Reflection from Layered Surfaces due to Subsurface Scattering Pat Hanrahan Wolfgang Krueger Department of Computer Science Department of Scientific Visualization Princeton University German National Research Center for Computer Science Abstract of incidence. Diffuse reflection is qualitatively explained as due to subsurface scattering [18]: Light enters the material, is absorbed The reflection of light from most materials consists of two ma- and scattered, and eventually exits the material. In the process of jor terms: the specular and the diffuse. Specular reflection may this subsurface interaction, light at different wavelengths is differ- be modeled from first principles by considering a rough surface entially absorbed and scattered, and hence is filtered accounting consisting of perfect reflectors, or micro-facets. Diffuse reflection for the color of the material. Moreover, in the limit as the light ray is generally considered to result from multiple scattering either is scattered multiple times, it becomes isotropic, and hence the di- from a rough surface or from within a layer near the surface. Ac- rection in which it leaves the material is essentially random. This counting for diffuse reflection by Lambert’s Cosine Law, as is qualitative explanation accounts for both the directional and col- universally done in computer graphics, is not a physical theory ormetric properties of diffuse materials. This explanation is also based on first principles. motivated by an early proof that there cannot exist a micro-facet This paper presents a model for subsurface scattering in layered distribution that causes equal reflection in all outgoing directions surfaces in terms of one-dimensional linear transport theory. We independent of the incoming direction [10]. derive explicit formulas for backscattering and transmission that The above model of diffuse reflection is qualitative and not can be directly incorporated in most rendering systems, and a gen- very satisfying because it does not refer to any physical param- eral Monte Carlo method that is easily added to a ray tracer. This eter of the material. Furthermore, there is no freedom to adjust model is particularly appropriate for common layered materials coefficients to account for subtle variations in reflection from dif- appearing in nature, such as biological tissues (e.g. skin, leaves, ferent materials. However, it does contain the essential insight: etc.) or inorganic materials (e.g. snow, sand, paint, varnished or an important component of reflection can arise from subsurface dusty surfaces). As an application of the model, we simulate the scattering. In this paper, we present a model of reflection of light appearance of a face and a cluster of leaves from experimental due to subsurface scattering in layered materials suitable for com- data describing their layer properties. puter graphics. The only other work in computer graphics to take CR Categories and Subject Descriptors: I.3.7 [Computer this approach is due to Blinn, who in a very early paper presented Graphics]: Three-Dimensional Graphics and Realism. a model for the reflection and transmission of light through thin Additional Key Words and Phrases: Reflection models, integral clouds of particles in order to model the rings of Saturn[2]. Our equations, Monte Carlo. model differs from Blinn’s in that it is based on one-dimensional linear transport theory—a simplification of the general volume rendering equation [19]— and hence is considerably more general 1 Motivation and powerful. Of course, Blinn was certainly aware of the trans- An important goal of image synthesis research is to develop a port theory approach, but chose to present his model in a simpler comprehensive shading model suitable for a wide range of ma- way based on probabilistic arguments. terials. Recent research has concentrated on developing a model In our model the relative contributions of surface and subsur- of specular reflection from rough surfaces from first principles. face reflection are very sensitive to the Fresnel effect (which Blinn In particular, the micro-facet model first proposed by Bouguer did not consider). This is particularly important in biological tis- in 1759 [4], and developed further by Beckmann[1], Torrance & sues which, because cells contain large quantities of water, are Sparrow[26], and others, has been applied to computer graphics translucent. A further prediction of the theory is that the sub- by Blinn [2] and Cook & Torrance[8]. A still more comprehen- surface reflectance term is not necessarily isotropic, but varies in sive version of the model was recently proposed by He et al[12]. different directions. This arises because the subsurface scattering These models have also been extended to handle anisotropic mi- by particles is predominantly in the forward direction. In fact, it crofacets distributions[24, 5] and multiple scattering from complex has long been known experimentally that very few materials are microscale geometries[28]. ideal diffuse reflectors (for a nice survey of experiments pertaining Another important component of surface reflection is, however, to this question, see [18]). diffuse reflection. Diffuse reflection in computer graphics has al- We formulate the model in the currently emerging standard most universally been modeled by Lambert’s Cosine Law. This terminology for describing illumination in computer graphics [16, law states that the exiting radiance is isotropic, and proportional 11]. We also discuss efficient methods for implementation within to the surface irradiance, which for a light ray impinging on the the context of standard rendering techniques. We also describe surface from a given direction depends on the cosine of the angle how to construct materials with multiple thin layers. Finally, we apply the model to two examples: skin and leaves. For these examples, we build on experimental data collected in the last few PermissionPermission to to copy copy without without fee fee all all or or part part of of this this material material is is granted granted years, and provide pointers to the relevant literature. pprovidedrovided that that the the copies copies are are not not made made or or distributed distributed for for direct direct Another goal of this paper is to point out the large amount of ccommercialommercial advantage, advantage, the the ACM ACM copyright copyright notice notice and and the the title title of of the the recent work in the applied physics community in the application ppublicationublication and and its its date date appear, appear, and and notice notice is is given given that that copying copying is is by by of linear transport theory to modeling appearance. ppermissionermission of of the the Association Association for for Computing Computing Machinery. Machinery. To To copy copy ootherwise,therwise, or or to to republish, republish, requires requires a a fee fee and/or and/or specific specific permission. permission. ©1993©1993 ACM ACM-0-0-89791-89791-601-601--8/93/008/00158/93/008…$1.50…$1.50 165 1.0 Li Lr,s Lr,v θ 0.8 i θr θr layer1 !!! !!!!!!!!! !!! 0.6 !!!!!! !!!!!!!!! !!!!!! !!!!!!!!!!!!!!!!!!!!! !!!!!! θ d!!!!!!layer2!!!!!!!!!!!! ’ !!!!!!!!! !!! !!!!!!!!!!!! !!!!!!!!!!!!!!! 0.4 !!!!!!!!!!!!!!!!!!!!! θ !!!!!!!!!!!! Amplitude !!! !!!!!!!!! !!!!!! 666666666666666666666666666!!! !!! 0.2 666666666666666666666666666layer3 666666666666666666666666666 666666666666666666666666666θ θ 0.0 666666666666666666666666666t t Lri Lt,v 0 10 20 30 40 50 60 70 80 90 z Angle of incidence Figure 1: The geometry of scattering from a layered surface Figure 2: Fresnel transmission and reflection coefficients for a ray leaving air (n = 1:0) and entering water (n = 1:33). (θi; φi) Angles of incidence (incoming) (θr; φr) Angles of reflection (outgoing) Lt;v - transmitted radiance due to volume or subsurface scat- (θt; φt) Angles of transmission tering L(z; θ; φ) Radiance [W / (m2 sr)] Li Incident (incoming) radiance The bidirectional reflection-distribution function (BRDF) is de- Lr Reflected (outgoing) radiance fined to the differential reflected radiance in the outgoing direction Lt Transmitted radiance per differential incident irradiance in the incoming direction [23]. L+ forward-scattered radiance L backward-scattered radiance L (θ ; φ ) − r r r fr(θi; φi; θr; φr) BRDF fr(θi; φi; θr; θr) ≡ Li(θi; φi) cos θid!i ft(θi; φi; θt; φt) BTDF fr;s(θi; φi; θr; φr) Surface or boundary BRDF The bidirectional transmission-distribution function (BTDF) has a ft;s(θi; φi; θt; φt) Surface or boundary BTDF similar definition: fr;v(θi; φi; θr; φr) Volume or subsurface BRDF ft;v(θi; φi; θt; φt) Volume or subsurface BTDF Lt(θt; φt) ft(θi; φi; θt; θt) n Index of refraction ≡ Li(θi; φi) cos θid!i 1 σs(z; λ) Scattering cross section [mm− ] 1 Since we have separated the reflected and transmitted light into σa(z; λ) Absorption cross section [mm− ] 1 two components, the BRDF and BTDF also have two components. σt(z; λ) Total cross section (σt = σa + σs) [mm− ] W Albedo (W = σs ) σt fr = fr;s + fr;v d Layer thickness [mm] p(z; θ; φ; θ0; φ0; λ) Scattering phase function ((θ0; φ0) to (θ; φ)) ft = fri + ft;v If we assume a planar surface, then the radiance reflected from Table 1: Nomenclature and transmitted across the plane is given by the classic Fresnel coefficients. 2 Reflection and Transmission due to Layered 12 Surfaces Lr(θr; φr) = R (ni; nt; θi; φi θr; φr)Li(θi; φi) 12 ! Lt(θt; φt) = T (ni; nt; θi; φi θt; φt)Li(θi; φi) As a starting point we will assume that the reflected radiance Lr ! from a surface has two components. One component arises due to where surface reflectance, the other component due to subsurface volume R12(n ; n ; θ ; φ θ ; φ ) = R(n ; n ; cos θ ; cos θ ) scattering. (The notation used in this paper is collected in Table 1 i t i i r r i t i t ! 2 2 and shown diagramatically in Figure 1.) 12 nt nt T (ni; nt; θi; φi θt; φt) = 2 T = 2 (1 R) ! ni ni − Lr(θr; φr) = Lr;s(θr; φr) + Lr;v (θr; φr) where: where R and T are the Fresnel reflection formulae and are de- L - reflected radiance due to surface scattering r;s scribed in the standard texts (e.g.