1

Light Scattering from FilamentsArno Zinke and Andreas Weber,Member IEEE

Institut fur Informatik II, Universitat Bonn, Romerstr. 164, 53117 Bonn, Germany

Abstract— Photo realistic visualization of a huge number ofindividual filaments like in the case of hair, fur or knitwear, isa challenging task: Explicit rendering approaches for simulatingradiance transfer at a filament get totally impracticable withrespect to rendering performance, and it is also not obvioushow to derive efficient scattering functions for different levelsof (geometric) abstraction or how to deal with very complexscattering mechanisms. We present a novel uniform formalism forlight scattering from filaments in terms of radiance, which we callBidirectional Fiber Scattering Distribution Function (BFSDF).We show that previous specialized approaches, which have beendeveloped in the context of hair rendering, can be seen asinstances of the BFSDF. Similar to the role of the BSSRDF forsurface scattering functions, the BFSDF can be seen as a generalapproach for light scattering from filaments which is suitablefor deriving approximations in a canonic and systematic way.For the frequent cases of distant light sources and observers wededuce an efficient far field approximation (Bidirectional CurveScattering Distribution Function, BCSDF). We show that on thebasis of the BFSDF parameters for common rendering techniquescan be estimated in a non ad-hoc, but physically based way.

Index Terms— Three-Dimensional Graphics and Realism –Shading, Light Scattering, Hair Modeling, Cloth Modeling

I. I NTRODUCTION

Physically based visualization of filaments like hair, fur,yarns, or grass can drastically improve the quality of a render-ing in terms of photo realism. As long as the light transportmechanisms for light scattering are understood, rendering asingle filament is usually not very challenging. In this case thefilament could be basically modeled by long and thin volumeshaving specific optical properties like index of refractionor absorption coefficient. However, for the frequent case ofa scene consisting of a huge number of individual fibers(like hair, fur or knitwear) this explicit approach becomesimpracticable with respect to rendering performance. It is alsonot clear a priori, how to deal with very complex or unknownscattering mechanisms or how efficient visualizations of fibersbased on measurement data could be implemented.

Although progress has been made in rendering particularfibers with particular lighting settings no general formulationhas been found. As a result some more or less accurate modelsfor specific fibers and illumination conditions were developed,especially in the realm of hair rendering [1], [2], [3], [4], [5].If the viewing or lighting conditions change such models mayfail or have to be adopted.

Our Contribution

1) Introduction of novel general concept for radiance trans-fer at a fiber: In this paper we introduce a novel concept

Accepted for publication byIEEE Transactions on Visualization andComputer Graphics; c© IEEE.

similar to Bidirectional Scattering-Surface Reflectance Dis-tribution Functions (BSSRDF) and Bidirectional ScatteringDistribution Functions (named BSDF) for surfaces in orderto describe the radiance transfer at a fiber. The basic idea isto locally approximate this radiance transfer by a scatteringfunction on the minimum enclosing cylinder of a straightinfinite fiber, which is a first order approximation with respectto the curvature of the filament. We call this approximation aBidirectional Fiber Scattering Distribution Function (BFSDF),which can be seen as a special BSSRDF that is not definedon the surface, but on the fiber’s local minimum enclosingcylinder. (See section IV).

2) Systematic derivation of less complex scattering func-tions: Based on the BFSDF we then derive further lesscomplex scattering functions for different levels of (geometric)abstraction and for special lighting conditions, cf. Fig. 1. Withthe help of such scattering functions efficient physically basedvisualizations—adapted to a desired rendering technique orquality—are possible. Furthermore the BFSDF allows for sys-tematic comparisons and classifications of fibers with respectto their scattering distributions, since it provides a uniformand shape independent radiance parameterization for filaments.(See sections V, VI and VII.)

3) General framework for existing models for hair render-ing: Moreover, we show that the existing models developedfor hair rendering [1], [2] can be integrated in our framework,which also gives a basis to discuss their physical plausibility.(See section VIII.)

4) Derivation of analytical solutions and approximationsfor special cases:We furthermore address some special caseswhere analytical solutions or approximations are available, e.g.a general BCSDF for opaque fibers (mapped with a BRDF) isderived. In this context we also derive a flexible and efficientnear field shading model for dielectric fibers that accuratelyreproduces the scattering pattern for close ups but can becomputed much more efficiently than particle tracing (or anequivalent) that was typically used to capture the scatteringpattern correctly. (See section IX.)

5) Integration into existing rendering systems:We are ableto propose approximations of the BFSDF, which translate it tosome position dependent BSDF and preserves subtle scatteringdetails thus allowing an integration into existing renderingkernels that can cope with BSDF and polygons only. (Seesection X.)

II. PREVIOUS WORK

Bidirectional Scattering Distribution Functions (BSDF) re-late the incoming irradiance at an infinitesimal surface patch tothe outgoing radiance from this patch [6]. It is an abstract op-tical material property which implies a constant wave length, a

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Fig. 1. Overview of the BFSDF and its special cases. The effectivedimensions of the scattering functions are indicated on the left.

light transport taking zero time and being temporally invariantas well. Having the BSDF of a certain material, the localsurface geometry and the incoming radiance, the outgoingradiance can be reconstructed. This BSDF concept works wellfor a wide range of materials but has one major drawback:Incoming irradiance contributes only to the outgoing radiancefrom a patch, if it directly illuminates this patch. Therefore ithas to be generalized to take into account light transport insidethe material. This generalization is called a Bidirectional Sur-face Scattering Distribution Function—BSSRDF [6]. Severalpapers have addressed some special cases mainly in the contextof subsurface scattering [7], [8].

Further scattering models like BTF or even more gener-alized radiance transfer functions have been introduced toaccount for self-shadowing and other mesoscopic and macro-scopic effects [9].

Marschner et al. [2] presented an approach specialized forrendering hair fibers. They very briefly introduce a novel curvescattering function defined with respect to curve intensity(intensity scattered per unit length of the curve) and curveirradiance (incoming power per unit length). Unfortunately nohint about how it can be derived for other types of fibersthan hair is given. One major restriction of the scatteringmodel is that both viewer and light sources have to be distantto the hair fiber. Therefore it is not suitable for renderingcomplex illumination like for example indirect illuminationand hair-hair scattering, which is especially important for lightcolored hair [3]. Close-ups remain a problem, too. To fix thoseproblems the scattering model was extended to a specializednear field approach [3].

Another application of fiber rendering is the visualization ofwoven cloth, where optical properties of a single yarn have tobe taken into account. A method for predicting a correspondingBSDF based on a certain weaving pattern and on dielectricfibers was developed by Volevich et al. [10]. Groller et al.proposed a volumetric approach for modeling knitwear [11].They first measure the statistical density distribution of the

Fig. 2. Left: Incoming and outgoing radiance at a filament.Right: An”intuitive” variable set for parameterizing incoming and outgoing radiance atthe minimum enclosing cylinder. In this example the local cross section shapeof the filament is pentagonal. The vector~u denotes the local tangent (axis),sthe position along this axis. The anglesα andβ span a hemisphere over thethe surface patch ats, with α being measured with respect to~u within thetangent plane andβ with respect to the local surface normal~n. Note that~Drepresents both incoming and outgoing directions.

cross section of a single yarn with respect to the arrangementof individual yarn fibers and then translate it along a threedimensional curve to form the entire filament. The resultsare looking quite impressive, but the question how to dealwith yarns with more complex scattering properties is notaddressed. A similar idea was presented by [12]. Here allcomputations base on a structure called lumislice, a lightfield for a yarn-cross-section. However, the authors do notaddress the problem of computing a physically based lightfield according to the properties of a single fiber of the yarn.

Adabala et al. [13] describe another method for visualizingwoven cloth. Due to the limitations of the underlying BRDF(Cook-Torance microfacet BRDF) realistic renderings of yarnswith more complex scattering properties remain a problem.In the realm of applied optics light scattering from straightsmooth dielectric fibers with constant and mainly circularor elliptical cross sections were analyzed in a number ofpublications [14], [15], [16], [17], [18]. Schuh et al. [19]introduced an approximation which is capable of predictingscattering of electro magnetic waves from curved fibers, too.All these approaches are of very high quality, but are eithermore specialized in predicting the maxima positions of thescattering distribution or rather impractical for rendering pur-poses because of their high numerical complexity.

III. B ASICS AND NOTATIONS

A. Motivation and General Assumptions

We want to apply the basics of the radiometric concepts forsurfaces in the realm of fiber optics. The general problem isvery similar: How much radiancedLo scattered from a singlefiber one would observe from an infinitesimal surface patchdAo in direction(αo, βo), if a surface patchdAi is illuminatedby an irradianceEi from direction(αi, βi) (cf. Fig. 2, Left).

This interrelationship could be basically described by aBSSRDF at the surface of the fiber. Such a BSSRDF wouldin general depend on macroscopic deformations of the fiber,how the fiber is oriented and warped. As a consequence local(microscopic) fiber properties could not be separated fromits global macroscopic geometry. Furthermore, all radiometricquantities must be parameterized with respect to the actual

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surface of the filament, which may be not well defined—ase.g. in the case of a fluffy wool yarn.

Therefore we define a BSSRDF on the local infinite min-imum enclosing cylinder rather than on the actual surface ofthe fiber in order to make the parameterization independent ofthe fiber’s geometry. Thus the radiance transfer at the fiber islocally approximated by a scattering function on the minimumenclosing cylinder of a straight infinite fiber, which means afirst order approximation with respect to the curvature. Wecall this function a Bidirectional Fiber Scattering DistributionFunction (BFSDF). This approximation is accurate, if theinfluence of curvature to the scattering distribution can beneglected. This is the case e.g. if

• the filament’s curvature is small compared to the radiusof its local minimum enclosing cylinder or

• most of the incoming radiance only locally contributes tooutgoing radiance.

The latter condition is satisfied for instance by opaque wires,since only reflection occurs and therefore no internal lighttransport inside the filament takes place. Hence the scatteringis a purely local phenomenon and the curvature of the fiberplays no role at all. The former condition for instance holdsfor hair and fur. In this case substantial internal light transporttakes place, but since the curvature is small compared to theradius and since the light gets attenuated inside the fiber,the differences compared to a straight infinite fiber may beneglected. The higher the curvature and the more internallight transport takes place, the bigger the potential error thatis introduced by the BFSDF.

B. Notations

Before technically defining the BFSDF we will give anoverview of our notations, which follow [2] in general.

The local filament axis is denoted~u and we refer to theplanes perpendicular to~u as normal planes. All local surfacenormals lie within the local normal plane. All incoming andoutgoing radiance is parameterized with respect to the localminimum enclosing cylinder with radiusr oriented along~u and is therefore independent of the actual cross sectiongeometry of the filament (Fig. 3 Left).

We introduce two sets of variables, which are suitable forcertain measurement or lighting settings. The first parameterset describes every infinitesimal surface patchdA at theenclosing cylinder by its tangential positions and its azimuthalposition ξ within the normal plane (cylindrical coordinates).The directions of both incoming and outgoing radiance arethen given by two spherical anglesα and β which span ahemisphere atdA oriented along the local surface normal ofthe enclosing cylinder (Fig. 2 Right, Fig. 3 Right).

Typically this very intuitive way to parameterize bothdirections and positions is not very well suited to actualproblems. In particular, this is the case for data acquisition andmany “real world” lighting conditions. Hence, we introducea second set of variables according to [2]. This set can besplit into two groups, one parameterizing all azimuthal featureswithin the normal plane(h, ϕ) and another parameterizing alllongitudinal features(s, θ). The angleθ denotes the inclination

Fig. 3. Left: Parameterization with respect to the normal plane and filamentaxis (tangent)~u. The vector ~D′ denotes the projection of a direction~D ontothe normal plane. The angleθ ranges from−π/2 to π/2, with θ = π/2 if ~uand ~D are pointing in the same direction.Right: Azimuthal variables withinthe normal plane. The parameterh is an offset position at the perimeter. Notethat parallel rays have sameϕ (measured counterclockwise with respect to~u) but differenth.

of a direction with respect to the normal plane,ϕ its azimuthalangle with respect to a fixed axis~v andh an offset position ofthe surface patch at the minimum enclosing cylinder, measuredwithin the normal plane. The variables is the same asintroduced before and always means the position along theaxis of the minimum enclosing cylinder (Fig. 3).

We will now derive some useful transformation formulaefor variables of both sets.

The projection of the angle of incidence at the enclosingcylinder onto the normal planeγ′ is given by the followingtwo equations:

cos γ′ =cos β

cos θ(1)

γ′ = |arcsinh| (2)

Thus for the spherical angleβ we have

β = arccos(cos θ

√1− h2

)(3)

The following relation holds betweenξ, ϕ andh, cf. Fig. 3:

ξ =

ϕ + arcsinh 0 ≤ ϕ + arcsinh < 2π2π + ϕ + arcsinh ϕ + arcsinh < 0ϕ + arcsinh− 2π ϕ + arcsinh ≥ 2π

(4)

The spherical coordinateα equals the angle between theprojection of ~D onto the local tangent plane and~v. Since theangle between~D and~v is given byπ/2−θ and the inclinationof ~D with respect to the tangent plane isπ/2−β one obtains:

cos α = cos (π/2− θ) / cos (π/2− β)= sin θ/ sinβ (5)

=sin θ√

1− (1− h2) cos2 θ

Furthermore, the actual angle depends on the sign ofh:

α =

arccos

(sin θ√

1−(1−h2) cos2 θ

), h ≤ 0,

2π − arccos(

sin θ√1−(1−h2) cos2 θ

), h > 0.

(6)

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IV. B IDIRECTIONAL FIBER SCATTERING DISTRIBUTION

FUNCTION — BFSDF

We now technically define theBidirectionalFiberScatteringDistribution Function (BFSDF) of a fiber. The BFSDF relatesincident flux Φi at an infinitesimal surface patchdAi to theoutgoing radianceLo at another position on the minimumenclosing cylinder:

fBFSDF (si, ξi, αi, βi, so, ξo, αo, βo) :=dLo (si, ξi, αi, βi, so, ξo, αo, βo)

dΦi (si, ξi, αi, βi). (7)

Using the BFSDF the local orientation of a fiber and the in-coming radianceLi, the total outgoing radiance of a particularposition can be calculated by integrating the irradiance overall surface patches and all incoming directions as follows:

Lo (so, ξo, αo, βo) =+∞∫−∞

2π∫0

2π∫0

π/2∫0

fBFSDF (si, ξi, αi, βi, so, ξo, αo, βo)

Li (si, ξi, αi, βi) sin βi cos βi dβi dαi dξi dsi. (8)

For physically based BFSDF the energy is conserved, thusall light entering the enclosing cylinder is either absorbed orscattered. In case of perfectly circular symmetric filaments theBFSDF in addition satisfies the Helmholtz Reciprocity:

fBFSDF (si, ξi, αi, βi, so, ξo, αo, βo) =fBFSDF (so, ξo, αo, βo, si, ξi, αi, βi) . (9)

If the optical properties of a fiber are constant alongs, thenthe BFSDF does not depend onsi andso but on the difference∆s := so − si. Thus the dimension of the BFSDF functiondecreases by one and the scattering integral reduces to thefollowing equation:

Lo (so, ξo, αo, βo) =+∞∫−∞

2π∫0

2π∫0

π/2∫0

fBFSDF (∆s, ξi, αi, βi, ξo, αo, βo)

Li(si, ξi, αi, βi) sin βi cos βi dβi dαi dξi d∆s.(10)

This special case is an appropriate approximation for thegeneral case if the following condition is satisfied:

• The cross section shape and material properties varyslowly or at a very high frequency with respect tos.

Good examples for high frequency detail are cuticula tiles ofhair fibers or surface roughness due to the microstructure ofa filament. Since the latter assumption is at least locally validfor a wide range of different fibers we will restrict ourselvesto this case in the following.

The BFSDF and its corresponding rendering integral wasintroduced with respect to the “intuitive variable set” first.We now derive a formulation for the second variable set.

Fig. 4. Left: Cross section of three very densely packed fibers. Theproblematic intersection of two enclosing cylinders is marked by an ellipse.

Fig. 5. Right: A slice at the minimum enclosing cylinder illuminated byparallel light and viewed by a distant observer.

Reparameterizing the scattering integral yields:

Lo (so, ho, ϕo, θo) =+∞∫−∞

1∫−1

2π∫0

π/2∫−π/2

∣∣∣∣ ∂ (ξi, αi, βi)∂ (hi , ϕi, θi)

∣∣∣∣fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) Li (si, hi, ϕi, θi)√

1− cos2 θi (1− h2i )

√1− h2

i cos θi dθi dϕi dhi d∆s.

(11)

The above equation can be finally simplified to the followingequation:

Lo (so, ho, ϕo, θo) =+∞∫−∞

1∫−1

2π∫0

π/2∫−π/2

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo)

cos2 θiLi (si, hi, ϕi, θi) dθi dϕi dhi d∆s. (12)

This form of the scattering integral is especially suitablewhen having distant lights, i.e. parallel illumination, sinceparallel rays of light have sameϕi and θi but differenthi. Consider for example directional lighting, which can beexpressed by the radiance distributionL(ϕ′i, θ

′i, ϕi, θi) =

E⊥(ϕ′i, θ′i)δ(ϕi−ϕ′i)δ(θi− θ′i)/ cos θi with normal irradiance

E⊥(ϕ′i, θ′i). Substituting the radiance in the scattering integral

by this distribution yields:

Lo (ho, ϕo, θo) = E⊥(ϕ′i, θ′i) cos θ′i

+∞∫−∞

1∫−1

fBFSDF (∆s, hi, ϕ′i, θ

′i, ho, ϕo, θo) dhid∆s.(13)

All formulae derived in this section hold for a single filamentonly. But a typical scene consists of a huge number ofdensely packed fiber assemblies. Since incoming and outgoingradiance was parameterized with respect to the minimum en-closing cylinder, those cylinders must not intersect. Otherwiseit can not be guaranteed that the results are still valid (Fig. 4).If a cross section shape closely matches the enclosing cylinderthe maximum overlap is very small. Hence, computing thelight transport at a projection of the overlapping regions ontothe enclosing cylinder can be seen as a good approximationof the actual scattering.

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V. D IELECTRIC FIBERS

Since most filaments consist of dielectric material it isvery instructive to discuss the scattering at dielectric fibers.For perfectly smooth dielectric filaments with arbitrary butconstant cross section the dimension of the BFSDF reducesfrom seven to six. This is a direct consequence of the fact thatlight entering at a certain inclination will always exit at thesame inclination (according to absolute value), regardless ofthe sequence of refractions and reflections it undergoes.

The incident light is reflected or refracted several timesbefore it leaves the fiber. The amount of light being reflectedresp. refracted is given by Fresnel’s Law. At each surfaceinteraction step the scattered intensity of a ray decreases—except in case of total reflection. If absorption inside the fibertakes place, then loss of energy is even more prominent. Sincean incoming ray of light does not exhibit spatial or angu-lar blurring, the BFSDF is characterized by discrete peaks.Therefore, it makes sense to analyze only those scatteringcomponents with respect to their geometry and attenuationwhich

• can be measured outside the fiber and• have an intensity bigger than a certain given threshold.

As a result the BFSDF can be approximated very wellby the distribution of the strongest peaks. Then the effectivecomplexity further reduces to three dimensions, since allscattering paths and thereby all scattering components are fullydetermined byhi, ϕi andθi which are needed to compute theattenuation.

Having n such components the BFSDF may be factorized:

fBFSDF(∆s, hi, ϕi, θi, ho, ϕo, θo)

≈n∑

j=1

f jBFSDF(∆s, hi, ϕi, θi, ho, ϕo, θo)

=δθ(θi + θo)

cos2 θi

n∑j=1

δjs(∆s− λj

s(ho, ϕo, θo))

δjϕ(ϕi − λj

ϕ(ho, ϕo, θo))

δjh(hi − λj

h(ho, ϕo, θo))aj(ho, ϕo, θo). (14)

The first factorδθ says that intensity can only be measured,if θo = −θi. Since no blurring of the scattered ray of lightoccurs, all other factors describing the scattering geometryare formalized by Dirac delta distributions, too. Because raysentering the enclosing cylinder atsi propagate into longi-tudinal direction, they usually exit the enclosing cylinder atanother longitudinal positions. The functionsλj

s account forthis relationship betweensi ands. The two functionsλj

ϕ andλj

h characterize the actual scattering geometry (the paths of theprojection of the ray paths onto the normal plane. The factoraj is the attenuation for thej-th component. These attenuationfactors include Fresnel factors as well as absorption. Due toBravais Law [2] allλj and aj can be directly derived fromthe 2D analysis of the optics of cross section of a fiber withinthe normal plane together with a modified index of refraction(depending on the incoming inclinationθi, see next section).

Note that the factor(cos θi)−2 in front of the equationcompensates for

• the integration measure with respect toθi,hi andϕi whichcontributes a factor of(cos θi)−1 and

• the cosine of the angle of incidence which gives another(cos θi)−1.

Even though this kind of BFSDF is for smooth dielectricmaterials with absorption and locally constant cross sectiononly, its basic structure is very instructive for a lot of othertypes of fibers, too. For example internal volumetric scat-tering or surface scattering—due to surface roughness—donot change this basic structure in most cases but result inspatial and angular blurring, cf. Fig. 7. By replacing theδ-distributions in the BFSDF for smooth dielectric cylinders—cf. eqn. (15)—with normalized lobes (like Gaussians) centeredover the peak similar effects can be simulated easily.

The presence of sharp peaks in the scattering distribution isnot only a characteristic of dielectric fibers with constant crosssection, but of most other types of (even not dielectric) fiberswith locally varying cross section shape, too. If the variation isperiodic and of a very high frequency these peaks are typicallyshifted or blurred in longitudinal directions, compared to afiber without this high frequency detail. Nevertheless BravaisLaw only holds for constant cross sections. Hence eqn. (14)has to be adopted to other cases. At least it can be seen asa basis for efficient compression schemes and a starting pointfor analytical BFSDFs.

A. Dielectric Cylinders

As an interesting analytical example—which was alreadypartially discussed in [14], [19] and [2]—we now analyzethe scattering of a cylindric fiber made of colored dielectricmaterial with radiusr. We restrict ourselves to the firstthree scattering components (Fig. 6), dominating the visualappearance and denote them according to [2]:

• The first backward scattering component, a direct (white)surface reflection (R-component)

• The first forward scattering component, light that is twotimes transmitted through the cylinder (TT-component)

• The second backward scattering component, light whichenters the cylinder and gets internally reflected (TRT-component)

Hence, the resulting BFSDF is a superposition of three inde-pendent scattering functions, each accounting for one of thethree modes:

f cylinderBFSDF ≈

δθ

cos2 θi(δR

s δRϕ δR

h aR + δTTs δTT

ϕ δTTh aTT

+δTRTs δTRT

ϕ δTRTh aTRT). (15)

Neglecting the attenuation factorsaR, aTT, andaTRT thisBFSDF can be derived by tracing the projections of rays withinthe normal plane (Fig. 6 Left). Rays entering the cylinder atan offsethi always exit at−hi. Furthermore intensity canonly be measured, if the relative azimuthϕ = M [ϕo − ϕi]equalsϕR for the R-component,ϕTT for the TT-component

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Fig. 6. Left: Azimuthal scattering geometry of the first three scattering modes (R,TT,TRT) within the normal plane for a fiber with circular cross section. Therelative azimuths ar denotedϕR, ϕTT andϕTRT . Here the projection of the angle of incidence onto the normal plane isγ′

i and the angle of refractionγ′r

respectively. Note that the outgoing ray leaves the fiber always at−hi. Right: Longitudinal scattering geometry of the first three scattering modes (R,TT,TRT)at a dielectric fiber. Note that the outgoing inclinationsθR,θTT and θTRT always equal−θi. The difference of the longitudinal outgoing positions withrespect to incoming positionsi are denotedsR, sTT andsTRT.

or ϕTRT for the TRT-component. Here, the operatorM mapsthe difference angleϕo − ϕi to an interval(−π, π]:

M [α] =

α, −π < α ≤ πα− 2π, π < α ≤ 2πα + 2π, −2π ≤ α ≤ −π

(16)

For the relative azimuths the following equations hold, cf.Fig. 6 Left:

ϕR = 2γ′i (17)

ϕTT = M [π + 2γ′i − 2γ′t] (18)

ϕTRT = 4γ′t − 2γ′i (19)

Due to Bravais Law all azimuths are calculated from 2D-scattering within the normal plane with help of a modifiedindex of refraction

n′ =

√n2 − sin2 θi

cos θi

(instead of the relative refractive indexn) and Snell’s Law[2]. The projected (signed) angle of incidence light is denotedγ′i and equalsarcsin hi. Its corresponding (signed) angle ofrefraction isγ′t = arcsin(hi/n′).

Because rays entering the cylinder at a positionsi propagateinto longitudinal s-direction, they leave the fiber at anotherpositionssTT andsTRT respectively. In case of direct surfacereflection (R-mode) the light is reflected at incoming positionsi. The two positionssTT and sTRT are directly relatedto the length covered by the ray inside the cylinder. Forthe TT component this lengthl can be calculated from itsprojection onto the normal planels, and the longitudinal angleof refractionθt (see Fig. 6):

l =ls

cos θt=

2r cos γ′tcos θt

(20)

with θt = − sgn(θi) arccos((n′/n) cos θi). This length dou-bles for rays of the TRT-mode.

Therefore for the positionssTT and sTRT the followingholds:

sTT = − sgn θi

√l2 − l2s = ls tan θt (21)

sTRT = 2ls tan θt (22)

Putting all geometric analysis together the geometry termsλintroduced in the previous section are:

λRs = 0; λTT

s = sTT;λTRTs = sTRT

λRh = λTT

h = λTRTh = −ho

λRϕ = ϕo + 2 arcsinho

λTTϕ = ϕo + M [π + 2 arcsin(ho)− 2 arcsin(ho/n′)]

λTRTϕ = ϕo + 4 arcsin(ho/n′)− 2 arcsinho (23)

For smooth dielectric cylinders the R component gets attenu-ated by Fresnel reflectance only:

aR = Fresnel(n′, n′, γ′i). (24)

According to [2] the indexn′ is used to calculate the re-flectance for perpendicular polarized light, whereas anotherindex of refractionn′ = n2/n′ is used for parallel polarizedlight.

For the two other modes (TT, TRT) a ray gets attenuated ateach reflection rsp. refraction event. Hence for the correspond-ing Fresnel factorsFresnelTT and FresnelTRT the followingequations hold:

FresnelTT = (1− aR)(1− Fresnel(1/n′, 1/n′, γ′t)) (25)

FresnelTRT = FresnelTTFresnel(1/n′, 1/n′′, γ′t) (26)

If a ray enters the cylinder, as is the case for both the TTand TRT modes, absorption takes places. For homogenousmaterials the attenuation only depends on the length of theinternal path—i.e. the absorption length—and the absorptioncoefficient σ, which is a wave length dependent materialparameter. For the TT component the absorption length equalsl and a ray of the TRT mode covers twice this distance insidethe cylinder. Note that the absorption length given in [2](which wasl = (2 + 2 cos γ′t)/ cos θt) is wrong.

Hence the total attenuation factors with absorption are

aTT = FresnelTT e−σ l, (27)

aTRT = FresnelTRT e−2σ l. (28)

Note that higher order scattering can be analyzed in astraight forward way, since analytical solutions for the cor-responding geometry termsλ are available.

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Fig. 7. The left diagram shows the distribution of scattered light at a perfectsmooth dielectric circular cylinder obtained with particle tracing. The intensityof the scattered light is computed with respect toξ and∆s at the cylindricshell: the darker the grey, the higher the outgoing flow through the surface. Asingle light source is illuminating a small rectangular surface patch ats = 0from directionϕ = π andθ = 0.2. The direct surface reflection component(R), the forward scattered component (TT) and the first order caustic (TRT)are clearly and sharply visible. The right image shows the same scene buta dielectric cylinder with volumetric scattering. As a consequence both TTand TRT components are blurred. Note that the structure of the scatteringdistribution (i.e. the position of the peaks and their relative intensities) doesnot change.

VI. FAR FIELD APPROXIMATION AND BCSDF

“Real world” filaments are typically very thin and longstructures. Hence, compared to its effective diameter bothviewing and lighting distances are typically very large. There-fore all surface patches along the normal plane have equal(ϕo, θo) and the fibers cross section is locally illuminated byparallel light (of constant radiance) from a fixed direction(ϕi, θi). Furthermore adjacent surface patches of the localminimum enclosing cylinder have nearly the same distanceto both light source and observer.

When rendering such a scene the screen space width of afilament is less or in the order of magnitude of a single pixel.In this case fibers can be well approximated by curves havinga cross section proportional to the diameter of the minimumenclosing cylinderD.

That allows us to introduce a simpler scattering formalismto further reduce rendering complexity which we call far fieldapproximation, cf. [3].

Using similar quantities as Marschner et al. [2] (curveradiance and curve irradiance) we show how an appropriatescattering function can be derived right from the BFSDF. Thecurve radiancedLo is the averaged outgoing radiance alongthe width of the fibers minimum enclosing cylinder times itseffective diameter (Fig. 5).

Assuming parallel light from direction(ϕi, θi) being con-stant over the width of a fiber this curve radiance can becomputed by averaging the outgoing radiance of all visiblepatches atso (over the width of the fiber) and weighting themwith respect to their relative screen space coverage (projectedarea to screen space) by a factor ofcos (ξo − ϕo):

dLo (ϕi, θi, ϕo, θo) :=D

π

ϕo+π/2∫ϕo−π/2

cos (ξo − ϕo)

dLo(ϕi, θi, ξo, αo(θo, ho(ξo, ϕo)), βo(θo, ho(ξo, ϕo)))dξo.

(29)

Notice that for the integration with respect toξo one has totake into account its transition of from zero to2π properly.Alternatively one gets the following equation by integrationwith respect toh:

dLo (ϕi, θi, ϕo, θo) =D

2

1∫−1

dLo (ho, ϕi, θi, ϕo, θo) dho. (30)

The factorD amounts for the fact that the projected area toscreen space is proportional to the width of a fiber.

Since we assume constant incident radianceLi(ϕi, θi), dLo

may be substituted by:

dLo (ϕi, θi, ho, ϕo, θo) = cos2 θiLi(ϕi, θi)dϕidθidho

1∫−1

∞∫−∞

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) d∆sdhi

(31)

which yields:

dLo (ϕo, θo, ϕi, θi) =D

2Li(ϕi, θi) cos2 θidϕidθi

1∫−1

1∫−1

∞∫−∞

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) d∆sdhidho

(32)

This scattered radiancedLo is proportional to the incomingcurve irradiancedEi:

dEi (ϕi, θi) := DLi(ϕi, θi) cos2 θidϕidθi. (33)

Thus, a new bidirectional far-field scattering distributionfunction for curves that assumes distant observer and distantlight sources can be defined. It relates the incoming curveirradiance to the averaged outgoing radiance along the thewidth (curve radiance). We will call itfBCSDF (BidirectionalCurve ScatteringDistribution Function):

fBCSDF (ϕi, θi, ϕo, θo) :=dLo (ϕo, θo, L (ϕi, θi))

dEi (ϕi, θi). (34)

By comparing the two equations (32) and (33) we obtain:

fBCSDF (ϕi, θi, ϕo, θo) =12

1∫−1

1∫−1

∞∫−∞

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) d∆sdhi dho. (35)

Finally, in order to compute the total outgoing curve radi-ance one has to integrate all incoming light over a sphere withrespect toϕi andθi:

Lo (ϕo, θo) = D

2π∫0

π/2∫−π/2

fBCSDF (ϕi, θi, ϕo, θo)Li (ϕi, θi) cos2 θidθi dϕi.(36)

This equation matches the rendering integral given in [2,equation 1] in the specific context of hair rendering. Thus theBCSDF is identical to the fiber scattering function very briefly

8

introduced in [2]. In contrast to [2] our derivation shows theclose connection between the BFSDF and the BCSDF. In ourformalism the BCSDF is just one specific approximation ofthe BFSDF and can be computed directly from eqn. (35).

Besides of the advantage of being less complex than theBFSDF the BCSDF can help to drastically reduce samplingartifacts which would be introduced by subtle BFSDF detaillike very narrow scattering lobes. Nevertheless there are somedrawbacks. First of all local scattering effects are neglectedwhich restricts the possible fields of application of the BCSDFin case of close ups or indirect illumination. Furthermore, sincethe fiber properties are averaged over the width, the entirewidth (at least in the statistical average) has to be visible.Otherwise undesired artifacts may occur. Finally the BCSDFis not adequate for computing indirect illumination due tomultiple fiber scattering in dense fiber clusters, since the farfield assumption is violated in this case.

VII. F URTHER SPECIAL CASES

Although a distant observer and distant light sources arecommonly assumed for rendering, there may be cases wherethese two assumptions may be not valid. For those situationsfurther approximations of the BFSDF are straight forward andcan be derived similarly to the BCSDF. In particular we nowdiscuss the following two special cases:

• a close observer and locally constant incident lighting:near field scattering with constant incident lighting

• a distant observer with locally varying incident lighting:curve scattering with locally varying incident lighting

A. Near Field Scattering with Constant Incident Lighting

Although a BCSDF may be a good approximation fordistant observers, it fails when it comes to close ups. Sincethe outgoing radiance is averaged with respect to the widthof a filament the intensity does not vary and local scatteringdetails can not be resolved. Now, if incident illumination canbe assumed locally constant (like in the case of distant lightsources) the outgoing radiancedL(ϕi, θi, ho, ϕo, θo) can becomputed from eqn. 31. Integration with respect to incidentdirection (ϕi,θi) yields:

Lo (ho, ϕo, θo) =

2π∫0

π/2∫−π/2

fnearfieldLi cos2 θidθidϕi. (37)

with

fnearfield (ϕi, θi, ho, ϕo, θo) =1∫

−1

∞∫−∞

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) d∆sdhi.

(38)

A practical example of a near field scattering function fordielectric fibers is given in section IX-B.

B. Curve Scattering with Locally Varying Incident Lighting

Although locally constant incident lighting may be assumedin most cases of direct illumination, this situation may changein case of indirect illumination. For the sake of completenesswe give the result for curve scattering with varying lighting,too. For the rendering integral one obtains:

Lo (ϕo, θo) =

D

∞∫−∞

1∫−1

2π∫0

π/2∫−π/2

fvaryingCSDFLi cos2 θidθidϕidhid∆s

(39)

with

fvaryingCSDF (∆s, hi, ϕi, θi, ϕo, θo) =

12

1∫−1

fBFSDF (∆s, hi, ϕi, θi, ho, ϕo, θo) dho. (40)

VIII. P REVIOUS SCATTERING MODELS

Especially in the context of hair rendering several scatteringmodels for light scattering from fibers have been proposed. Wenow show, how they can be expressed in our notation whichfor instance allows systematic derivations of further scatteringfunctions basing on the corresponding BFSDF and BCSDF.Furthermore we discuss their physical plausibility.

A. Kajiya & Kay’s model

One of the first simple approaches to render hair, which isstill very commonly used, was presented by Kajiya & Kay [1].It assumes a distant observer, since the outgoing radiance isconstant over the width of a fiber and basically accounts fortwo scattering components: a scaled specular ad hoc Phongreflection at the surface of the fiber, centered over the specularcone and an additional colored diffuse component. The diffusecoefficient Kdiffuse is obtained from averaging the outgoingradiance of a diffuse BRDF over the illuminated width of thefiber, which produces significantly different results comparedto the actual solution derived in section IX-A.1 (see Fig. 11).The phenomenological Phong reflection fails to predict thecorrect intensities due to Fresnel reflectance (cf. section IX-A.2).

Kajiya & Kay’s model can be represented by a BCSDF asfollows:

fKajiya & KayBCSDF = KPhong

cosn(θo + θi)cos θi

+ Kdiffuse. (41)

B. The model of Marschner et al. (2003)

A much more sophisticated far field model for light scat-tering from hair fibers was proposed by Marschner et al. [2].It bases on significant measurements of light scattering fromsingle hair filaments and implies again a distant observer anddistant light sources. In [2] it is shown that all important fea-tures of light scattering from hairs can be basically explainedby scattering from cylindrical dielectric fibers made of coloredglass accounting for the first three scattering components

9

(R, TT, TRT) introduced in section V already. The BCSDFrepresenting the basic model of Marschner et al. (withoutsmoothing the caustic in the TRT component) can be directlyderived from its underlying BFSDF, which is given by:

fgeneratorMarschnerBFSDF =

δ(ho + hi)cos2 θd

(g(θo + θi −∆θR, wθR)aR(θd, hi)

δ(∆s)δ(ϕi − λϕR)

+g(θo + θi −∆θTT, wθTT)aTT(θd, hi)

δ(∆s− λTT

s

)δ(ϕi − λϕ

TT)

+g(θo + θi −∆θTRT, wθTRT)aTRT(θd, hi)

δ(∆s− λTRTs )δ(ϕi − λTRT

ϕ )). (42)

This BFSDF can be seen as a generator for the basicmodel proposed in [2]. Note that it matches the BFSDF for acylindric dielectric fiber with normalized Gaussiansg(x, w) :=1/(

√2πw) exp(−x/(2w2)) replacing the delta distributions

δθ. This accounts for the fact that no fiber is perfectly smooth,and the light is scattered to a finite lobe around the perfectspecular cone. Additional shifts∆θR,TT,TRT account for thetiled surface structure of hair fibers, which lead to shiftedspecular cones compared to a perfect dielectric fiber. Fora better phenomenological match in [2] it is furthermoreproposed to replaceθi with θd = (θo − θi)/2. Hence thisBFSDF is just a variation of eqn. (15) and the complexderivation of the basic model (for a smooth fiber) given in[2] can be reduced to the following general recipe:• Build a BFSDF by writing the scattering geometry in

terms of products of delta functions with geometry termsλϕ and adding an additional attenuation factor.

• Derive the corresponding BCSDF according to eqn. (35).All ray density factors given in [2] as well as multiplescattering paths for the TRT-component are a directconsequence of theλϕ-terms within theδ-functions.

Note that for performing the integration offgeneratorMarschnerBFSDF

in (35) symbolically, all delta functions of the formδ(ϕi −λG

ϕ (ho)) have to be transformed to make them directly de-pendent on the integration variableho. This is achieved byapplying the following rule:

1∫−1

fG(ho)δ(ϕi − λGϕ (ho))dho =

n−1∑i=0

fG(hio)

|dλGϕ

dho(hi

o)|

with G ∈ {R,TT,TRT}. Here the expressionfG(ho) sub-sumes all factors depending onho (except theδ-function itself)and hi

o denotes the i-th root of the expressionϕi − λGϕ (ho).

For both the R and the TT component there exists alwaysexactly one root. For the TRT component one needs a casedifferentiation, since it exhibits either one or three roots. Thisexactly matches the observation of [2] that one or three differ-ent paths for rays of the TRT component occur. Although it isbasically possible to compute the rootshi

o for all componentsdirectly, it is—for the sake of efficiency—useful to apply theapproximation given by [2, equation 10].

As an example we now have a look at the direct surfacereflection component (R-component). According to eqn. (42)

one has:

fRBFSDF =

aR(θd, hi)cos2 θd

δ(ho + hi)g(θo + θi −∆θR, wθR)δ(∆s)δ(ϕi − λϕ

R)

Inserting this BFSDF into eqn. (35) for the correspondingBCSDF yields:

fRBCSDF =

aR(θd,φ2 ) cos(φ

2 )g(θo + θi −∆θR, wθR)

4 cos2 θd. (43)

This equation matches the formula given in [2, equation 8](for p = 0, inserting the ray density and attenuation factors).

Moreover, in [2] this basic BCSDF is extended to accountfor elliptic fibers and to smooth singularities in a post pro-cessing step. Since the BCSDF is computed for a perfectsmooth cylinder, there are azimuthal anglesφc, for whichthe intensities of the TRT-components go to infinity. Thesesingularities (caustics) are unrealistic and have to be smoothedout for real world fibers. The method used in [2] is to firstremove the caustics from the scattering distribution and toreplace them by Gaussians (with roughly the same portionof energy) centered over the caustics positions. Four differentparameters are needed to control this caustic removal process.

Note that this rather awkward caustic handling for theTRT component performed in [2] as well as a problematicsingularity in the ray density factor of the TT-component foran index of refraction approaching the limit of one can beavoided, if another BFSDF for non-smooth dielectric fibers isused, cf. section IX-B.

Usually hair has not a circular but has an elliptic crosssection geometry. Since even mild eccentricitiese especiallyinfluence the azimuthal appearance of the TRT component,this fact must not be neglected. In [2] the following simpleapproximation is proposed: Substitute an indexn∗ for theoriginal refractive indexn depending onϕh = (ϕi + ϕo)/2for the TRT-component as follows:

n∗(ϕh) =12

((n1 + n2) + cos(2ϕh)(n1 − n2))

wheren1 = 2(n− 1)e2 − n + 2n2 = 2(n− 1)e−2 − n + 2.

IX. EXAMPLES OF FURTHERANALYTIC SOLUTIONS AND

APPROXIMATIONS

In general it cannot be expected that for complexBFSDF/BCSDF there are analytical solutions—a situation thatis similar to the one for BSSRDF/BSDF. Nevertheless, in thefollowing we derive analytical solutions or at least analyticalapproximations for some interesting special cases.

A. Opaque Circular Symmetric Fibers

Perfectly opaque fibers with a circular cross section (likewires) can be modeled by a generalized cylinder togetherwith a BRDF characterizing its surface reflectance. For thiscommon case we now show, how both BFSDF and BCSDFcan be derived. Since no internal light transport takes place,light is only reflected from a surface patch, if it is directly

10

illuminated. Thus the BFSDF simply writes as a product ofthe BRDF of the surface and two additional delta distributionslimiting the reflectance to a single surface patch:

fBFSDF = fBRDFδ(ξo − ξi)δ(∆s) (44)

or with respect to the other set of variables:

fBFSDF = fBRDFδ(φ + arcsinho − arcsinhi)δ(∆s) (45)

with φ = M [ϕo−ϕi] being the relative azimuth. To derive theBCSDF the complex (first) delta term has to be transformedinto a form suitable for integration with respect tohi first.Applying the properties of Dirac distributions one obtains

fBFSDF = fBRDF| cos(φ + arcsinho)|δ(hi − sin (φ + arcsinho))δ(∆s). (46)

According to equation (35) for the BCSDF of a circularfiber mapped with an arbitrary BRDF the following holds:

fBCSDF =12

1∫−1

1∫−1

∞∫−∞

fBRDF| cos (φ + arcsinho)|δ(hi − sin (φ + arcsinho))δ(∆s)d∆sdhidho (47)

=12

cos φ∫−1

fBRDF |hi=sin (φ+arcsin ho)

| cos (φ + arcsinho)|dho (48)

Some exemplary results are shown in figure 11.1) Lambertian Reflectance:Many analytical BRDFs in-

clude a diffuse term accounting for Lambertian reflectance.The BCSDF approximation for such a Lambertian BRDF(fLambertian

BRDF = kd) component can be directly calculated fromeqn. (48):

fLambertianBCSDF =

kd

2

cos φ∫−1

cos (φ + arcsinho)dho (49)

with kd denoting the diffuse reflectance coefficient. To solvethis integral we replace the integrand by the absolute value ofits Taylor series expansion aboutho = 0 up to an order oftwo:

fLambertianBCSDF ≈ kd

2

cos φ∫−1

| cos φ− ho sinφ− h2o cos φ

2|dho.

(50)

For the resulting approximative BCSDF, which has a relativeerror of less than five percent compared to the original, thefollowing equations holds:

fLambertianBCSDF ≈ kd

2

|56

cos φ + cos2 φ− 16

cos4 φ +12

sinφ− 12

sinφ cos2 φ|.(51)

2) Fresnel Reflectance:A second very important feature ofopaque fibers like metal wires or coated plastics is Fresnelreflectance. We analyzed such a surface reflectance in thecontext of scattering from dielectric fibers in section V-A(R-component) and the corresponding BCSDF was alreadyderived in section VIII-B, cf. eqn. (43). In fact the BRDFis averaged over the width of a fiber, the BCSDF of a fiberwith a narrow normalized reflection lobe BRDF (instead of adirac delta distribution) can be very well approximated by thisBCSDF for smooth fibers. Some exemplary scenes renderedwith a combination of a Lambertian and a Fresnel BRDFcan be seen in Fig. 11. The BCSDF approximation producesvery similar results compared to the precise BRDF (BFSDF)solution, but required much less computation time.

B. A Practical Parametric near field Shading Model for Di-electric Fibers with Elliptical Cross Section

In the following we derive a flexible and efficient near fieldshading model for dielectric fibers according to section VII-A. This model accurately reproduces the scattering pattern forclose ups but can be computed much more efficiently thanparticle tracing (or an equivalent) that was typically used tocapture the scattering pattern correctly.

As a basis, we take the three component BFSDF given ineqn. (42) that was also used to derive the BCSDF correspond-ing to the basic model proposed in [2], cf. section VIII-B.Due to surface roughness and inhomogeneities inside the fiberthe scattering distribution gets blurred (spatial and angularblurring), cf. Fig. 7. In order to account for this effect wereplace allδ-distributions with special normalized Gaussiansg(I, x, w) := N(I, w) exp(−x/(2w2)) with a normalization

factor of N(I, w) := 1/I∫−I

exp(−x/(2w2))dx. The widthsw

of these Gaussians control the “strength of blurriness”. Fur-thermore, we add a diffuse component approximating higherorder scattering.

With these modifications we obtain the following moregeneral BFSDF for non-smooth dielectric fibers:

fdielectricBFSDF =

g(1, ho + hi, wh)cos2 θd

(g(π

2, θo + θi −∆θR, wθ

R)aR(θd, hi)

g(∞,∆s, ws)g(π, ϕi − λϕR, wR

ϕ )

+g(π

2, θo + θi −∆θTT, wθ

TT)aTT(θd, hi)

g(∞,∆s− λTT

s , ws

)g(π, ϕi − λϕ

TT, wTTϕ )

+g(π

2, θo + θi −∆θTRT, wθ

TRT)aTRT(θd, hi)

g(∞,∆s− λTRTs , ws)g(π, ϕi − λTRT

ϕ , wTRTϕ ))

+kdδ(∆s) (52)

Note that the awkward caustic handling done in [2] canbe avoided by calculating the corresponding BCSDF out offdielectricBFSDF , cf. section VIII-B. Here, the appearance of the

caustics is fully determined by the azimuthal blurring widthwϕ

TRT. The more narrow this width is, the closer the causticpattern matches the one of a smooth fiber. All azimuthal

11

TABLE I

SUMMARY OF ALL PARAMETERS OF THE NEAR FIELD SHADING MODEL.

notation meaning∆θR,TT,TRT longitudinal shifts of the specular coneswθ

R,TT,TRT widths for longitudinal blurringwϕ

R,TT,TRT widths for azimuthal blurringn index of refractionσ index of absorptionkd diffuse coefficiente eccentricity

scattering of such a BCSDF can be derived efficiently bynumerical integration in a preprocessing pass. The result is atwo dimensional lookup table with respect toθd andϕ whichis then used for rendering and which replaces the complexcomputations proposed in VIII-B. Some exemplary compar-isons between BFSDF and BCSDF-renderings are presentedin the appendix (Fig. 10).

Assuming locally constant illumination a near field scatter-ing function fdielectric

nearfield can be derived from the BFSDF bysolving eqn. 38 in section VII-A forfdielectric

BFSDF . Assuming anarrow widthwh, i.e. a small spatial blurring, this approxi-mately yields:

fdielectricnearfield = 2kd

+1

cos2 θd(g(

π

2, θo + θi −∆θR, wθ

R)aR(θd, ho)

g(π, φ + 2 arcsinho, wRϕ )

+g(π

2, θo + θi −∆θTT, wθ

TT)aTT(θd, ho)

g(π, φ + M [π + 2 arcsinho − 2 arcsin (ho/n′)], wTTϕ )

+g(π

2, θo + θi −∆θTRT, wθ

TRT)aTRT(θd, ho)

g(π, φ + 4 arcsin (ho/n′)− 2 arcsinho, wTRTϕ )). (53)

This result can be directly used for shading of fibers witha circular cross section. Although elliptic fibers could bemodeled by calculating the correspondingλ terms, we proposeto use the following more efficient approximation instead:Change the diameter of the primitive used for rendering (a halfcylinder or a flat polygon) according to the projected diameterand apply the first order approximation for mild eccentricitiesproposed in [2] (cf. section VIII-B).

Results obtained with this model are given in the appendix(Fig. 13, 14, 15, 11, 10). Notice especially the results shownin Fig. 14, in which comparisons to photographs are givenshowing how well the scattering patterns of the renderings arematching the originals.

Interestingly, this near field shading model is computa-tionally less expensive than the far field model of [2]. Itcan be evaluated more than twice as fast (per call) com-pared to an efficient reimplementation of this far field model.Furthermore an extension of the model to all higher orderscattering components is possible, since analytical expressionsfor corresponding geometry termsλ are available.

X. I NTEGRATION INTO EXISTING RENDERING SYSTEMS

Existing rendering kernels can cope with BSDF and poly-gons only. Therefore we propose approximations of the

Fig. 8. A simple example of a position dependent BSDF approximation forthe BFSDF.Left: BFSDF snapshot of a perfect dielectric cylinder (R,TT,TRT)Right: Concentrating this BFSDF to the incident position gives a BSDFdepending on the incident offsethi on a flat ribbon.

BFSDF, which translate it to some position dependent BSDFand preserve subtle scattering details.

A. BSDF Approximation of the BFSDF

In most cases internal light transport is limited to a verysmall region around the incident position. Furthermore, thediameter of a typical fiber is very small compared to otherdimensions of a scene. Hence, concentrating the BFSDF to theinfinitesimal surface patch in which incident light penetratesthe fiber, introduces only very little inaccuracies (see Fig. 8).This approach may be formalized by integrating an BFSDFwith respect∆s andξo:

fBSDF[hi](ϕi, θi, ϕo, θo) :=

∞∫−∞

2π∫0

fBFSDFdξod∆s (54)

The result is some position dependent BSDFfBSDF[hi](ϕi, θi, ϕo, θo) varying over the width of thefiber. With this approximation a fiber can be modeled verywell by a primitive having the same effective cross sectionlike the minimum enclosing cylinder, e.g. flat ribbons or halfcylinders always facing the light and camera.

B. BCSDF as BSDF

If the BCSDF is used for rendering, its transformation toa BSDF is trivial: in fact nothing has to be changed. OnlydiameterD has to be neglected in the rendering integral, ifthe fibers are modeled by primitives geometrically accountingfor their effective cross sections.

XI. I NDIRECT ILLUMINATION AND MULTIPLE

SCATTERING

For light colored fibers indirect illumination—mostly dueto multiple fiber scattering—significantly contributes to theoverall scattering distribution and must not be neglected. Foran example we refer to [3], which addresses the importanceof global illumination in case of blond hair. A Monte Carlorendering of light blond hair is presented in Fig. 9.

Because a BFSDF already comprises all internal lighttransport of a single filament, no further simulation of internalscattering has to be performed. Therefore sampling a BFSDFfor multiple filament scattering is more efficient than simu-lating the entire filament scattering—including internal lighttransport–during rendering.

12

However, although global illumination due to multiplefiber scattering can be basically accelerated by sampling theBFSDF—cf. section X— this direct approach may still beimpracticable with respect to rendering time.

Especially in the field of interactive hair rendering severalvery efficient rendering techniques—such as deep or opacityshadow maps and pixel blending—have been proposed toapproximate effects of multiple fiber scattering [20], [21],[22], [23]. In a forthcoming publication we will show howour theoretical framework can be applied to obtain physicallybased results using these techniques.

XII. C ONCLUSION AND FUTURE WORK

In this paper we derived a novel theoretical frame workfor efficiently computing light scattering from filaments. Incontrast to previous approaches developed in the realm of hairrendering it is much more flexible and can handle many typesof filaments and fibers like hair, fur, ropes and wires. Approxi-mations for different levels of (geometrical) abstraction can bederived in a straight forward way. Furthermore, our approachprovides a firm basis for comparisons and classifications offilaments with respect to their scattering properties. The basicidea was to adopt basic radiometric concepts like BSSRDFand BSDF to the realm of fiber rendering. Although theresulting radiance transfer functions are strictly speaking validfor infinite fibers only, they can be seen as suitable localapproximations.

Different types of scattering functions for filaments were de-rived and parameterized with respect to the minimum enclos-ing cylinder. The Bidirectional Fiber Scattering DistributionFunction (BFSDF) basically describes the radiance transfer atthis enclosing cylinder. It is an abstract optical property of a fil-ament which can be estimated by measurement, simulation oranalysis of the scattering distribution of a single filament. TheBFSDF can be seen as the basis for all further approximations,cf. Fig. 1. We believe that we have derived scattering functionsfor most of the special cases that will occur in practice. Forexamples of rendering results based on various of the derivedscattering functions we refer to the appendix.

However, the theoretical analysis given in this paper shouldbe seen as a starting point and further work is requiredto investigate many important problems which arise in thecase of very complex “real world filaments” such as globalillumination for fiber based geometries. As an example werefer to [3] were the importance of multiple fiber scatteringfor light colored hair was addressed. It plays an importantrole for the right perception of the hair color and must not beneglected (see also Fig. 9). Efficient compression strategiesare needed, since the BFSDF is an eight dimensional functionsuch as the general BSSRDF. Moreover, if the BFSDF is givenin a purely numerical form, the four dimensional renderingintegral itself is very costly to evaluate. Therefore, the BFSDFshould be approximated by series of analytical functions whichcan be efficiently integrated. Fortunately, many scatteringdistributions have very few intensity peaks which can bewell approximated with existing compression schemes (e.g.Lafortune Lobes [24], Reflectance Field Polynomials [25],

Linear Basis Decomposition [26], clustering and factorization[27], [28], [29]) developed in the realm of BRDF and BTFrendering.

Another important challenge is the reconstruction of theBFSDF or BCSDF from macroscopic image data, whichfrequently are more easily obtainable than systematic mea-surements at the fiber level. We presume that using specialmodeling knowledge which is coded by the usage of a lowerdimensional scattering functions such as the BCSDF mightbe crucial for the practical solvability of inverse problemsinvolving the BFSDF.

Acknowledgements.We are grateful to the anonymous ref-erees for many detailed suggestions. We would like to thankR. Klein for the possibility to use a robot camera system for thephotographs given in Fig. 14, and to R. Sarlette and A. Hennesfor their technical support. The presented work is partiallysupported byDeutsche Forschungsgemeinschaftunder grantWe 1945/3-2.

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Arno Zinke received his M.Sc. degree in computerscience (Dipl.-Inform.) from the University of Bonn,Germany, in 2004, where he is currently pursuingthe PhD degree in computer science. His researchinterests include physically-based simulation andrendering.

Andreas Weber received his M.Sc. degree in math-ematics (Dipl.-Math.) in 1990 and his PhD degree(Dr. rer. nat.) in computer science in 1993 from theUniversity of Tubingen, Germany. After being post-doctoral researcher at the University of Tubingen,Germany, Cornell University, Ithaca, NY, USA, andthe Fraunhofer-Institut for Computer Graphics inDarmstadt, Germany, he joined the University ofBonn, Germany, as a professor of computer science.He has published more than 50 research papers incomputer graphics and computer algebra. A focus

of his current research is physics-based simulation and animation. He is amember of the IEEE.

APPENDIX

All renderings given in the appendix or as supplementalmaterial are rendered with a proprietary ray tracer.

Unless stated differently “BFSDF rendering” always meansthat for direct illumination the near-field shader (see sec-tion IX-B) was used and indirect illumination was computedfrom an importance sampled BFSDF using Monte-Carlo pathtracing.

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Fig. 9. Two different strands of light blond hair fibers.Top row: BFSDFrenderings with full global illumination.Bottom row: renderings of the samescenes according to Marschner et al. for direct (local) illumination. (Theparameters were set to match our original BCSDF approximation of theBFSDF). This example clearly shows the evidence of multiple fiber scatteringfor the right perception of the hair color.

Fig. 10. Different strands of translucent dielectric fibers illuminated bythree distant point light sources. The left two leftmost images were renderedwith a hair like material and the two rightmost images show synthetic fibers.Direct illumination was computed according to section IX-B. The BCSDFwas computed from the BFSDF for dielectric fibers (eqn. (52)).

Fig. 11. Top row: Various physically based (i.e. energy conserving) BCSDFrenderings of opaque fibers illuminated by three distant point light sources.The BCSDFs were generated according to Section IX-A from the followingBRDF types: Lambertian (left); Fresnel+(1−Fresnel)×Lambertian (leftmiddle); Fresnel for aluminium (right middle); Fresnel for gold (right). TheseBCSDF examples are given in a purely analytical form, so that parameters—like index of refraction—can be easily changed on the fly, e.g. for the“gold” and “aluminium” examples, which use the same BCSDF with differentmaterial properties.Middle row: BFSDF renderings of the same scenes.Notice how well the BCSDF renderings already match the BFSDF renderings.Bottom row: Some exemplary renderings with the phenomenological modelof Kajiya& Kay. Lighting and geometry was not changed. The diffusecomponent was computed according to [1] from the diffuse coefficient of theBRDF. The width of the specular lobe (if present) as well the peak intensitywas set to match the ones of the original physically based renderings settingthe Fresnel factor to one. Nevertheless, the results differ substantially from theBCSDF resp. BFSDF renderings. Note that the BCSDF images were computedapproximately one order of magnitude faster, since16× less eye-rays per pixelwere needed to achieve a similar rendering quality.

Fig. 12. Various BFSDF rendering results.

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Fig. 13. BFSDF renderings of a hair style consisting of about 90000 individual hair fibers (left: brown; middle: blond; right: dark red dyed) illuminated byfive point light sources.

Fig. 14. An oval made of rough plastic illuminated by one point light source. The upper row shows photographs and the second row renderings with avariant of the shader presented in section IX-B. The parameters are set to roughly match the original photographs. Note the similarity of both positions andintensities of the highlights for different viewing angles. It took less than five seconds to render each frame with simple ray tracing.

Fig. 15. Various BFSDF examples with a more complex illumination. The images were rendered in about 30 seconds each.