRDF values should be performed. Out-of-planemeasurements were made at a variety of incident and re-flected directions for both coarse and fine metallic flakesamples—the results are listed in Tables 1 and 2. As canbe seen from these figures, there is a very good correspon-dence between the measured and modeled BRDF values,indicating that the NEF Beard-Maxwell model is appro-priate for representing these metallic surfaces.

6 Conclusion

The NEFDS provides a collection of surface reflectiondata that can be useful for work in realistic image synthe-sis. Although the database was developed for applicationin the remote sensing field, it contains materials of impor-tance to computer graphics researchers. The database hasa good user interface, it is well documented, and it canbe extended with additional measurements. The protocolfor performing these measurements is well defined and itmakes use of existing radiometric instruments.

The NEFDS incorporates a modified form of theBeard-Maxwell reflection model. This is a physics basedreflection model that can be fit to measured data. Thisgives it the advantage of a first principles physics deriva-tion coupled with a measurement scheme for determiningthe model parameters. The key insight in the model is thatall first surface reflectances can be calculated by measur-ing zero-bistatic first surface reflectances. The principallimitation of the model is its inaccuracy at grazing angles.

The acceptance of sophisticated reflectance models incomputer graphics depends upon the availability of datanecessary to use of the models. Rendering systems suchas Radiance are capable of producing photo realistic pic-tures. With the addition of real-time shading techniques,even personal computers can make acceptable interactivepictures of surfaces with complex reflectance properties.Just as the need for geometric data has spawned a 3Dmodeling industry, databases containing the reflectancecharacteristics of common materials will be of increasingimportance in computer graphics. NEFDS is an exampleof such a database.

7 Acknowledgments

The authors would like to acknowledge the significantcontributions made by one of the original co-developersof the NEF-BM model and the NEF database, MichaelMetzler of ISciences LLC. Metzler wrote software thatallowed us to directly access the data in the NEF databasewithout going through the user interface. Metzler wasalso instrumental in specifying the protocol that was em-ployed to make the measurements as well as performingthe parameter estimation from the measurement data de-scribed in Section 5.4. Maria Nadal of the National In-stitute of Standards and Technology (NIST) acquired thedata, some of which is shown in Figure 9, that allowedus to synthesize pictures of the metal flake samples. Thescenes shown in Figures 4 and 5 were modeled and ren-dered by Peter Walker. Funding for this research was pro-vided by NIST under the program administration of FernHunt.

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