Volume xxx, (1999 ) number yyy pp. 000{000Re ectance Models with Fast Importance SamplingL�aszl�o Neumann, Attila Neumann, L�aszl�o Szirmay-KalosDepartment of Control Engineering and Information Technology,Technical University of Budapest,Budapest, M}uegyetem rkp. 11, H-1111, HUNGARYEmail: szirmay@iit.bme.hu, neumann@mail.datanet.huAbstractWe introduce a physically plausible mathematical model for a large class of BRDFs. The new modelis as simple as the well-known Phong model, but eliminates its disadvantages. It gives a good visualapproximation for many practical materials: coated metals, plastics, ceramics, retro-re ective paints,anisotropic and retro-re ective materials, etc. Because of its illustrative properties it can be used easilyin most commercial software and because of its low computational cost it is practical for virtual reality.The model is based on a special basic BRDF de�nition, which meets the requirements of reciprocity andof energy conservation. Then a class of BRDFs is constructed from this basic BRDF with di�erentweight functions. The de�nition of such weight functions requires the user to specify the pro�le ofthe highlights, from which the weight function is obtained by derivation. It is also demonstrated howimportance sampling can be used with the new BRDFs.Keywords: Rendering, re ection models, BRDF, albedo, anisotropy, retro-re ective models, coatedmetals, energy mirror, Monte-Carlo quadrature, importance sampling.1. IntroductionA central problem in photorealistic rendering is thestudy of the optical material models characterised byBi-directional Re ectance Distribution Functions, orBRDFs.BRDF models can be classi�ed according to howthey correspond to the real physical phenomenon oflight-surface interaction.Global illumination algorithms require that theBRDFs do not violate physics. Such shading modelsmust satisfy both reciprocity and energy balance, andare called physically plausible1. Physically plausiblemodels can be constructed empirically using simplefunctions and enforcing properties of the reciprocityand energy balance, or they can be derived from moredetailed physical models describing the microstructureof surfaces or the scattering of electromagnetic waves2.The classical Lambert model of di�use re ectionand the Torrance-Sparrow model3 of specular re ec-tion are physically founded. Other physically based

models used so far are the Cook-Torrance4 and He-Torrance5 models, which are good approximations formetals and plastics but are very expensive computa-tionally. The Cook-Torrance model has been gener-alised for anisotropic surfaces by Kajiya6. The Cook-Torrance model has also been simpli�ed and made ap-propriate for practical implementations by Ward7 andSchlick20.The original Phong8, Phong-Blinn9 and Whitted10models, on the other hand, are empirical models.These models are not physically plausible since theydo not ful�l the law of reciprocity and the principleof the conservation of energy. The physically plausi-ble Oren-Nayar model11 characterises di�erent mate-rials with an appropriate retro-re ective component,including ceramics, walls, foams, etc. Painted coat-ings are often �tted to the Beard-Maxwell empiri-cally based model12. The NIST database also appliesthis model to represent measurement data. Recently,Lafortune et.al.13 proposed the combination of generalcosine models to obtain a versatile family of BRDFs.c The Eurographics Association 1999. Published by BlackwellPublishers, 108 Cowley Road, Oxford OX4 1JF, UK and 238 MainStreet, Cambridge, MA 02142, USA.

2 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance SamplingBased on the preliminary version of this paper14, To-bler et.al.15 proposed a new BRDF that is particularlygood for producing artistic contour e�ects.During rendering, the BRDF models are usuallyused for two di�erent purposes. On the one hand, inlocal illumination formulae the output radiance in adirection is computed from the incident radiance andfrom the surface orientation, which requires the eval-uation of the BRDF for a single view vector and lightvector pair. On the other hand, to improve Monte-Carlo random walk methods by applying importancesampling, directions should be generated that followa probability density of the BRDF multiplied by a co-sine function.A practically useful BRDFmust support both tasks.Unfortunately, physically based BRDFs contain manycomplex formulae, thus not only applying accurate im-portance sampling is hopeless, but even their simpleevaluation is rather time consuming.This is one reason why the Phong model is still incommon use despite of its de�ciencies. In global il-lumination algorithms, it is usually modi�ed to sat-isfy reciprocity16; 1 and to allow simple importancesampling17. The main disadvantage of the physicallyplausible Phong model is evident for \grazing angles"where it becomes dark. If the exponent parameter ishigh, this darkening is true from two di�erent aspects.At grazing angles both the radiance and the total re- ected power sharply decrease. The missing power be-comes noticeable in images generated by global illumi-nation algorithms.This paper intends to develop a class of BRDF mod-els. The proposed models are physically plausible, butnot physically based. It can be viewed as a mathe-matical construction that is simple to compute and isparticularly e�cient for importance sampling.Speci�c members of the new class can be de�ned ina new way. The computational cost of the new BRDFsis comparable to that of the Phong model. The modelis able to approximate the visual properties of met-als, plastics, re ective, retro-re ective and anisotropicmaterials.2. Basic BRDFThe complex models to be proposed by this paper arebased on a simple, elementary BRDF model. This ba-sic BRDF is just a mathematical tool, we do not in-tend to use it directly for rendering.In the formulae the classical notations are used: Land V are the unit vectors to the light source andto the viewpoint, N is the unit normal vector of thesurface element. The L0 and V0 vectors are the results

of mirroring the L and V vectors across the normalvector N. The subscript P indicates the orthogonalprojection of the given unit vectors - or their end-points - onto the base plane as shown in �gure 1.L

N

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OV Rp p

pFigure 1: Geometry of the BRDF de�nitionThe basic BRDF is based on simple geometric con-structs, namely on circles on the base plane. Beforegiving the de�nition of the proposed basic BRDF, twonotations are introduced.Firstly, let C(L0P ; R) denote the common part of twocircles, namely the unit circle with centre in O calledthe base circle and the circle of radius R and centreL0P .Secondly, let the m(L;V) metric be the distance ofthe L0P and VP points. It can be shown thatm(L;V) = jVP � L0P j = jh� (N � h)Nj (1)where h = L+V. It must be emphasised that h is notthe same as the half-way unit vector denoted usuallyby H, since h is not normalised.Note that function m is symmetric, that ism(L;V) = m(V;L)since the distance of L0P and VP is the same as thedistance between V0P and LP .Using these notations, the basic BRDF correspond-ing to parameter R (0 � R � 2) is de�ned by:f�r (R;L;V) = ( c(R) if m(L;V) � R;0 otherwise. (2)where c(R) is a constant depending on R.Note that m(L;V) � R means that the projectionof V is in the circle of radius R, which is centred bythe mirror of projection L, i.e.VP 2 C(L0P ; R):Thus the BRDF also depends on R, which is shown byincluding it as the �rst parameter. The star characteris to emphasise the fact that this is a basic BRDF fromwhich other BRDFs will be constructed.c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 3R<1 R=1 R>1

c(R) = 1/R c(R) = 1/2 π πFigure 2: Possible relations to the base circleThe basic BRDF is a constant c(R) if the projectionof the mirroring of L onto the centre of the directionalsphere and the projection of V are not further thanR.

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Basic BRDF at 0 and 60 incident degrees

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Figure 3: Basic BRDF for R = 0:2The computation of the basic BRDF function isquite simple:BasicBRDF(R;V;N;L)h = L + Vm2 = jh� (N � h)Nj2if (m2 � R2) then return c(R)else return 0end2.1. ReciprocityThe basic BRDF is reciprocal, which is represented byidentity f�r (R;L;V) = f�r (R;V;L)since according to de�nition (2) it is equivalent tom(L;V) = m(V;L):

2.2. Energy conservation2.2.1. Albedo functionThe power re ected into the half-space from the pointlight source of unit irradiance situated in direction Lis called the direction-dependent albedo, or the direc-tional hemiospherical re ectance:a(L) = Z fr(L;V) (N �V) d!V: (3)To satisfy energy conservation, the re ected powercannot exceed the incoming power, that isa(L) � 1: (4)The albedo of the basic BRDF can be easily deter-mined. In formula (3) the multiplication with (N�V) =cos!V corresponds to the projection onto the baseplane. Thus, according to de�nitions (2) and (3) thealbedo is proportional to the area of C(L0P ; R), that isaR(L) = c(R) � Area(C(L0P ; R)): (5)The area of C(L0P ; R) is maximum when L = N, i.e.L0P = O. If R < 1, this means that the circle is nottruncated. If R � 1, then the common part coincideswith the base circle. SinceArea(C(L0P ; R)) � Area(C(O; R)); (6)if in equation (2) we choosec(R) = 1Area(C(O; R)) = 1(� �min(1; R2)) ; (7)then the albedoaR(L) = Area(C(L0P ; R))Area(C(O; R)) (8)will not exceed 1, that is, the basic BRDF will conserveenergy.R

t

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Figure 4: Computation of the intersection of two cir-clesc The Eurographics Association 1999

4 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance SamplingTo compute Area(C(L0P ; R)), the intersection of twocircles must be determined. Let the distance of thetwo circles be d. If d < 1 � R or d > 1 + R, then thesolution is trivial. For the intersecting case, we canuse the following formulae that uses the notation of�gure 4: Area(C(L0P ; R)) = t1 + t2 =1 � (�� cos� � sin�) +R2 � (� � cos � � sin �): (9)The angle � and � can, in turn, be determined usingthe theorem of cosine angles:d2 + 12 � 2d � cos� = R2d2 +R2 � 2d � R � cos � = 1: (10)The direction-dependent albedo of the basic BRDFwill reach the imposed maximum of 1 for vertical illu-mination, i.e. for L = N.

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Albedo of the basic BRDF in the range of R = [0..1]

R=0.2R=0.4R=0.6R=0.8

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Figure 5: Albedo of the basic BRDF for small R val-ues (R � 1)When R < 1 , then the albedo equals to the ratioof the area of the possibly truncated circle and thearea of the complete circle of radius R. For R � 1 itis the ratio of the area of the possibly truncated basecircle and the area of the whole base circle (�). Thus,we always have aR(N) = 1, furthermore it is easy toshow that 0:391 < 23 � p32� � aR(L) � 1holds. The minimum point is at horizontal illumina-tion ((N � L) = 0) and for R = 1.For small values of R, the albedo is generally 1,and even at the margins of the base circle it hardlydecreases under half. This is due to the fact that in

this case the circle is generally not truncated, only forillumination at grazing angles.The R = 0 situation is the marginal case corre-sponding to a special mirror. In this case C(R) be-comes a Dirac-delta function.The R = 2 case corresponds to the ideal di�usemodel, where also a(L) � 1.

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Albedo of the basic BRDF in the range of R = [1..2]

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Figure 6: Albedo of the basic BRDF for large R values(1 � R � 2)2.2.2. Mean albedoIn addition to the direction-dependent albedo de�nedby equation (3), we can also introduce the notion ofthe mean albedo. This direction-independent and di-mensionless value is equal to the \re ectivity" for dif-fuse materials.For non-di�use materials, it can be de�ned as theweighted mean value of direction-dependent albedos:amean = 1� � Z a(L) (N � L) d!L; (11)where L is the running vector of the directional hemi-sphere and the weighting factor is (N � L) = cos!L.In the case of uniform illumination on the half-space, the value of the mean albedo is equal to theratio of the re ected and the incident power. Figure 7illustrates the values of amean for the basic BRDFs asa function of R. The mean albedo is minimal at R = 1.The BRDF and the albedo characterise a materialfrom two di�erent aspects. The BRDF is used to de-scribe the \re ected radiance" and it is a scalar whichdepends on both L andV. The albedo depending onlyon direction L serves for the description of the \re- ected power". The mean albedo (which does not de-pend on L and V) characterises the average re ectiv-ity of the BRDF. c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 5

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Mean albedo of the basic BRDF in the range of R = [0..2]

Figure 7: Mean albedo of the basic BRDFExamining these characteristics, we can see the es-sential di�erence between the behaviour of the idealmirror obtained by the new BRDF and that obtainedas a limit case from the (N �H)n Phong-Blinn modelwithout di�use component18.For the ideal metal mirror, both the mean and thedirection dependent albedos are 1, that is amean = 1and a(L) � 1.However, for the \Phong-mirror", which is the re-sult of letting n go to in�nity, we have 1 � a(L) � 0and amean = 0:5. In the case of a Phong mirror, there ected energy tends to zero for grazing angles.The mirror case of the new model is not equivalentto a metal mirror since at grazing angles the metal-lic mirrors re ect the total energy by increasing theBRDF by a factor of 1=(N�L), while in the new modelthe BRDF is constant and the solid angle of the re- ectance is increased by 1=(N�L). Thus the new modelis a ideal \energy mirror" but not ideal \radiance mir-ror" for higher incident angles.2.3. Importance samplingSuppose that we want to calculate the radiance of apoint in direction V due to the illumination of thescene. The input and output radiances [Wm�2sr�1]are denoted by Lin and Lout, respectively. The radi-ance in the viewing direction V is:Lout(V) = Z Lin(L) fr(L;V) (N � L) d!L: (12)This integral can be approximated by the Monte-Carlo method. Using importance sampling to reducethe variance, the incoming L1;L2; : : : ;LM directionsare not uniformly distributed, but their probability

density distribution function is at least approximatelyproportional to the integrand. Since Lin(L) is usuallynot known a-priori, we can sample L with probabilityproportional to theR(L;V) = fr(L;V) (N � L)weighted BRDF function. This function has an inter-esting illustrative content, according to the \inversesituation". If there were a light source in the viewingdirection, then the distribution of the energy on thehalf-space after the re ection would follow the func-tion R.Function R � d!L also de�nes the conditional prob-ability that a photon is re ected into the solid angled!L given that it comes from V 19.Due to the fact that fr is reciprocal, we have:a(V) = Z fr(L;V) (N � L) d!L; (13)which is exactly the value of the albedo function inviewing direction V. For sampling of L1;L2; : : : ;LMdirections, we have to use thedV(L) = fr(L;V) � (N � L)a(V) d!L (14)probability density function. The approximation of theintegral in equation (12) is:Lout(V) = a(V) �E �Lin(L)� � a(V)M � MXk=1 Lin(Lk):(15)In the case of isotropic BRDFs the albedo functionscan be tabulated in a one-dimensional table. Whencalculating multiple interre ections, the main compu-tational problem is the generation of the directionsfrom equation (14) following distribution dV.2.4. Importance sampling for the basic BRDF2.4.1. Sampling in special casesFor the introduced basic BRDFs, we can easily gener-ate directions with a distribution of equation (14).Let us consider �rst a circle of radius R < 1 and as-sume that this circle does not intersect the base circle.In this case C(V0P ; R) is identical with the circle withcentre in V0P and of radius R. The value of the albedois 1 because the circle is not truncated.Let us generate M number of 2D points inside thiscircle with a uniform distribution. These will be the(Lk)P (k = 1; : : : ;M) projection points in the planeof projection. Since the projection automatically in-cludes the \cosine" factor in d(L), re-projecting thesec The Eurographics Association 1999

6 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Samplingpoints onto the unit hemisphere's surface, we obtainthe L1;L2; : : : ;LM directions of the desired distribu-tion.N

VL

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Figure 8: Sampling when R < 1 and the circles donot intersectFor a given unit viewing vector V, the projectionon the base plane and the mirroring are realised inthe following way:V0P = �VP = �(V� (N �V)N): (16)The formula for re-projecting the LP sample pointwhich is inside the circle of centre V0P and of radiusR onto the hemisphere is:L = LP +p1� (LP � LP ) �N: (17)N

VL

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Figure 9: Sampling when R � 1 and the circles donot intersectThe situation is similar if R > 1 and the circle con-tains the complete unit base circle. Now the uniformlydistributed points can be generated on the whole basecircle. This case corresponds to the generation of di-rections following a \cosine distribution" for di�usematerials.2.4.2. Generating uniformly distributedpoints in a circleTo generate uniformly distributed points inside a cir-cle, we can either use an area preserving mapping ofthe square onto the circle, or apply rejection sampling.In rejection sampling the points are generated in asquare that encloses the circle and those points thatare outside the circle are ignored.

2.4.3. Sampling in the general caseIn the cases discussed so far, the domain where thesample points are generated was a complete circle andthe value of the albedo was 1. In the general case theC(V0P ; R) region is the common part of two circles.To realise importance sampling, we have to generateuniformly distributed points inside this region.N

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Figure 10: Sampling when the circles intersectTo solve this problem, rejection sampling can be ap-plied. It means that uniform samples are generated inone of the circles and then checked whether or not theother circle contains this sample. If it does not, thesample is rejected and a new sample is generated.To reduce the number of rejected samples, thesmaller circle is worth selecting as a domain for sam-ple generation and the bigger as the region of rejectionsampling.The following subroutine �nds an L sample fromdirection V using a basic BRDF of parameter R, andalso provides the albedo to weight the radiance ob-tained using direction L:DOUBLE GenerateDirectionWithBasicBRDF(R;V;N;L)V0P = �(V � (N �V)N)found = TRUErepeatif R < 1 thenGenerate point LP in C(V0P ; R)if LP is outside C(O; 1) then found = FALSEelseGenerate point LP in C(O; 1)if LP is outside C(V0P ; R) then found = FALSEendifuntil foundL = LP +p1� (LP � LP ) �Na = C(V0P ; R)=(� �min(1; R2))return aend2.4.4. Sampling with Russian rouletteIn the case of highly re ective materials R is small andthus C(V0P ; R) is typically a complete circle. The com-c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 7putation of the common part of two circles is requiredonly at the margins.For other materials, truncation of the circles mustbe considered several times. However, if we use theRussian roulette, we have to check only whether ornot the points are inside the circle, the albedo (thearea of the common part) should not be computedexplicitly. LetCmin(V0P ; R) = (C(V0P ; R); if R < 1;C(O; 1); otherwise: (18)Let us generate in the Cmin(V0P ; R) region exactlyM uniformly distributed 2D points. Now, the weight-ing factor of (Lk)P (k = 1; : : : ;M) will be 0 or 1 (i.e.accepted or rejected) according to whether or not thepoint is inside the \other circle" too. The approxima-tion of the half-space integral is:Lout(V) = E �Lin(L)� � 1M � MXk=1 Lin(Lk); (19)where only the accepted points are considered in thesum, that is, which have a weighting factor of 1. Thismeans that the number of terms can be at most M ,but is often less. This method is somewhat simpler,but its variance is not the possible minimum becauseit does not carry out the exact importance sampling.The weight factors of the elementary rays are 0 or 1,only their expected value is equal to the aR(V) albedo.The algorithm that generates either direction Lfrom direction V, or reports that the sample shouldbe rejected by returning a zero value is the following:BOOL GenerateDirectionWithBasicBRDF(R;V;N;L)V0P = �(V � (N �V)N)if R < 1 thenGenerate point LP in C(V0P ; R)if LP is outside C(O; 1) then return 0elseGenerate point LP in C(O; 1)if L0P is outside C(V0P ; R) then return 0endifL = LP +p1� (LP � LP ) �Nreturn 1end3. Mixture of basic BRDFs3.1. General descriptionThe basic BRDF is not a realistic material model byitself, but only a mathematical construction since it isdiscontinuous (�gure 3). Therefore we intend to create

Figure 11: Metallic spheres generated by Monte-Carlo ray tracing with 50 samples per pixel (top: im-portance sampling according to the cosine angle only(51 min); middle: Russian roulette based importancesampling of section 2.4.4 (46 min); bottom: normalimportance sampling of section 2.4.3 (54 min)real BRDFs using a mixture of the basic ones. Letthe weighting function be p(R) that is expected to be\normalised" (R p(R)dR = 1).The mixture BRDF is then:fr(L;V) = 2Z0 p(R) � f�r (R;L;V) dR: (20)Weighting function p(R) can be considered as aprobability density function of one variable in the0 � R � 2 interval.The mixture BRDFs evidently meet the require-ments of reciprocity and energy conservation. So theBRDFs are physically plausible models if their value isc The Eurographics Association 1999

8 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Samplingnon-negative. A su�cient but not necessary conditionis that the weighting function is non-negative.The singularities of Dirac-delta type cases must beexamined separately, namely when the integral of p(R)in the respective point is a �nite, non-zero value, andp(R) is \in�nite".Two types of singularities are important in BRDFmodelling. The R = 0 case corresponds to a specialideal mirror, and R = 2 corresponds to the ideal dif-fuse model. Separating these, we have:2Z0 p(R)dR = pMirror+pDi�use+ 2�0Z+0 p(R) dR = 1 (21)3.2. Albedo for a given p(R) functionSince mixing is a linear operation, the albedo ap andthe mean albedo apmean of the mixture BRDF can becomputed as follows:ap(L) = 2Z0 p(R) � aR(L) dR; (22)apmean = 2Z0 p(R) � (aR)meandR: (23)According to equation (21) for the albedos of themirror and of the Lambertian model, we have:aMirror = pMirror; aDi�use = pDi�use: (24)4. Modi�ed importance sampling for a givenp(R)4.1. Reducing the general case to basicBRDFsUsing equations (12) and (20), we transform the half-space integral into a double integral and then reversethe order of integration:Lout(V) = Z Lin(L) fr(L;V) (N � L) d!L =Z Lin(L) � 24 2Z0 p(R) f�r (R;L;V) dR35 (N � L) d!L =2Z0 p(R) � 24Z Lin(L) f�r (R;L;V) (N � L) d!L35 dR =

2Z0 p(R) � LoutR (V) dR: (25)This integral is approximated by a sum usingMonte-Carlo quadrature. For the calculation of therequired Rk values, we apply importance sampling ac-cording to the p(R) function. Using the approximatevalue of the integral (12), we have:Lout(V) = 2Z0 p(R) � LoutR (V) dR � 1M � MXk=1 LinRk (V);(26)similarly to equation (25), with the exception thathere each incoming direction is represented by a dif-ferent radius Rk.Let the cumulative probability distribution functionof p be the following function:F (R) = RZ0 p(t) dt; (0 � R � 2; 0 � F (R) � 1):(27)In practice, we may calculate F in a closed form, or us-ing numerical methods. Importance sampling requiresthe inverse of F , which is usually computed numer-ically and the resulting values are stored in a su�-ciently large one-dimensional array.The algorithm of generating a direction L to direc-tion V is as follows:BOOL GenerateDirectionWithMixtureBRDF(V;N;L)Generate a random value r in [0; 1]R = F�1(r)return GenerateDirectionWithBasicBRDF(R;V;N;L)end5. Generalisation of the basic BRDF5.1. Generalisation of the L! L0 mirroringtransformationThe L0 vector in de�nition (2) was obtained from theoriginal L vector by mirroring. Instead of mirroring,basic BRDFs can be de�ned using other L ! T (L)transformations too. To ful�l the requirement of reci-procity, the T (L) transformation has to satisfy thefollowing identity: T (T (L)) = L: (28)This requirement is satis�ed by mirroring since mir-roring two times gives back the original point.c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 9Another trivial transformation is the identity, thatis T (L) = L.5.2. The retro-re ective modelFor the T (L) = L transformation, the m(L;V) metricfrom de�nition (2) becomes:m(L;V) = jLP �VP j = jg � (N � g)Nj; (29)where g = L �V. For such models, the radiance ofthe re ected light is maximum in the incidence direc-tion. This model is called the retro-re ective model.In practice, materials exhibiting such properties arethe beaded screens or the signal paints used on thehighways. The marginal cases of this retro-re ectivemodel are those mirror-constructions which are usedin laser-based telemeters that behave practically asideal mirrors, but re ect exactly in the incident direc-tion.5.3. Generalisation of the metricIn de�nition (2) the m(L;V) metric was an Euclideandistance in the projection plane. The metric can begeneralised as follows:m(L;V) = jjV; T (L)jj: (30)In equation (30) jj�; �jj is an arbitrary metric fortwo unit vectors on the hemisphere. For the Euclideanmetric on the base plane and for basic BRDFs de�nedin equation (2), we have:jjV; T (L)jj = jVP � T (L)P j: (31)5.4. Anisotropic modelsThe use of anisotropic models can often substantiallyimprove realism in rendering, especially in the case ofmetals6; 20. Formula (31) provides useful generalisa-tion possibilities for practice. Using an \elliptic norm"on the base plane, anisotropic BRDFs are obtained.Let us de�ne an orthonormal co-ordinate system onthe surface element representing the given BRDF, byconsidering the (u;v) vectors which are perpendicu-lar to each other and to the N unit normal vector too.The metric m can be de�ned as follows:m(L;V) =r�BA � (z � u)�2 + (z � v)2; (32)where z = VP � T (L)P and B=A � 1 is the ratioof the axes of the ellipse, and the major and the mi-nor axis of the ellipse are taken along vectors u andv, respectively. This metric m contracts the distancecomponent along vector u. When B=A = 1, we havethe same case as in equation (1).

Let E(Q;v; b) denote the common part of the ellipsewith centre in Q, with a minor axis of length b and ofthe unit base circle. In this case:VP 2 E(L0P ;v; b)() L0P 2 E(V0P ;v; b); (33)which means that the model meets the requirement ofreciprocity.The maximum value of b is determined for that casewhen the E(Q;v; b) ellipse contains the complete unitbase circle, for every Q inside the base circle. Themaximal b = 2 case happens for illumination whichis along the minor axis and is horizontal. The valueof b can be in [0; 2], the length of the major axis in[0; 2A=B].Parameter b corresponds to the earlier R parameter.The elementary BRDF depends now only on b:f�r (b;L;V) = ( c(b); if VP 2 E(L0P ;v; b);0 otherwise. (34)The albedo of the BRDF in formula (34) is:ab(L) = Area(E(L0P ;v; b))Area(E(O;v; b)) : (35)The albedo is maximal for L = N. The constant inequation (34) for every value of b is taken so that fora vertical illumination the albedo function is 1.The computation of this constant can be separatedinto two cases. When b � B=A, i.e. when the majoraxis is less than 1, the base circle contains the whole el-lipse with centre Q = O. On the other hand, when themajor axis is greater than 1, the intersections shouldbe computed.c(b) = ( 1=(ab�); if b � B=A;1=Area(E(O;v; b)); if B=A < b � 2:(36)In importance sampling, the anisotropic model canbe used similarly to the isotropic model. Points aregenerated inside a circle that has radius b, then a scal-ing transformation is applied that transforms the cir-cle to the ellipse.The anisotropic model can also be used to de�nemixture BRDFs:fr(L;V) = 2Z0 p(b)�f�r (b;L;V) db = 2Zm(L;V) p(b)�c(b) db(37)Anisotropic models can be de�ned with other normsc The Eurographics Association 1999

10 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance SamplingΒ

Α

u

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Figure 12: Elliptical BRDFs

Figure 13: A scene containing anisotropic metalstoo, for example with the p-th power norm:m(L;V) = h�BA (z � u)�p + (z � v))pi1=p (38)6. BRDF de�nition with a given scalarfunctionA BRDF maps two vector variables to a scalar value,thus always involves dimension reduction. The exactphase where the dimension reduction happens canvary in di�erent BRDFs. The dimension reduction alsomeans that each BRDF can be associated with equiv-alence classes of vector pairs, in a way that two vec-tor pairs provide the same BRDF value if they arein the same class. In Phong-Blinn BRDFs, for ex-ample, all L;V pairs are equivalent if they have thesame N �H scalar product. In the new model, the di-mensional reduction takes place when computing them(L;V) = jh� (N � h)Nj metric.In this section we shall examine how a BRDF canbe de�ned with an arbitrary function of one variable

and how its corresponding p(R) function can be deter-mined. Let us consider an arbitrary, non-negative �(m)scalar function de�ned in the [0; 2] interval, which willde�ne the shape of the re ection lobes or the charac-teristics of the BRDF. Using this, we de�ne a f�r (L;V)function:f�r (L;V) = �(m(L;V)) = �(m); m 2 [0; 2] (39)Function �(m) is multiplied with an appropriateCmax constant in order to ful�l energy conservationrequirement, thus thefr(L;V) = fCmax�r (L;V) = Cmax � f�r (L;V)function will be a plausible BRDF.Let us recall de�nition (20) of the mixture BRDFs.The lower limit of the integral can be modi�ed, sinceaccording to de�nition (2), if R < m(L;V), the valueof fr(R;L;V) is zero. When R � m, its value is thec(R) de�ned by equation (7). Thus, the following for-mula is equivalent to formula (20) for the de�nition ofthe mixture BRDF:fr(L;V) = 2Zm(L;V) p(R) � f�r (R;L;V) dR =2Zm(L;V) p(R) �c(R) dR = 2Zm(L;V) p(R) �� �min(1; R2) dR:(40)Computing the derivative according to variable m,we obtain:p(R) = �Cmax � d�(m)dm (R) � � �min(1; R2): (41)The derivative of function � must be taken with aminus sign, since m is the lower limit of the integra-tion.The Cmax factor should be set to make p(R) a proba-bility density and consequently the BRDF energy con-serving, thus1Cmax = � 2Z0 d�(m)dm (R) � � �min(1; R2) dR (42)for decreasing � functions.The maximum of the albedo of this BRDF is 1. Toobtain darker materials, the BRDF can be multipliedwith a factor that is less than 1.In a software implementing this approach � canbe provided interactively by the user as an arbitrarycurve using a graphics interface (the curve de�nesc The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 11a one variable function). Only the ratio of the idealmirror component (polishing) must be speci�ed sepa-rately (see equation (21)).Note that � is not required to be monotone decreas-ing, but the condition of non-negativity must be im-posed in any case. For � functions having monotone in-creasing parts, the computation of the maximal albedoand of the normalising factor is more di�cult thanequation (42) and can be accomplished by numericalmethods using formula (22).6.1. Analytically integrable casesIn this section we will present some monotone decreas-ing � functions which are suitable for applications andfor which the Cmax multiplier from equation (42) canbe derived analytically.Let G be the primitive function of �, and H be theprimitive of G (with arbitrary constants!) i.e. G = R �and H = R G .Then, it can be easily deduced that:Cmax = 12�(G(1)�H(1) +H(0)) : (43)This formula is true even if �(2) > 0.6.1.1. Exponential functionThis section uses the exponential function as a scalarfunction to de�ne the highlight pro�le:�(m) = e�s�m: (44)The normalisation factor isCmax = s22�(1� (1 + s)e�s) ; (45)and the BRDF isf (s)r (L;V) = s22�(1� (1 + s)e�s) � e�s�m: (46)Using formula (41), we can obtain the following ex-pression for the required weighting function p:p(R) = �Cmax � d�(m)dm (R) � � �min(1; R2) =s3 � e�s�R �min(1; R2)2(1� (1 + s)e�s) : (47)The probability densities, the inverse of the prob-ability distribution functions, the albedos, and theshapes of the BRDF lobes are shown in �gure 14, �g-ure 15, �gure 16 and �gure 17, respectively, for di�er-ent s parameters.For s > 10, the model is already highly specular. A

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Figure 15: Inverse of the probability distributionfunctions F�1(r) for di�erent s values

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Figure 16: Albedos of the mixed BRDFs for di�erents valuesc The Eurographics Association 1999

12 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Samplingvery good approximation of the above formula for thisparameter domain is:f (s)r (L;V) = s22� � e�s�m(L;V): (48)when the corresponding p weighting function is:p(R) = s22 �R � e�s�R �min(1; R2): (49)Let us consider formula (40) and an m < 1 value:fr(L;V) = 2Z0 p(R) � c(R) dR = Im;1 + I1;2 =� � 1Zm p(R) �R2 dR+ � � 2Z1 p(R) dR: (50)In the case of a vertical illumination we havem � 1 in-dependently ofV, thus even for the horizontal viewingdirection there will be a non-zero \di�use part". Thisis component I1;2 in equation (50) which is the inte-gral over the [1; 2] interval. Thus, for the truly specularmaterials not only �(2) = 0 but �(m) = 0 for m � 1must be imposed.To illustrate this, the table 1 contains the corre-sponding values of the Cmax and s parameters of thefr(L;V) = Cmax � e�s�m function, I1;2 and aDi�use =pDi�use = Cmax � � � e�2s.7. Real materials7.1. Metals and the new modelThe BRDFs of metals increase towards in�nity forgrazing angles. This is due to the fact that for smoothmetals the product of the BRDF and the cosine termis approximately constant, thus the incident and theoutgoing radiance is approximately equal. The pro-posed model does not follow this behaviour since themaximum of the BRDF values is constant for any in-cident angle.To further explain this, let us compare the be-haviour at the limiting mirror case of the Phong, ofthe new model and of the real metals. Assume thatthe Fresnel function is approximately 1 (this is a goodapproximation for silver mirror, for instance), and theshininess | which is represented by s in the newmodel and by the exponent in the Phong model |goes to in�nity. If the observer looks at the mirror per-pendicularly and the illumination is also perpendicu-lar, then both the new model and the Phong modelprovide the same radiance and re ected power as a

real metal. However at grazing angles, while a real mir-ror would re ect the incoming radiance and the out-put power would be equal to the input power, for thePhong model both the radiance and the output powertend to zero. Interestingly, the output radiance of thenew model also goes to zero at grazing angles, but theoutput power equals to the input power at arbitrar-ily close to the horizontal illumination. Thus, the newmodel is an energy mirror in this limiting case, butis not an ideal radiance mirror. This is \better" thanthe Phong model which is neither radiance nor energymirror. The radiance mirror property means that theillumination of point light sources are re ected likereal metals even at grazing angles. The energy mir-ror property, on the other hand, means that the dis-tributed (e.g. skylight) illumination is re ected in away as real metals since for uniformly distributed il-lumination, the perceived radiance is proportional tothe albedo (equation (13).Phong new BRDF real metal(n!1) (s!1) mirrorsamean 1/2 1 1a(�! 90�) 0 1 1Lout(�! 90�) 0 0 Linfr(� ! 90�) constant constant 1Table 2: Comparison the limiting case of the Phongmodel, of the new model and of real metal mirrors atgrazing angles7.2. Coated metals, metallic paintsWe concluded that the new model is not a perfectmetallic model because the BRDF of metals goes toin�nity at grazing angles while the BRDF of the newmodel is bounded but its re ection lobe gets wider inorder not to absorb all energy. Considering real ma-terials, metals coated by a lacquer layer21 behave likethis. Since in lacquer the attenuation of the power is anexponential function of the length travelled, for higherincident angles, both the re ected radiance and powerwill tend to zero at grazing angles. If the lacquer alsocontains the particles of the basic metal, then the be-haviour of the material will be in between the metaland lacquered metals. Absorption, scattering and re- ection can occur simultaneously in the lacquer layer.Thus for higher incident angles the metal particles,especially those that are close to the surface, will beresponsible for re ecting the incoming photons. Theobject acts as energy mirror even if the re ected ra-diance decreases. Consequently the new model seemsc The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 13s Cmax I1;2 aDi�use0 1=� 0 1 LAMBERTIAN0.25 0.3754 0.20316 0.715270.5 0.4411 0.33071 0.509791. 0.6023 0.44002 0.256082 1.0718 0.39401 0.061674 2.8032 0.15834 0.002958 10.216 0.01076 0.0000016 40.743 0.00001 0.00000 HIGHLY SPECULAR32 162.97 0.00000 0.0000064 651.89 0.00000 0.00000128 2607.5 0.00000 0.00000256 10430. 0.00000 0.00000512 41721. 0.00000 0.00000 ALMOST MIRRORTable 1: Parameters of the fr = Cmax � e�s�m BRDF

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Figure 17: BRDF using exponential weight functionc The Eurographics Association 1999

14 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Samplingto be appropriate for representing coated metals andmetallic paints.Since the di�use component of the metals is practi-cally negligible, the metallic paints are approximatedas follows Cmax � �(m) � Fresnel(�0); (51)where Cmax � �(m) is a BRDF. For metals, the �(m)function has a quite small di�use component. We canuse, for example, the �(m) = (s2=2�) � e�s�m function,assuming that s > 10.The wavelength dependent Fresnel function de-scribes the re ection of the optically smooth materialas a function of the incident angle.In the Cook-Torrance model, as well as in othermirror-particle models, for a given (L;V) pair, theexpectation of the normal vectors of the re ectingfacets is the H vector, thus the angle of incidence is�H = arccos(N �H).This approach is slightly modi�ed in the followingway. The distance of the HP = (VP + LP )=2 vec-tor from the O origin is used to evaluate the Fresnelfunction.jHP�Oj = jLP �VP j2 = jg � (N � g)Nj2 = m(L;V)2 :(52)where them(L;V) metric is described by formula (29)belonging to the retro-re ective model.Let us de�ne angle �0 in the following way:�0 = arcsin�m(L;V)2 � : (53)In conclusion, the proposed new model is:fr;metallic(L;V) = s22� � e�s�jh�(N�h)Nj � Fresnel�(�0):(54)where �0 = arcsin(jg� (N � g)Nj=2).The model is the product of a re ective and a retro-re ective model. Importance sampling can be per-formed e�ciently on the re ective factor, the retro-re ective factor modi�es the weights of the elementaryrays as a multiplier.7.3. Plastics and polishingUsing a real refraction index in the Fresnel term in-stead of the complex index of metals, the proposedmodel can also be used for modelling the specular re- ection of plastics. Since the di�use re ection of plas-tics is also signi�cant, the BRDF function is de�nedas a sum of the di�use and specular terms. To meetthe requirement of energy conservation, the sum of the

Figure 19: A simple scene with a cube (s = 2) cylin-der (s = 4) and sphere (s = 8)albedos of the di�use and specular part must be lessthan 1. The BRDF is thenfr;plastics = adi�use� +(1�adi�use)�Cmax��(m)�Fresnel(�0):(55)Certain materials, as for example lacquered objects,re ect sharp mirror images for large viewing angles13. To simulate this phenomenon, any of the proposedBRDFs can be supplemented by an ideal mirror in thefollowing way:fr;mirrorextension = fr(L;V)+(1�a(L))�fr;idealmirror(L;V):(56)Note that the extension is limited by the \missingalbedo" of the original BRDF. The BRDF of the idealmirror is a Dirac function:fr;idealmirror(L;V) = Fresnel(�0) � �(L0 �V)cos � : (57)7.4. Generalised Lambertian modelsThe �(m) function can be monotone increasing as well.In this case the re ected radiance will have smallerslope as the incident angle changes than the cosinefunction obtained from the Lambertian model, whichconforms to the measurements using a certain classof materials, including ceramic, wall, foam, cloth, etc.Due to the local interre ections - especially for greatincident angles - these materials re ect more lightbackwards, than forwards in the mirror direction.Oren et. al.11 also proposed a model for these typesof materials. Here a di�erent approach is presented.To obtain such a model, let us assume that the lightsource is placed in the viewpoint. An appropriate func-tion of the re ected radiance that has a \smoother"slope than the Lambertian model is cos1�d � where �c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 15

Figure 18: Scene rendered using Russian roulette including a golden tank (s = 10), silver base plane (s = 4),aluminium (s = 6), silver (s = 8), copper (s = 15) and golden (s = 20) spheresis the angle of the light and of the normal vectors andd is the smoothness parameter. In the limit case ofd = 0 the di�use case is obtained, for the d = 1 limitcase we get a disk of constant radiance, which is likethe full moon. This selection leads to the following �function:�(m) = �1� �m2 �2��d=2 ; 0 � d � 1: (58)Unfortunately, these models do not satisfy the prin-ciple of the conservation of energy, so we have to mod-ify de�nition (58). The simplest way of correcting thisis to limit the BRDF with a constant. Another possi-ble approach is to consider the MacLaurin's series ofthe function, retaining only a few terms, such that thetruncated series will be less than the original function.As an example, let us consider the case when thefunction is clipped by a constant:fr(L;V) = Cmax �min �1� �m2 �2��d=2 ; �max! ;(59)where 0 � d � 1.Table 3 has been generated using a clipping wherethe function was taken to be constant above m = 1:9,that is �max = �(1:9). The Cmax value was determinedsuch that the maximum value of the albedo is 1. Forthe above model, this is reached always for m = 2.The minimum of the albedo is always in m = 0. So,contrary to the case of monotone decreasing functions,the pure di�use component is de�ned by the case cor-responding to m = 0.The model de�ned by formula (59) can always be

Figure 20: Objects (d = 1, d = 0:75, d = 0:5, d =0:25) rendered using generalised Lambertian BRDFscomputing only the direct illumination e�ects. In theupper image the light source is at the view position, inthe lower image there are two point light sourcesdescribed as the sum of a Lambertian model with analbedo (re ectivity) of aDi�use and of a complementarymodel. The albedo of the latter is zero form = 0 and itis a monotone increasing function of m. Formula (59)has the same behaviour as the model de�ned by equa-tion (58), except for the m > 1:9 domain. In the caseof 1:9 � m � 2, the re ected radiance for the abovementioned sphere will have a decrease proportional tothe cosine function at [0; 1].The \full moon" limit case means that for a spherec The Eurographics Association 1999

16 Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Samplingd Cmax fmax adi�use0 1=� 1.0 1.0 LAMBERTIAN0.1 0.30884 1.123440 0.976890.2 0.29939 1.262117 0.953520.3 0.28998 1.417912 0.929890.4 0.28061 1.592939 0.906020.5 0.27128 1.789571 0.881930.6 0.26200 2.010475 0.857640.7 0.25278 2.258647 0.833180.8 0.24363 2.537454 0.808560.9 0.23456 2.850676 0.783811.0 0.22556 3.202563 0.75895 "FULL MOON"Table 3: Parameters obtained for the generalised Lambertian modelilluminated from the viewpoint the radiance will beconstant, only at the margins will decrease with a co-sine character. The width of the \cosine-ring" can becontrolled by modifying �max. It is advisable to choosethe value of d in formula (59) to be less than 1.For the above model, a good approximation of theimportance sampling is to generate directions (witha \cosine" distribution) like in the case of the Lam-bertian model and to use an appropriate multipliercorrection factor for each ray.8. Conclusions and future workThe paper presents a new, simple and uni�ed de�-nition for obtaining a good visual approximation formany materials. The new model class can be used eas-ily in the majority of the commercial programs. Byallowing the interactive de�nition of the �(m) high-light pro�le functions, BRDF characteristics can bespeci�ed in a new manner.Due to their simplicity, the computation cost of thenew models is low. According to our running timemeasurements using Heckbert's BRDF viewer22, thecomputational time of the new model is practicallyequal to the time of the Phong-Blinn model and halfof the time of the Ward model. Importance sampling,on the other hand, is simpler and faster for the newmodel than for previous models.One of the problems to be solved by further re-search is to investigate which of the physically plausi-ble BRDFs (ful�lling reciprocity, energy conservationand non-negativity) can be de�ned by suitable(T (L);m(�; �); p(R))triplets (see section 5). Another important question iswhich materials can be modelled as the sum, product

and other generalisations of the introduced re ectiveand retro-re ective models.9. AcknowledgementsThis work has been supported by the NationalScienti�c Research Fund (OTKA ref.No.: F015884,T029135) and the Austrian-Hungarian Action Fund(ref.No.: 29�o4 and 32�ou9). The authors thank EricLafortune for the implementation of the generalisedLambertian model and for the rendering of images 19and 20, and Paul Heckbert for providing his BRDFeditor which has been extended and used to createthe BRDF curves.References1. R. Lewis. Making shaders more physically plau-sible. In Rendering Techniques '93, pages 47{62,1993.2. P. Beckmann and A. Spizzichino. The Scatteringof Electromagnetic Waves from Rough Surfaces.MacMillan, 1963.3. K. Torrance and M. Sparrow. O�-specular peaksin the directional distribution of re ected thermaldistribution. Journal of Heat Transfer | Trans-actions of the ASME, pages 223{230, May 1966.4. R. Cook and K. Torrance. A re ectance modelfor computer graphics. Computer Graphics, 15(3),1981.5. X. He, K. Torrance, F. Sillion, and D. Greenberg.A comprehensive physical model for light re ec-tion. Computer Graphics, 25(4):175{186, 1991.6. J. T. Kajiya. Anisotropic re ection models. InComputer Graphics (SIGGRAPH '85 Proceed-ings), pages 15{21, 1985.c The Eurographics Association 1999

Neumann, Neumann, Szirmay-Kalos / Re ectance Models with Fast Importance Sampling 177. G. Ward. Measuring and modeling anisotropicre ection. Computer Graphics, 26(2):265{272,1992.8. B. T. Phong. Illumination for computer generatedimages. Communications of the ACM, 18:311{317, 1975.9. J. F. Blinn. Models of light re ection for com-puter synthesized pictures. In Computer Graph-ics (SIGGRAPH '77 Proceedings), pages 192{198,1977.10. T. Whitted. An improved illumination model forshaded display. Communications of the ACM,23(6):343{349, 1980.11. M. Oren and S. Nayar. Generalization of Lam-bert's re ectance model. Computer Graphics(SIGGRAPH '94 Proceedings), pages 239{246,1994.12. J. Beard and J. R. Maxwell. Bidirectional re- ectance model validation and utilization. Tech-nical Report AFAL-TR-73-303, 1973.13. E. Lafortune, S. Foo, K. Torrance, and D. Green-berg. Non-linear approximation of re ectancefunctions. Computer Graphics (SIGGRAPH '97Proceedings), pages 117{126, 1997.14. L. Neumann and A. Neumann. A new class ofbrdf models with fast importance sampling. Tech-nical Report TR-186-2-96-24, Institute of Com-puter Graphics, Vienna University of Technology,1996. www.cg.tuwien.ac.at/.15. R. F. Tobler, L. Neumann, M. Sbert, and W. Pur-gathofer. A new form factor analogy and its ap-plication to stochastic global illumination algo-rithms. In Rendering Techniques '98, 1998.16. D. S. Immel, M. F. Cohen, and D. P. Greenberg.A radiosity method for non-di�use environments.In Computer Graphics (SIGGRAPH '86 Proceed-ings), pages 133{142, 1986.17. E. Lafortune and Y. D. Willems. Using the mod-i�ed phong re ectance model for physically basedrendering. Technical Report RP-CW-197, De-partment of Computing Science, K.U. Leuven,1994.18. L. Neumann and A. Neumann. Radiosity and hy-brid methods. ACM Transactions on Graphics,14(3):233{265, 1995.19. L. Szirmay-Kalos. Theory of Three DimensionalComputer Graphics. Akad�emia Kiad�o, Budapest,1995. http: //www.iit.bme.hu/~szirmay.20. Ch. Schlick. A customizable re ectance model

for everyday rendering. In Fourth Eurograph-ics Workshop on Rendering, pages 73{83, Paris,France, 1993.21. L. Neumann and A. Neumann. Photosimulation:Interre ection with arbitrary re ectance modelsand illumination. Computer Graphics Forum,(8):21{34, 1989.22. P. Heckbert. Brdf viewer. 1997. http://www.cs.cmu.edu/afs/cs.cmu.edu/user/ph/www/src/illum.23. P. Poulin and A. Fournier. A model foranisotropic re ection. Computer Graphics,24(4):273{281, 1990.24. M. Minnaert. The reciprocity principle in lunarphotometry. Astrophysical Journal, 93:403{410,1941.25. E. Ken. Re ectance phenomenology and model-ing tutorial. 1994. http://www.erim.org.26. B. Hapke. A theoretical photometric function forthe lunar surface. Journal of Geophysical Re-search, 68(15), 1963.27. R. Hall. Illumination and Color in ComputerGenerated Imagery. Springer-Verlag, New York,1989.28. I. De�ak. Random Number Generators and Simu-lation. Akad�emia Kiad�o, Budapest, 1989.29. L. Neumann, A. Neumann, and L. Szirmay-Kalos. New simple re ectance models for met-als and other specular materials. TechnicalReport TR-186-2-98-17, Institute of ComputerGraphics, Vienna University of Technology, 1998.www.cg.tuwien.ac.at/.30. L. Neumann, A. Neumann, and L. Szirmay-Kalos.Analysis and pumping up the albedo function.Technical Report TR-186-2-98-20, Institute ofComputer Graphics, Vienna University of Tech-nology, 1998. www.cg.tuwien.ac.at/.31. L. Neumann, A. Neumann, and L. Szirmay-Kalos.Compact metallic re ectance models. ComputerGraphics Forum (Eurographics'99), 18(3):161{172, 1999.32. L. Neumann, A. Neumann, and L. Szirmay-Kalos.Re ectance models by pumping up the albedofunction. Machine Graphics and Vision, 8(1):3{18, 1999.c The Eurographics Association 1999