Modeling human hair is one of the great challenges ... can be seen in the color plates at the end of the pa- per. ... ders with a colored interior and a surface com-.
Photo-Realistic Rendering of Blond Hair A. Zinke, G. Sobottka, A. Weber Institut f¨ur Informatik II, Universit¨at Bonn, Germany Email: {zinke, sobottka, weber}@cs.uni-bonn.de
Abstract We present a method for photo-realistic hair rendering, which is based on light scattering from (human) hair fibers. In contrast to existing approaches our method gives realistic results also for light hair types and for close-ups. These results are obtained by using a “near-field” model taking into account different scattering components and indirect illumination. Our near-field model is a generalization of previously published methods that can be seen as far-field models. Nevertheless our more general shading model can be used computationally as efficiently as the previous ones.
1
Introduction
Modeling human hair is one of the great challenges of computer graphics. Although a lot of progress has been made in the simulation of hair dynamics [3], [1], modeling of hair geometries [6], and in the development of interactive hair styling tools [2], [12] it is still not possible to manage the complexity of human scalp hair in real-time. In the realm of rendering the partially contradicting overall goals of realism versus rendering time have led to the introduction of a variety of hair rendering techniques considering both backlighting and self shadowing effects and making intense use of pixel-blending, shadow buffers, deepshadow maps, opacity maps or textures: [4], [9], [13], [7], [8], [10]. A recent milestone with respect to the goal of photo-realism has been [11]. In this article Marschner et al. propose a model for light scattering from human hair fibers. It is in particular suitable as a shading model for direct illumination by distant light sources and a distant viewer, as it uses a “far-field” scattering model. However, for lights close to the hair, close-ups or indirect illumination it is not a priori guaranteed that VMV 2004
all of the approximations are still valid. When hairhair scattering or local scattering phenomena have to be taken into account the scattering model has to be generalized. This generalization is especially necessary for the photo-realistic rendering of blond hair types, for which effects of indirect illumination are much more prominent than for darker hair types. Our Contribution: In this paper we present an adequate generalized model for local scattering effects, to which we will refer as near-field scattering model. This model is the theoretical foundation of our photo-realistic hair rendering framework, which achieves high quality for close-ups and indirect illumination, suitable for blond hair types. Furthermore, we show that for light hair types the effects of indirect illumination from neighboring hair fibers can result in visible effects up to a recursion depth of about 5. We can use our approach to identify situations in which common techniques for simulating transparency effects—pixel blending or (deep) shadow maps—will not differ substantially from the results of a more accurate model. For these cases we present an approximation for indirect illumination. Rendering results of some exemplary hair scenes can be seen in the color plates at the end of the paper.
2
Preliminaries
For a hair fiber with elliptical cross-section the local normalized tangent is denoted ~ u and points into the direction of the hair tip; incoming and outgoing directions ω ~ i and ω ~ r may then be locally expressed in polar coordinates. Thus, it is possible to write ω ~ i = (θi , ϕi ) and ω ~ r = (θr , ϕr ) where θi and θr are the angles of inclination to the hair tangent (0◦ is perpendicular, −90◦ parallel to the fiber’s axis). Stanford, USA, November 16–18, 2004
Figure 1: Left: Notations; the normal plane is colored in blue. Right: The three scattering components. The dashed lines indicate the scattering directions of a smooth dielectric cylinder. The azimuths around the hair are denoted ϕi and ϕr and measured within a plane perpendicular to the hair. This plane, defined by the major (~v ) and minor (w) ~ axes of the elliptical cross section, is called normal-plane and includes all local surface normals. The three vectors ~ u, ~v and w ~ form a basis of the euclidian three-dimensional space. In this context, the azimuths are measured so that ~v is 0◦ and w ~ is 90◦ . Fig. 1a) gives a sketch of this situation. If eccentricity equals one, we have a circular cross section and ~v and w ~ must be chosen such that (~ u, ~v , w) ~ forms an orthogonal basis. With these definitions one can define the following helpful variables: • the inclination difference angle θd : θd = (θi − θr )/2 • the relative azimuth φ: ϕ = ϕr − ϕi The angle γi is the angle of incidence (between surface normal and incoming direction) and, according to Snell’s law, γt = arcsin (sin γi /n2 ) is the angle of the refracted ray. Their projections onto the normal plane are called γi0 and γt0 . We assume the hair having a constant index of refraction n2 (for human hair usually 1.55) and the surrounding media being vacuum (n1 = 1.0).
3
tures observed during measurements are modeled by the first three scattering components called R (direct reflection), T T (two times transmitted) and T RT (two times transmitted and one internal reflection) of a smooth dielectric cylinder, since they contain the biggest fraction of scattered power (see Fig. 1b)). Consider a generalized cylinder with uniform cross section along a fixed axis. The idea is to simulate far-field scattering using a special model called curve scattering. In this model hair fibers are treated as a 3D-curves with a cross section proportional to its effective diameter which indicates that a thicker fiber intercepts more light. This implies that viewer and light sources are distant (when compared to the dimension of the hair radius) to the hair. As a result of Bravais’ law it has been shown that the scattering function of R, T T and T RT can be decomposed into two parts, one analyzing the scattering geometry of the normal plane and another longitudinal scattering term. Furthermore, a ray that enters a cylinder at a particular inclination to the axis will always exit at the same inclination. The azimuthal scattering function N consists basically of three different expressions: N = AF S, where A is an absolute absorption coefficient, F is the Fresnel attenuation and S a ray-density factor. By replacing n2 with a modified index of refraction 1/2 n02 (n2 , θi ) = (n22 − sin2 θi ) / cos θi , the usual Snell’s law of refraction still holds for the projected angles: sin γt0 = sin γi0 /n02 . The law of reflection and Fresnel’s formulas can also be used in conjunction with these angles to calculate N .
Previous work on light-scattering from human hair fibers
Marschner et al. [11] propose an approximation of hair fibers as transparent dielectric circular cylinders with a colored interior and a surface composed of rough, tilted scales. All important fea666
respect to θi and θr and can be modeled by normalized Gaussians. These Gaussians will be written as MR , MT T , and MT RT . The corresponding widths βR , βT T , and βT RT are empiric properties. To achieve a better coherence with experimental measurements the scattering function was finally generalized by replacing θi with θd in MR,T T,T RT . In the T RT -component n2 is replaced by n∗2 (n2 , ϕh ), which gives a simple qualitative approximation for scattering from elliptic cross sections with mild eccentricities.
In the following we consider a circular cross section leading to a simpler form of scattering functions as we exploit symmetries. The following important results base on the theory deduced in [11]. 1) The relative azimuths are directly related to the angle of incidence: ϕR = −2 γi0 , ϕT T = 2 (γt0 − γi0 ) + π, ϕT RT = 4γt0 − 2γi0 2) The Fresnel attenuation may be computed as FR = F (γi0 , n02 , n02k ), FT T = 1 − F (γi0 , n02 ) 1 − F (γt0 , 1/n02 , 1/n02k ) , FT RT = 1 − F (γi0 , n02 , n02k ) 0 0 0 1 − F (γt , 1/n2 , 1/n2k ) F (γt0 , 1/n02 , 1/n02k ),
4
A Generalized Scattering Model
To support near-field scattering from generalized dielectric cylinders with arbitrary but locally constant cross section and fixed axis we extend the model of Marschner et al. [11]. We enclose the fiber’s local shape with the minimum enclosing cylinder with radius R and axis ~ u, to allow for a geometry independent parametrization. Our scattering function depends on the local geometry of the hair, i.e., the positions where incoming and outgoing rays intersect the enclosing cylinder can be parameterized with respect to incoming direction ω ~ , h0 = Rh = R sin γp0 and s (Fig. 2 and 3).
where F are the usual Fresnel formulas which have to be calculated separately for parallel (with refractive index n02k = n22 /n02 ) and perpendicular (with n02 ) polarized light. ~ R = ~1, 3) The absolute absorption is A ~ T T = (exp(−lσaR ), exp(−lσaG ), exp(−lσaB )), A ~ T RT = A ~ 2T T with the absorption length A 0 l(γi , θi , n2 ) = ls / cos θt , i.e., the distance covered by a ray inside the cylinder (T T ). It doubles for the T RT -mode. Its projection onto the normal plane p is given by ls = R 2 + 2 cos 2γt0 = 2R cos γt0 with R being the effective radius of the fiber; θt = arccos (n02 /n2 cos θi ) is the inclination angle of the refracted rays within the cylinder. The relative absorption coefficient is denoted by σa and is separately given for the three color components red, green, and blue. 4) The ray density S is given by SR = |2dϕR /dh|−1 , ST T = |2dϕT T /dh|−1 , ST RT = |2dϕT RT /dh|−1 p . For ST RT singularities occur at γc0 = ± arcsin 4 − n0 22 /3, which have to be smoothed out. These singularities are called caustics and are a consequence of the idealized model (smooth cylinder). The caustics are first removed from the intensity distribution and then replaced by Gaussians (with the same portion of energy) centered over γc0 . Generally, human hair fibers can not be treated as perfect-smooth cylinders. Since the hair surface (cuticula) is tiled, the scattering inclinations differ from those of a smooth cylinder: The R mode is displaced towards the root, T T and T RT towards the tip. These shifts are denoted αR , αT T , and αT RT (Fig. 1b)). The inside of the hair fiber (cortex) consists of inhomogeneous matter. Therefore, the longitudinal scattering modes are blurred with
Figure 2: Longitudinal parameter s. The scattering function is the quotient of outgoing radiance dL and incoming irradiance dE scattered from an infinitesimal small surface. 0 ˆ ωi , h0 , ω S(~ i ~ i , hr , sr − si ) =
dL(~ ωr , h0r , sr ) dE(~ ωi , h0i , si )
.
Due to Lambert’s law the irradiance is proportional to the cosine of the angle of incidence: 0
0
0
0
dE(~ ωi , si , hi ) = Li (~ ωi , si , hi ) cos(γi (hi , ϕi , θi )) dhi dsi .
666
(thus, s increases or decreases), intensity can only be measured if: si =
sr l tan θ sgnθi + sr 2l tan tθ sgnθ t i + sr
R - mode T T - mode T RT - mode,
with absorption length l. The complete longitudinal functions then are MR = δ(θr + θi )δ(sr − si )/θi2 , MT T = δ(θr + θi )δ(sr − si + 2R cos γt tan θt sgnθi )/θi2 , MT RT = δ(θr + θi )δ(sr − si + 4R cos γt tan θt sgnθi )/θi2 .
Figure 3: Azimuthal parameter h0 . Note γi,r is the angle of incidence at the minimum enclosing cylinder and γi,r 0 its projection onto the normal plane, thus, the angle of incidence at the cross section perimeter.
The factor θi−2 accounts for the projected solid angle of the specular cone. In contrast to M , N depends on the cross section geometry. Since θ influences N only indirectly through n02 and absorption length, it can be derived independently in 2D. The azimuthal scattering function for a circular cross section is a product of Fresnel attenuation F , absolute absorption A, ray-density S 0 and two additional δ-factors. The δ-terms say that intensity can only be measured if h0r = −h0i and if the relative azimuth ϕ satisfies the equations already given in the section before. Thus, the azimuthal scattering functions for a circular cross section are the following ones.
Now the scattering integral can be written as follows. R
R
ω ~ i ∈Ω
s∈R
LrR(~ ωr , h0r , sr ) = 0 ˆ ωi , h0 , ω S(~ i ~ r , hr , sr − si )
h0 ∈[−R,R] i
Li (~ ωi , si , h0i ) cos(γi (h0i , ϕi , θi )) dh0i dsi dθi dϕi .
Every cross section geometry has its own unique γi (h0i , ϕi , θi ). We shall now consider the special case of a circular cross section. Using the relations γi0 = arccos(cos γi / cos θi ) and h0i /R = sin γi0 we obtain for the angle of local incidence γi : v u u cos γi = t1 −
h0i R
0 NR = FR (n02 , h0i , R)SR (n02 , hi ) δ(h0i + h0r )δ(ϕr − ϕi − (−2γi )), r,g,b NT T = AT T (γi0 , θt , n02 , σa ) 0 0 FT T (n02 , γi0 )ST T (n2 , hi ) 0 0 δ(hi + hr )δ(ϕr − ϕi − (2γt − 2γi + π)),
!2 cos θi .
r,g,b NT RT = AT RT (γi0 , θt , n02 , σa ) 0 0 FT RT (n02 , γi0 )ST RT (n2 , hi ) 0 0 δ(hi + hr )δ(ϕr − ϕi − (4γt − 2γi )).
Plugging this equation into the given scattering integral gives R
R
ω ~ i ∈Ω
s∈R
The two coefficients A, F are the same factors as are described in the previous section. S 0 is different from S and can be roughly computed from the quotient of the “thickness” of the outgoing and the incoming beams, since light from a small interval dh0i is scattered into dh0r . With dh0 ≈ Rdϕ we get:
LrR(~ ωr , h0r , sr ) = 0 ˆ ωi , h0 , ω S(~ i ~ r , hr , sr − si )
h0 ∈[−R,R] i s
Li (~ ωi , si , h0i )
1−
0 2 h i R
cos θi dh0i dsi dθi dϕi .
dh0 0 r SR ≈ dh0
Similar to [11] the scattering function can be approximately expressed as a superposition of the first three scattering modes, namely R, T T , and T RT . With respect to Bravais’ law we write each component as a product of two terms. One analyzing the azimuthal scattering geometry (N ) and another longitudinal term (M ).
−1 ≈
i R
dh0 0 r ST T ≈ dh0 i dh0 0 r ST RT ≈ dh0
R∂ϕ ∂h −1 i R = 2SR , ∂hi ∂h0 i
−1 −1 R∂ϕ T T ∂hi ≈ = 2ST T , 0 ∂hi ∂h i TT
−1 R∂ϕ T RT ∂hi ≈ = 2ST RT . 0 ∂h ∂h i i T RT i
ˆ i , θi , h0 , ϕr , θr , h0 , sr − si ) = S(ϕ i r NR (ϕi , h0i , ϕr , h0r )MR (θi , θr , sr − si )+ NT T (ϕi , h0i , ϕr , h0r )MT T (θi , θr , sr − si )+ NT RT (ϕi , h0i , ϕr , h0r )MT RT (θi , θr , sr − si ).
−1
For caustics removal we use the methods proposed in [11] and modify the ST0 RT factor. The modified coefficient is denoted ST00 RT . In this context, it is a very rough approximation, since the assumption that the cross section is illuminated with
For a perfect dielectric cylinder scattering occurs in the specular cone (θr = −θi ). Additionally, because the rays propagate into longitudinal direction 666
a homogenous irradiance at any incoming azimuth holds only for distant light sources. Recalling [11] the scattering function of a perfect-smooth dielectric cylinder can be generalized to simulate light scattering from (human) hair fibers: a) because of cuticula roughness and inhomogeneous cortex, the δ-distributions are replaced by lobes centered over the δ-peaks. We simply use Gaussians and denote them m, u, v and w. The widths of mR,T T,T RT are equal to the widths of the Gaussian (βR,T T,T RT ) defined in the previous section. Due to the tiled cuticula the scattering components are longitudinally shifted by αR,T T,T RT . All other related parameters are empiric properties, too. b) θi is substituted with θd in the longitudinal functions MR,T T,T RT . This scattering function is a realistic approximation for nearly circular cross sections. In principle, every cross section can be analyzed by changing the azimuthal functions N and γi to match the corresponding scattering geometry. But for an elliptic cross section with mild eccentricity some effects can be qualitatively simulated by replacing the index of refraction n2 with n∗2 in the T RT -term. Combining all these results gives a realistic model for light scattering from (human) hairs with curvature being small compared to the hair radius (cf. Appendix 1)). The approach works fine even for more oblique incidence (s rapidly increases with respect to θt ) since in such a case both the T T - and the T RT -component contributes a marginal portion of energy to the intensity distribution. Though, when implemented it usually achieves only a poor rendering performance. One possibility to improve the performance is to simplify the model, e.g., by neglecting the s-dependence. Some rendering results of such a simplified model are shown in Fig. 7b).
5
with respect to this variable we have Z1
avg
AF SR,T T ,T RT =
AR,T T ,T RT 0
0
FR,T T ,T RT SR,T T ,T RT dh.
avg The averaged factors are called AF SR,T T,T RT and depend on outgoing inclination θr , index of refraction and relative absorption coefficients only (cf. Appendix 2)). These integrals can be solved numerically. However, for a more efficient implementation, we use the fact that these integrals can be approximated well by polynomials or products of polynomials and exponentials, cf. Fig. 4. The curve progression of the T RT -graph component strongly depends on the method used to remove the caustics. For hair with nearly circular cross section that is lit from behind or front these simplifications are very accurate. Therefore, in these cases it can be expected that pixel blending and simple (deep) shadow/opacity buffers could be used without much loss in quality—provided the parameters are chosen in accordance with the average attenuation coefficients.
6
Results
Blond Hair: We have implemented our near-field scattering model within a standard ray tracer which has not been optimized for speed yet. In Fig. 5 a comparison of the result of rendering a blond hair strand for different methods is given, which all use ray-tracing with real self shadowing. For the corresponding parameters the same values have been used in all examples, resulting in globally different hair-colors. All pictures were originally rendered with a resolution of 1 280 × 1 280 pixels (64 rays per pixel) on a Pentium-4 machine operating at 2.6 GHz. The rendering time was 7 min for Fig. 5a) which did not increase when using our near-field approach. The computational costs for considering the effects of indirect illumination are much higher: the rendering time of Fig. 5b) was about 150 min. However, as can be seen clearly in these examples the effects of indirect illumination are important for the photo-realistic rendering of blond hair. Fig. 7 also demonstrates that a recursion depth of up to 5 might
An Approximation of Computing Attenuation Coefficients
In case of indirect illumination, most local scattering effects arising from hair geometry can be globally approximated because of statistical reasons. The idea is to average all attenuation coefficients from light scattered from the hair and potentially radiating into viewing direction. For nearly circular hair |h0i | ≈ |h0r | and |θi | ≈ |θr |. Therefore, by parameterizations with h = sin γi and integration 666
Figure 4: Left: AF S avg of the R-component for n2 = 1.8 (red line), n2 = 1.55 (green line) plus an approximation (dashed line), n2 = 1.3 (blue line); the averaged ray density factor (magenta line); Center: AF S avg of the T T -component for n2 = 1.55 and different relative absorption coefficients (relative to radius); Right: AF S avg of the T RT -component for n2 = 1.55 and different relative absorption coefficients (relative to radius). The caustics were removed with the method proposed in [11] using following parameters: wc = 15◦ , kG = 2.5, ∆η 0 = 0.3. visibly influence the global rendering result for a blond hair strand. As the neighboring hair fibers being the source of indirect illumination are close to the hair fibers, the general assumption of light sources being far away from the hair fibers used in [11] is certainly not justified in general. The problems of photo-realistically rendering blond hair-types with the far-field model of [11] is exemplified in Fig. 6. The same parameter setting that result in blond hair by the near-field model (plus hair-hair scattering) give a much darker hair color when using the model of [11]. The natural approach of getting lighter hair color fails: when the absorption coefficients are set to a lower value— we reduce them by a factor of 2 in Fig. 6c)—the overall result is not the one of a lighter hair color. Other possibilities to make the appearance of the hair lighter—such as using a much higher diffusion coefficient—would also result in a loss of photorealism.
to estimate scattered intensities. Especially, when using recursive methods it helps to set up the maximum recursion depth. For example, considering a very light hair with an absorption coefficient of 0.1 (relative to the hair radius) after 5 recursions one still gets an averaged outgoing radiance of up to 17% (AF STavg T at θi = 0 raised to the power of 5; see Fig. 4) of the incoming irradiance. Additionally, the averaged attenuation coefficients help to simplify hair geometries by evaluating, whether some hairs noticeable contribute to the scattered intensity distribution or not.
7
Conclusions
The combination of the model introduced in [11] for direct illumination (as a shader) together with averaged attenuation coefficients for hair-hair scattering gives almost photo-realistic results in many settings. However, there are scenes—especially closeups and light (blond) hair or certain light source settings—where our more accurate near-field model leads to noticeable better and more realistic visualizations. The effects of indirect lighting might visibly contribute to the result of the ray tracing up to a recursion depth of 5 in complicated situations (see Fig. 7a)). Thus, most of the rendering time is spent on intersection tests in general. Limiting the recursion depth will drastically improve on the rendering time (by sacrificing some photo-realism) as well as utilizing pixel blending or shadow/opacity maps
Using Averaged Attenuation Coefficients: Our approach gives reasonable results for both direct and indirect illumination respecting the model introduced in [11]. The more accurate version captures a lot of effects of local scattering and shows that hair in general cannot be treated as a transparent cylinder. However, under many circumstances averaged attenuation coefficients only depending on incoming inclinations provide good results for indirect illumination, too. Another advantage of this approximated coefficients approach is the possibility 666
[10, 5] (with averaged attenuation factors) as an approximation for hair-hair scattering. Using local surface normals of the given hairgeometries our near-field scattering model can be implemented as efficiently as the far-field approach developed in [11]. Somewhat surprisingly, the computational effort necessary for photo-realistic hair rendering is strongly dependent on the hair-color! Whereas for black hair the classic phenomenological model of Kajiya and Kay [4] gives (near) photo-realistic results, the photo-realistic rendering of brown hair requires the sophistication of the far-field model developed in [11], but with the possibility to restrict recursion depth of indirect illumination and to use approximations. Blond hair is even more difficult: The necessary recursion depth for indirect illumination can be up to 5, and the results from our nearfield model visibly differ from those of the far-field model.
avg
AF ST T (θr , n2 , σa ) = 1 2 1 − F (γi0 (h), n02 )
avg
AF ST RT (θr , n2 , σa ) = 1 − F (γi0 (h), n02 ) exp
1 − F (γt0 (h), 1/n02 )
cos γt0 (h,n02 ) −4Rσa dh. cos θ t
[2] Lieu-Hen Chen, Santi Saeyor, Hiroshi Dohi, and Mitsuru Ishizuka. A system of 3d hair style synthesis based on the wisp model. The Visual Computer, 15(4):159–170, 1999. [3] Sunil Hadap and Nadia Magnenat-Thalmann. Modeling dynamic hair as a continuum. Computer Graphics Forum, 20(3), 2001. [4] J. T. Kajiya and T. L. Kay. Rendering fur with three dimensional textures. In Proceedings of the 16th annual conference on Computer graphics and interactive techniques, pages 271–280. ACM Press, 1989. [5] Tae-Yong Kim and Ulrich Neumann. Opacity shadow maps. In Proc. of Eurographics Workshop on Rendering, pages 177–182, 2001. [6] Tae-Yong Kim and Ulrich Neumann. Interactive multiresolution hair modeling and editing. ACM Transactions on Graphics, 21(3):620–629, July 2002. SIGGRAPH 2002. [7] Waiming Kong and Masayuki Nakajima. Visible volume buffer for efficient hair expression and shadow generation. In IEEE Computer Animation, pages 58–65, 1999. [8] Waiming Kong and Masayuki Nakajima. Hair rendering by jittering and pseudo shadow. In Computer Graphics International 2000, pages 287–291, 2000. [9] Andre M. LeBlanc, Russell Turner, and Daniel Thalmann. Rendering hair using pixel blending and shadow buffers. The Journal of Visualizations and Computer Animation, 2(3):92– 97, 1991.
ˆT RT (~ S ωi , h0i , ω ~ r , h0r , sr − si ) = r,g,b AT RT (γi0 (h0i , R), θt (θd , n2 ), n02 (θd , n2 ), σa ) 0 FT RT (n2 (n2 , θd ), hi (h0i , R)) 00 0 0 ST RT (n2 (n2 , θd ), hi (hi , R)) mT RT (θr + θi , αT RT , βT RT ) uT RT (h0r , h0i ) vT RT (sr − si + 2Rsgnθi (cos γt tan(θt + 2α)+ cos γt tan θt )) 2 wT RT (ϕr − ϕi − (4γt − 2γi ))/θd , with θt = arccos (n02 /n2 cos θd ).
[10] Tom Lokovic and Eric Veach. Deep shadow maps. In Proceedings of the 27th annual conference on computer graphics and interactive techniques, pages 385–392, 2000. [11] Stephen R. Marschner, Henrik Wann Jensen, Mike Cammarano, Steve Worley, and Pat Hanrahan. Light scattering from human hair fibers. ACM Transactions on Graphics, 22(3):780–791, 2003. SIGGRAPH 2003. [12] Zhan Xu and Xue Dong Yang. V-hairstudio: An interactive tool for hair design. IEEE Computer Graphics and Application, 21(3):36–43, May/June 2001.
2) These imply the following average attenuation coefficients:
2
00 0 0 ST RT (h, θr , n2 )F (γt (h), 1/n2 )
[1] Yosuke Bando, Bing-Yu Chen, and Tomoyuki Nishita. Animating hair with loosely connected particles. Computer Graphics Forum, 22(3):411–411, 2003. Eurographics 2003.
ˆT T (~ S ωi , h0i , ω ~ r , h0r , sr − si ) = r,g,b AT T (γi0 (h0i , R), θt (θd , n2 ), n02 (θd , n2 ), σa ) FT T (n02 (n2 , θd ), hi (h0i , R)) 0 0 0 ST (n (n , θ ), h (h , R)) 2 i d 2 i T mT T (θr + θi , αT T , βT T ) uT T (h0r , h0i ) vT T (sr − si + 2R cos γt tan θt sgnθi ) 2 wT T (ϕr − ϕi − (2γt − 2γi + π))/θd ,
1
R1 0
References
ˆR (~ S ωi , h0i , ω ~ r , h0r , sr − si ) = FR (n02 (n2 , θd ), hi (h0i , R)) 0 SR (n02 (n2 , θd ), hi (h0i , R)) mR (θi + θr , αR , βR ) uR (h0r , h0i )vR (sr − si ) 2 wR (ϕr − ϕi − (−2γi ))/θd ,
(θr , n2 ) =
t
1) The total scattering functions for each component are:
avg
1 − F (γt0 (h), 1/n02 )
cos γt0 (h,n02 ) exp −2 Rσa dh, cos θ
Appendix
AF SR
q q 2 1−h2 n0 2 2 −h q q 2 02 0 1−h − n −h2 2
R1
[13] Tzong-Jer Yang and Ming Ouhyoung. Rendering hair with back-lighting effects. In Proc. of CAD/Graphics’97, pages 291–296, 1997.
Z1 p 0 0 1 − h2 F (γi (h), n2 ) dh, 0
666
Figure 5: Blond hair strand. Left: Direct illumination; far-field model, no hair-hair-scattering (the model of [11]); Center: Direct illumination + hair-hair scattering: near-field model. Right: Close-ups (top: nearfield model, bottom: far-field model). The hair model consists of 2 500 single hairs with 100 cylindric segments each.
Figure 6: Comparison between the model of Marschner et al. and our new near-field model for light hair types. The hair ball consists of 10 000 single hairs with 30 cylindric segments each. Left: The fiber assembly rendered with our near-field model; Center: The model of [11] is used for rendering; Right: Rendering using the model of [11] with absorption coefficients set to half the values of b).
Figure 7: Left: Blond hair strand rendered with our near-field model and increasing recursion depths (1–4). Right: Different hair scenes rendered with our near-field model.
666
