A Coupled Matte-Specular Reflection Model

Transcription

1 A Coupled Matte-Specular Reflection Model Peter Shirley University of Utah Helen Hu y University of Utah ric Lafortune z Cornell University Jon Blocksom x Cornell University Abstract Real surfaces often exhibit both specular (mirror-like) and matte behavior. In many scientific fields this behavior is approximated by the Lambertian-specular model, where a linear combination of Lambertian and ideal specular reflection is assumed. While this approximation is useful, it does not capture the dramatic surface appearance for acute and near-perpendicular viewing angles. The model presented in this paper does capture these angle-dependent behavior changes by allowing the matte and specular coefficients to co-vary. Furthermore, this modified matte-specular model uses a non-lambertian matte term to maintain reciprocity of the bidirectional reflectance distribution function (BRDF), an important point for many rendering engines. Finally, this model maintains the simplicity of the traditional models. An associated web page is shirley/coupled/coupled.html. 1 Introduction The optical behavior of dull, non-glossy surfaces, or matte surfaces, is often approximated using the Lambertian reflection model, which relies on a constant bidirectional reflectance distribution function (BRDF). While Lambertian behavior is known to be physically impossible [1], it is computationally convenient, and in many applications the errors that result from using the model to approximate matte reflection are not visually significant [7]. However, the Lambertian approximation is not appropriate for surfaces which have specular (mirror-like) characteristics in additional to matte, such as polished floors and finished woods. One way these matte and specular materials are traditionally modeled is with a linear combination of ideal specular and Lambertian reflection. Like the Lambertian approximation for matte surfaces, this Lambertian-specular model is computationally convenient and captures much of the qualitative optical behavior of shiny non-metals. However, unlike the Lambertian approximation, the Lambertian-specular model yields unrealistic images for many commonplace environments. In this paper, we present a modification of the Lambertian-specular model which does not fail for these situations and which retains the desirable computational simplicity. In the traditional Lambertian-specular model, when the specular term is much greater than the Lambertian term, the surface behaves like a mirror. When the Lambertian term is larger, the specular highlights become subtle and the surface looks matte. This surface behavior has been captured by many reflection models [8, 5, 12]. What is often overlooked is that in the real world the relative proportions of matte and specular appearance change with viewing angle. This effect can be seen in Figure 1, a photograph of a typical hallway. The mirror-like reflection increases as the viewing angle increases relative to the surface normal vector, e.g. when we view the shirley@cs.utah.edu y hhh@cs.utah.edu z eric@graphics.cornell.edu x jon@graphics.cornell.edu Figure 1: Photographillustrating appearanceof polishedfloor. Specularityincreases and matte behaviorfades toward the end of the hallway. Note the difference in appearance of the reflections of the blue doors. other end of the hallway. The reflection of the blue door at the end of the hall is stronger than the barely visible reflection of the near blue door. The reflection of the far blue door also shows that the matte component at that end must be small, or else the blue of the reflection would become desaturatedby mixing with the achromatic matte color of the floor. This effect, where the matte component and the specular components are inversely related, agrees with energy conservation, which is discussed later. In this paper, we introduce a simple model that captures the angular-dependent relationship of the matte and specular coefficients. Our model uses a physically-based specular coefficient derived from the Fresnel equations, and a heuristic matte component of the BRDF. To our knowledge, it is the first model that produces the matte/specular tradeoff while remaining reciprocal and energy conserving. Because the key feature of the new model is that it couples the matte and specular scaling coefficients, we will hereafter refer to it as the coupled model. The coupled model is intended for use in appearance-based applications where the traditional Lambertian-specular model is currently being used. The coupled model captures additional qualitative aspects of appearance with little extra computation and without complicating existing rendering libraries. While it is not meant to be a physically-based BRDF for high-precision scientific applications, we hope that the coupled model will be a useful empirical approximation for polished surfaces in the way the Lambertian model is useful for approximating matte surfaces. Section 2 of this paper provides an overview of the physics underlying the behavior of surfaces with both specular and matte characteristics. Section 3 describes several traditional matte-specular BRDF models and explains where they fail. Section 4 explains the derivation of the coupled model. This section also presents results of renderings created with the model. Section 5 concludesthe paper. Appendix A contains instructions on how to generate Monte Carlo sampling consistent with the coupled model.

2 R 1 R 2 R 3 < (1 R 1 ) < (1 R 2 ) < (1 R 3 ) (1 R 3 ) (1 R 1 ) (1 R 2 ) Figure 2: The fraction of energy specularly reflected (R) and energy transmitted ((1-R)) changes with incident angle, but at each angle the sum of the specularly reflected and transmitted energyequals the incident energy (). The length of the vectors indicates the amountof energyin each component. Thedirectionof the vectorindicates the direction of propagation. 2 Overview of physics Surfaces which have a glossy appearance are often a clear dielectric, such as polyurethane or oil, with some subsurface structure. The specular (mirror-like) component of the reflection is caused by the smooth dielectric surface and is independent of the structure below this surface. The magnitude of this specular term is governed by the Fresnel equations [1]. The qualitative effects of the Fresnel equations are illustrated in Figure 2, where R 1, R 2, and R 3 are the fractions of incident light specularly reflected, and the size of these fractions is shown by the length of the reflected vector. Note that the specular reflectance increases as the angle of incidence diverges from the surface normal. The light that is not reflected specularly at the surface is transmitted through the surface. There either it is absorbed by the subsurface or it is reflected from a pigment or a subsurface and transmitted back through the surface of the polish. This transmitted light forms the matte component of reflection. Since the matte component can only consist of as much light as is transmitted, it will naturally decrease in total magnitude for increasing angle (Figure 3). Figures 2 and 3 illustrate why the magnitudes of the matte and specular behaviors are coupled; because they partition the incident energy, one must decrease as the other increases. This inverse relationship is the fundamental behavior that the traditional Lambertianspecular model does not capture, as will be detailed in Section 3. Note that the preceding discussion deals only with geometrically smooth glossy surfaces. Reflection from rough surfaces is an important topic that we do not address. Although there are accurate, albeit complicated, models for rough surface reflection [5], there is currently no coupled matte-specular model for smooth surfaces, let alone rough ones. We address only the smooth case. Note also that the traditional Phong model is an modification of the traditional Lambertian-specular model, where the specular term is allowed to have some directional fuzziness. Therefore the phong model will not produce a perfectly clear specular reflection. We accomplish a fuzzy reflection effect in Section 4 by using explicit microgeometry which is smooth at the scale of reflection calculation. 3 Traditional BRDF Models The standard formal way to express reflection models for a given wavelength is with the five-dimensional spectral bidirectional reflectance distribution function (BRDF): ( ), where ( ) is the direction of the incident light, ( ) is the direction of the reflected light, and is the wavelength. Many renderers assume two physical constraints on the form of the BRDF, namely energy conservation and reciprocity. nergy conservation means that Figure 3: Viewingalongthesame incidentangles as infigure2, the fractionofenergy in the matte term is at most what is transmitted and thus varies with incidentangle. The energy of the matte component is spread over many outgoing directions, as indicated by the grey lines. The length of each grey line indicates the relative energy near that direction. for any incident ( ) and wavelength, the fraction of energy reflected, R( ) is at most one: R( ) = Z 2 Z 2 ( ) cos sin d d 1: (1) Reciprocity means that, for all angle pairs, the BRDF does not change when incident and outgoing directions are exchanged: ( )=( ): Lewis calls BRDF models that are faithful to these constraints physically plausible [6]. Many rendering algorithms rely on these properties, so it is important when using these renderers that BRDF models both conserve energy and be reciprocal. As an example we see that the Lambertian BRDF: L( )= Rd() is both reciprocal and energy conserving provided the reflectance R d() is less than one for all wavelengths. The traditional Lambertian-specular model uses two constants to modulate a constant and specular component [8]. In standard radiometric terms, this idea is expressed as the BRDF: ( )= Rd() + R s s( ) (2) where R d() is the hemispherical reflectance of the matte term, R s is the specular reflectance, and s is the normalized specular BRDF (a Dirac delta function on the sphere). quation 2 is a simplified version of the Lambertian-specular BRDF where R s is independent of wavelength. This independence causes a highlight that is the color of the luminaire, so a polished rather than a metal appearance will be achieved [1]. We do this intentionally because the coupled model developed in the next section is intended to simulate only smooth polished surfaces, and not the general case that includes metals. If a metal appearance is desired, then there should be no subsurface term, and the full Fresnel equations can be used directly. Ward suggests that in order to conserve energy, R d()+r s 1 [12]. However, such models with constant R s fail to show the increase in specularity for steep viewing angles (Figure 1). He et al. suggest using the Fresnel equation for the coefficient of the specular term [5], but do not address the subsurface term s angular behavior because this model is intended primarily to simulate surface physics. Shirley attempted to simulate the change in the matte appearance with angle by explicitly dampening R d() as R s increases [9]: Rd()(1 ; ( Rf ()) )= + R f () s( ) (3)

3 where R f () is the Fresnel reflectance for a polish-air interface. The problem with quation 3 is that it is not reciprocal, as can been seen by exchanging and which changes the value of the matte dampening factor becauseof the multiplication by (1;R f ()). The specular term is a scaled Dirac delta function, so it is reciprocal, but this does not make up for the (sometimes infinite) non-reciprocity of the matte term. Because Shirley s BRDF is not physically plausible, it will cause some rendering methods to have ill-defined solutions. 4 The Coupled Model To avoid choosing between physically plausible models and models with good qualitative behavior over a range of incident angles, we note that the Fresnel equations that account for the specular term, R f (), are derived directly from the physics of the dielectric-air interface. Therefore the problem must lie in the matte term. We could try to a full-blown simulation of subsurface scattering as implemented by Hanrahan and Krueger [4], but this technique is both costly and requires a detailed knowledge of subsurface structure, which is usually neither known nor easily measurable. Instead, we can modify the matte term to still be a simple approximation that captures the important qualitative angular behavior shown in Figure 1. Let us assume that the matte term is not Lambertian, but instead is some other function that depends only on, and : d( ). We discard behavior that depends on or in the interest of simplicity. We try to keep the formulas reasonably simple because the physics of the matte term is complicated and sometimes requires unknown parameters. An obvious candidate for the matte component d( ) that will be reciprocal is the separable form k()f ()f ( ) for some function k. This simple formula has been proven to have several computational advantages [3], and it suggests the model: ( )=k()f ()f ( )+R f () s( ) As discussed in Section 2, we know that the matte component can only contain energy not reflected in the surface (specular) component. This and quation 1 suggest the following constraint on d for each incident and : 2k()f ()Z 2 f ( ) cos sin d 1 ; R f (): (4) In the case of a perfectly reflecting matte component, quation 4 becomes an equality, and we can see that f () must be proportional to (1 ; R f ()). If we assume matte components that absorb some energy have the same directional pattern as this ideal, we get a BRDF of the form: ( ) = k()[1 ; R f ()][1 ; R f ( )] + R f () s( ): This is similar to a BRDF model used in the sensor community, although the constants used in that model do not have the normalization properties we desire [2]. We could now insert the full form of the Fresnel equations to get R f () and then use energy conservation to solve for constraints on k. However, since the unpolarized form of the Fresnel equations already introduces an approximation by averaging the reflectances for the two polarization axes, we lose little by using a fairly accurate (within 4% everywhere) approximation of the Fresnel reflectance for air-dielectric interfaces introduced by Schlick: R f () =R +(1; cos ) 5 (1 ; R ) (5) where R is the normal-incidence reflectance for the refractive index of the polish n: (n ; 1)2 R = (6) (n +1) 2 This implies that f () / (1 ; (1 ; cos ) 5 ). Applying quation 4 with an equality gives: 21(1 ; R) k = 2 The full coupled BRDF is then: ( )= 21(1 ; R ) 2 R +(1; cos ) 5 (1 ; R ) s( ): (8) (7) R d()1 ; (1 ; cos ) 5 1 ; (1 ; cos ) 5 + To test this new coupled model we assembled a simple test scene and created an approximate geometric model, where each tile has a displacement map that roughly corresponds to the subtle roughness of the real tiles. We took two photographs of the test scenes from different view angles, resulting in two different values of. We created rendered images for each viewpoint using both the traditional Lambertian-specular model (R s = :2) and the coupled model for the tiles. These images are shown for a mid-range viewing angle in Figure 4, and near grazing angle in Figure 5. The choice of R s =:2 was hand-tuned to produce what we judged to be the best qualitative appearance for the two viewing angles. Of course better appearance for one of the two viewing angles could be achieved at the expense of the other, but our chosen R s attempts to spread the error between the two viewpoints. Note that the behavior of the coupled model qualitatively corresponds with the real behavior. This can be seen in Figure 4, where the specular reflection is almost invisible in both the photo (Figure 4a) and in the image rendered with the coupled model (Figure 4b), but is clearly visible in the image rendered using the traditional model (Figure 4c). In the low-angle photograph (Figure 5a), the specular image is distinct and the matte behavior is less obvious, as the two different color tiles not looking as distinct as they do in Figure 5a. This qualitative behavior can be seen in the image renderered using the coupled model (Figure 5b), but not in the image rendered with the traditional model (Figure 5c). Also note that when using the traditional model, the specular component is often an overestimate, causing the highlights to be too intense sometimes, as is seen in Figures 4c and 5c. To cure this problem one could lower R s, but then images of non-luminaires would be barely visible. Another benefit of the coupled model is that R is easier to choosethan R s, becauser has a precise physical meaning for a particular incident angle. For reasonable values of refractive indices, the R is limited to approximately the range :3 to :6 (the value R = :5 was used for the figures). The value of R s is harder to choose because it must typically be tuned for viewpoint in static images, and tuned for a particular camera sequence for animations. Thus, the coupled model is easier to use in a hands-off mode. We did not attempt to mimic all subtleties of geometry exactly, so the reader should concentrate on the gross appearancefeatures in the rendered and photographic images of the model. These images were produced using a Monte Carlo path tracer. Details on how to generate random sample directions consistent with the coupled BRDF are given in the Appendix. 5 Conclusion We have described a modification of the ubiquitous Lambertianspecular reflection model which has qualitative advantages over the traditional Lambertian-specular form. The coupled model remains an empirical model, and it adds the visually important correspondence between specularity term and viewing angle. Our model applies only to smooth polished surfaces, which are important in many

4 Figure 4: a, b, c - A comparisonof the traditionaland new models to a photographat a mid-rangeviewing angle. a: photographat mid-rangeviewing angle. b: new coupledmodel used to render the tiles. c: traditional Lambertian-specularmodel used to render the tiles. Figure 5: a, b, c - A comparison of the traditional and new models to a photographat a low viewing angle. a: photographat a low viewing angle. b: new coupled model used to render the tiles. c: traditional Lambertian-specularmodel used to render the tiles. applications. It does not address polished surfaces with a microscopically rough coating. In applications where a displacement or bump map cannot be used to simulate surface roughness, and where reciprocity is not important, a rough coating can be simulated by allowing the specular term to become a spread reflection lobe. However, finding a physically-plausible model for rough polished surfaces that has visually appropriate behavior remains an open research problem. Appendix: Monte Carlo Sampling Given an incident angle and a BRDF, we often want to scatter a reflected ray whose direction follows the outgoing energy distribution indicated by the BRDF. In the case of the coupled BRDF presented in this paper, one can first choose probabilistically between a reflection governed by the matte term and the specular term. If the matte term of quation 8 is chosen, then we use a probability density function p that is proportional to the cosine-weighted BRDF for the given incident : p( ) /1 ; (1 ; cos ) 5 cos : Because p does not depend on we can choose uniformly given a canonical (uniform on [,1)) random number : =2. Because the differential measure on the sphere is sin d d, we can write down a one-dimensional probability density function q( ): q( ) /1 ; (1 ; cos ) 5 cos sin : Using standard techniques we can transform a secondcanonical random number into a ([11], pp ), but this requires solving for a root of a high-degree polynomial in cos : = 7 2 cos3 ; 21 4 cos cos5 ; 7 4 cos cos7 (9) The solution to this equation that gives a 2 [ 1) can be approximated by the following rational function: cos = 38295: : :4 +: : :4 2 : +1483:57 +1 (1) quation 1 is accurate for all within 1.2%. If more accuracy is needed, quation 1 provides an excellent initial guess for a Newton iteration to approximate quation 9 to machine precision in only a few iterations. Acknowledgements (To be fleshed out). Ken Torrance, Don Greenberg, Bruce Walter, Steve Marschner, Dan Kartch, Dave Weinstein, Dean Brederson, Lisa Durbeck. References [1] Robert L. Cook and Kennneth. Torrance. A reflectance model for computer graphics. Computer Graphics, 15(3):37 316, August ACM Siggraph 81 Conference Proceedings. [2] Ken llis. Reflectance phenomenology and modeling tutorial Report of the nvironmental Research Institute of Michigan. [3] Alain Fournier. Separating reflection functions for linear radiosity. In Proceedings of the Sixth urographics Workshop on Rendering, pages , June [4] Pat Hanrahan and Wolfgang Krueger. Reflection from layered surfaces due to subsurface scattering. Computer Graphics, pages , August ACM Siggraph 93 Conference Proceedings. [5] Xiao D. He, Kenneth. Torrence, François X. Sillion, and Donald P. Greenberg. A comprehensivephysical model for light reflection. Computer Graphics, 25(4): , July ACM Siggraph 91 Conference Proceedings.

5 [6] Robert Lewis. Making shaders more physically plausible. In Michael F. Cohen, Claude Puech, and Francois Sillion, editors, Fourth urographics Workshop on Rendering, pages urographics, June held in Paris, France, June [7] Gary W. Meyer, Holly. Rushmeyer, Michael F. Cohen, Donald P. Greenberg, and Kenneth. Torrance. An experimental evaluation of computer graphics imagery. ACM Transactions on Graphics, 5(1):3 5, January [8] Bui-Tuong Phong. Illumination for computer generated images. Communications of the ACM, 18(6): , June [9] Peter Shirley. Physically Based Lighting Calculations for Computer Graphics. PhD thesis, University of Illinois at Urbana-Champaign, January [1] Robert Siegel and John R. Howell. Thermal Radiation Heat Transfer. Hemisphere, Washington, third edition, [11] François X. Sillion and ClaudePuech. Radiosity & Global Illumination. Morgan- Kaufmann, San Francisco, [12] Gregory J. Ward. Measuring and modeling anisotropic reflection. Computer Graphics, 26(4): , July ACM Siggraph 92 Conference Proceedings.