Reflectance from Surfaces with Layers of Variable Roughness Hossein Ragheb Department of Computer Engineering Bu-Ali Sina University, Hamedan, Iran.
[email protected]
Abstract A new model for the scattering of light from layered surfaces with boundaries of variable roughness is introduced. The model contains a surface scattering component together with a subsurface scattering component. The former component corresponds to the roughness on the upper surface boundary and is modeled using the modified Beckmann model. The latter component accounts for both refraction due to Fresnel transmission through the layer and rough scattering at the lower layer boundary. By allowing independent roughness parameters for each surface boundary we can achieve excellent fits of the model to the measured BRDF data. We experiment with BRDF data from skin surface samples (human volunteers) and show that the new model outperforms alternative variants of the Beckmann model and the Lafortune et al. reflectance model.
1. Introduction Several reflectance models have been developed in the literature with the aim of accounting for both surface and subsurface scattering. In computer graphics, a layered surface structure was assumed by Hanrahan and Krueger [3] to develop a model for the subsurface scattering component. Linear transport theory was used, and no attempt was made to model surface scattering due to roughness. Their model was shown to give good results for biological tissues such as skin and plant leaves. Jensen et al. [5] have improved the model and compared it to the Lafortune et al. empirical reflectance model [8]. Matusik et al. [9] make empirical estimates of the BRDF (bidirectional reflectance distribution function) for several types of dielectrics and metals. Turning to the models used in computer vision, the Wolff model [11] can be interpreted as a Fresnel corrected variant of Lambert’s law. Here the outgoing Lambertian radiance is multiplied by two Fresnel transmittance terms. Similar Fresnel terms are used in the complex reflectance model of He et al. [4] which attempts to account for a number of effects including subsurface scattering. Further publications about the reflectance properties of specific materials (velvet, skin, fabrics, etc.) exist in optical physics literature [6, 7]. Our aim in this paper is to develop a model for scattering
Edwin R. Hancock Department of Computer Science University of York, York, YO10 5DD, UK.
[email protected]
of light from layered rough surfaces. Our study is limited to the case where the contribution from particles within the translucent surface layer in the scattered radiance is small. The main effects considered are rough scattering effects from the upper and lower layer boundaries. Although these assumptions may appear simplistic, they do lead to a model which gives remarkable agreement with experiment. Our future plan is to extend the model to incorporate more realistic assumptions concerning the surface structure. Hence, the model contains a single surface scattering component and a single subsurface scattering component. The former component models the effects of roughness on the upper surface boundary. Single surface scattering is well modeled by the Vernold-Harvey modification to the Beckmann model [10, 1]. Here we assume that the rough surface scattering is governed by the modified Beckmann model. The latter component accounts for rough surface scattering from the lower surface-layer boundary. Here our model accounts for the effects of radiance transmission through the surface layer and rough scattering effects at the lower layerboundary. The transmission effects are governed by the Fresnel theory and Snell’s law, while rough scattering is accounted for using the modified Beckmann model. Depending on the type of surface, the relative contributions from the two components can vary dramatically for different materials. This behavior is conveniently captured by allowing a balance parameter that governs the relative contributions of the two components to the total diffuse scattered radiance. We also allow for two independent roughness slope parameters. These account for the roughness of the upper and the lower surface boundaries. Hence we not only obtain a flexible model, but we also achieve excellent agreement between the model predictions and the measured data.
2. Surface and subsurface scattering model Here we assume that there are two major components for diffusely scattered light from a layered rough surface. The first component Lsf o arises from surface scattering which we model using the modified Beckmann model. The second component Lsb o arises from subsurface scattering. The outgoing radiance is a combination of these two components.
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When the parameter η is used to control the relative balance of the model components, the model is given by Lo (θi , θs , φs , σ/T, σ /T , n, η) = sb ηLo (θi , θs , φs , σ /T , n) + (1 − η)Lsf o (θi , θs , φs , σ/T ) (1) The physical meaning of this expression is that the incident light is scattered either by the surface or through subsurface interactions. Depending on the surface material, the relative importance of the two components may vary dramatically. As we use normalized values for the scattered rasb diance components, i.e. Lsf o and Lo in this paper, we make use of the parameter η to obtain a measure of their relative contribution to the outgoing radiance Lo . Other model parameters are described below.
2.1. Single surface scattering component
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To account for rough surface scattering from the upper and lower boundaries of the layer, we make use of the Beckmann-Kirchhoff theory [1]. It has been shown that the modified Beckmann model proposed by Vernold-Harvey [10] provides improved predictions of scattered radiance for a wide range of incidence and scattering angles. When the surface correlation function is Gaussian, according to the modified Beckmann model, the diffuse scattered radiance is LVo H−G (θi , θs , φs , σ/T ) = KG [cos(θi )/vz2 (θi , θs )] × 2 exp{−T 2 vxy (θi , θs , φs )/[4σ 2 vz2 (θi , θs )]}
of real-world surfaces that satisfy these conditions such as human skin, plant tissue, fabrics, layered stones and tiles. To derive the expression for Lsb o , we use the scattering geometry outlined in Fig. 1. While some authors [5, 3] have used a phase function to account for the distance between the two points corresponding to the two components sb Lsf o and Lo (where light beams leave the upper surface boundary), we assume that both points are in the same approximately flat area which is the tangent plane to the rough surface. Our experiments show how the contribution of the subsurface scattering component can be important even for very-rough surfaces depending on the medium material.
(2)
If the correlation function is exponential rather than Gaussian, then the modified Beckmann model is given by LVo H−E (θi , θs , φs , σ/T ) = KE [cos(θi )/vz2 (θi , θs )] × 2 {1 + [T 2 vxy (θi , θs , φs )/σ 2 vz2 (θi , θs )]}−3/2 (3) In Eqs. (2-3) vx (θi , θs , φs ) = k(sin θi − sin θs cos φs ), vy (θs , φs ) = −k(sin θs sin φs ), vz (θi , θs ) = −k(cos θi + 2 cos θs ), vxy (θi , θs , φs ) = vx2 (θi , θs , φs ) + vy2 (θs , φs ) and k = 2π/λ where λ is the wavelength. The distribution of surface height is assumed to be Gaussian for both cases. The ratio σ/T is the surface slope parameter which controls the model behavior for different roughness conditions. For these variants of the model it is sufficient to estimate σ/T rather than the surface roughness σ and the correlation length T separately. Note that according to the model geometry φi = π on the local tangent plane. The coefficients KG and KE are both proportional to (T /σ)2 and may easily be used to normalize the corresponding expressions.
2.2. Single subsurface scattering component In this section we develop our model for the subsurface scattering component for layered rough surfaces. The assumptions are as follows. The mean surface level of the lower boundary is assumed to be almost parallel to that of the upper boundary. We assume that there is no scattering within the layer itself. Multiple scattering effects from deeper layers are also ignored. There are good examples
θs
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Figure 1. Scattering geometry for the layered rough surface under study: Li is the radiance of the incident light, Lsf o is the outgoing radiance due to the upper boundary and Lsb o is that due to the lower boundary (see text).
In general the normal vector used in the subsurface scattering theory is the local surface normal, while in the Beckmann model it is the normal to the mean surface level. Here we assume that the normal vector used in the analysis of subsurface scattering is identical to that used in the Beckmann model, i.e. the normal to the mean surface level. Hence θi , θs and φs are identical in both geometries (the scattering azimuth angle φs is not shown in Fig. 1). Model derivation details: From Snell’s law, it follows that solid angles are compressed by the factor cos(θi )/n2 cos(θi ), and hence we can write cos(θi )dω = (1/n2 )[cos(θi )dω] (4) To model the subsurface scattering component(s) one way would be to return to the Kirchhoff integral and find the field transmitted into the dielectric. However, Hanrahan [3] developed a model for single scattering from layered surfaces using the linear transport theory. Our approach, on the other hand, is similar to that followed by Wolff [11] who derived an expression for diffuse reflectance from smooth surfaces. Wolff [11] used the Chandrasekhar diffuse reflection law to account for the diffuse scattering of transmitted light inside the dielectric. Hence, the outgoing diffuse radiance predicted by the Wolff model is given by olf f LW (θi , θs , n) = Li cos(θi )[1 − f (θi , n)] × o (5) {1 − f (sin−1 [(sin θs )/n], 1/n)}dω where n is the refractive index of the dielectric, the scaling factor is very nearly constant, and f (θi , n) is the Fres-
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nel reflection function for unpolarized light as appears in [11]. If light with radiance Li is incident upon a dielectric through a solid angle dω at an angle θi relative to the surface normal, then the incident energy flux per unit horizontal area is Li cos(θi )dω. The transmitted energy flux per unit horizontal area, measured in the dielectric will be Li [1 − f (θi , n)] cos(θi )dω. Hence, using Eq. (4), the transmitted radiance under mean surface level is found to be ˆ i (θi , n) = n2 Li [1 − f (θi , n)] L (6) VH Since we assume the modified Beckmann model Lo for the scattering inside the dielectric, the reflected radiance inside the dielectric, which results from incident light through infinitesimal solid angles dω, dω , will be 2 ˆ sb L o (θ , θs , φs , σ /T , n) = n [1 − f (θi , n)] × i
[LVo H (θi , θs , φs , σ /T )/ cos(θi )] cos(θi )dω
(7)
Here σ /T is the slope parameter due to the roughness on the lower surface boundary. It is independent from σ/T due to the upper surface boundary. Using Eq. (4), we have ˆ sb (θ , θ , φs , σ /T , n) = [1 − f (θi , n)] × L o
i
s
(8) LVo H (θi , θs , φs , σ /T )[cos(θi )/ cos(θi )]dω sb ˆ scattered at angle θ from inGiven radiance L o s side the dielectric, the scattered energy flux through the subsurface solid angle dω per horizontal area is [1 − ˆ sb cos(θ )dω . Hence the transmitted radiance f (θs , 1/n)]L o s into air (at scattered angle θs ) will be Lsb o (θi , θs , φs , σ /T , n) = [1 − f (θs , 1/n)] × ˆ sb L o (θi , θs , φs , σ /T , n)[cos(θs )dω / cos(θs )dω] ˆ sb = (1/n2 )[1 − f (θs , 1/n)]L o (θi , θs , φs , σ /T , n)
(9)
Substituting Eq. (8) into Eq. (9), we find that the scattered radiance by the subsurface scattering component is VH Lsb o (θi , θs , φs , σ /T , n) = Lo (θi , θs , φs , σ /T ) × (10) [1 − f (θi , n)][1 − f (θs , 1/n)]dω An identical correlation function is considered both on the upper and on the lower surface boundaries. We use the corresponding variants of the modified Beckmann model. Since Li and dω are constants during our experiments with each surface sample, the incident radiance Li is absorbed into the coefficients KG and KE of LVo H−G and LVo H−E .
3. Experiments We evaluate our model on publicly available data-sets of BRDF measurements for skin samples of human volunteers. In our experiments, we exclude BRDF measurements which occur in the direction of the specular spike. This allows us to study the diffusely scattered radiance more conveniently. To compare theory with experiment, we convert the tabulated BRDF measurements ν(θi , φi , θs , φs ) into normalized outgoing radiance LD o (θi , φi , θs , φs ) estimates. This is a straightforward task since the incidence angle θi and the brightness of the light-source Li (which is constant for
each surface sample) are tabulated. The relationship between the BRDF and the normalized outgoing radiance is Lo (θi , φi , θs , φs ) = ν(θi , φi , θs , φs )Li cos(θi )dω. We fit the models under study to the radiance data by varying the model parameters (i.e. σ/T , σ /T and η) to minimize the root-mean-squared error ∆RM S . The search for the best-fit parameters is simple and involves testing the values of σ/T , σ /T and η over a grid. There are 100 equally spaced values of η in the interval [0, 1], and 50 equally spaced values of σ/T or σ /T in the interval [0.1, 2.1] for the model variants with a Gaussian correlation function and in the interval [0.1, 4.1] for those with an exponential correlation function. K 100 D k k 2 1/2 (11) { [Lo (θi , ...) − LM ∆RM S = o (θi , ...)] } K k=1
Here LM o is the normalized radiance value predicted by the model, LD o that obtained from the BRDF data, and k runs over the index number of the BRDF measurements used (K). We do not estimate the index of refraction and use n = 1.37 for the human skin. Note that for dielectrics n has a very weak dependance on the wavelength of light. For the sample studied we show scatter plots of the predicted normalized radiance as a function of the measured normalized radiance for the models investigated (Fig. 2). The better the data is clustered around the diagonal straight line, the better the agreement between experiment and theory. The acronyms used to refer to the models studied are: single surface scattering models (VH-G and VH-E), single surface and subsurface scattering models (RL-G and RL-E) for which the surface scattering component is identical to the corresponding variant of the modified Beckmann model (LVo H−G or LVo H−E ) given by Eqs. (2) and (3), while the subsurface scattering component (Lsb o ) is given by Eq. (10). Cornell skin sample (Mahesh): The Cornell PCG group [2] has provided some BRDF data sets for human skin acquired using their image-based measurement technique, and using a digital camera. The data sets contain red, green, and blue channels collected with different imaging geometry. This makes them difficult to combine. Here we only experiment with the data for the red channel which seems to be the principal spectral band for human skin. We use the measurements corresponding to the forehead of a 23-year-old south Asian (Indian) male (Mahesh). As an empirical reflectance model, we choose the Lafortune et al. (LFTG) model [8] which is an improved version of the Phong model and is claimed to give accurate predictions of reflectance from human skin. To fit the LFTG model [8] to the data we use the model coefficients given in the Cornell PCG web pages which have been used to render the face corresponding to the Mahesh skin sample. The average estimates of the surface slope parameter for the 4 wavelengths (440, 490, 540, 590 nm) studied are: σ/T = 0.1450 and σ /T = 1.4975 for the RL-E, while σ/T = 0.1250 and σ /T = 0.7875 for the RL-G model variants. It is clear
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Figure 2. Mahesh skin sample: normalized radiance predicted by a model against those from data measurements; the LFTG model (left) and the RL-E model (right).
Sample plots in Figs. 2.a and 2.b contain K = 14855 measurements at the wavelength λ = 490nm. There are three human skin samples provided by the Cornell PCG (Mahesh, Steve and Grace). We have tested all of them and obtained reasonable fits of the new model to the data. We have chosen the Mahesh skin sample as it is the darkest and hence gives the poorest results among these three samples. Nonetheless, the results shown here show impressive performance of the new model. Fig 2.a is for the LFTG model, which does not provide a good fit. Fig 2.b is for the LRE model, which gives the best fit of the models tested (including VH-G, VH-E, LR-G and LR-E). Note that neither the Lambertian model nor the Wolff model can give comparable predictions of radiance for this surface, but are not shown here because of space limitations. In Fig. 3 we study the behavior of the models as the wavelength is varied. From these figure it is clear that the lowest values of the RMS fit error are obtained at short wavelengths. In other words, the contribution from the subsurface scattering to the outgoing radiance decreases as the wavelength increases, and hence the balance parameter decreases (from η = 0.5 to η = 0.05). The reason for this is that for long wavelengths the assumption underpinning the Kirchhoff theory may be violated for the moderatelyrough surface of human skin. Additionally, the amount of light penetrating the surface decreases with wavelength. In Fig. 3.a the LFTG, VH-G and RL-G models are compared. In Fig. 3.b the LFTG model is compared with the VH-E and RL-E models. Note that in Fig. 3.b the RL-E model gives the lowest RMS fit error for all wavelengths studied, while in Fig 3.a the RL-G model only gives the lowest RMS fit error for wavelengths less than 550nm. In both plots the new model outperforms the modified Beckmann model for most wavelengths. However for long wavelengths the new model becomes equivalent to the modified Beckmann model, i.e. to its surface scattering component, since the
We have developed a reflectance model that can be applied to layered rough surfaces. The model accounts for rough scattering at both boundaries of the layer, and for refractive attenuation within it. The effects of rough scattering are captured using the modified Beckmann model, and refractive attenuation by the Fresnel theory. We consider two independent roughness parameters due to the upper and lower surface boundaries. From our experiments with BRDF data from human skin samples, the new model variant which gives the best overall performance is that which uses an exponential correlation function (RL-E). Since the fit of the model to the skin data is greatly improved compared to several reflectance models studied, the model also appears to have potential in the computer graphics domain. 0.2
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subsurface scattering component diminishes dramatically.
4. Conclusions
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from these values that the roughness slope parameter for the upper surface boundary tends to be smaller than that for the lower boundary. One explanation for this may be due to the fact that the upper (outer) surface boundary is subject to abrasion, and this has a smoothing effect.
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Figure 3. Mahesh skin sample: the RMS fit error ∆RM S against wavelength for the models with the Gaussian (left) and the exponential (right) correlation functions.
References [1] P. Beckmann and A. Spizzichino, The Scatter. of Electromag. Waves from Rough Surfaces, Pergamon, NY, 1963. [2] Reflectance Data, Cornell University Program of Computer Graphics, www.graphics.cornell.edu/online/measurements. [3] P. Hanrahan and W. Krueger, “Reflectance from Layered Surfaces due to Subsurface Scattering,” Computer Graphics, SIGGRAPH93 Proceedings, 1993, pp. 165-174. [4] X.D. He, K.E. Torrance, F.X. Sillion and D.P. Greenberg, “A Comprehensive Physical Model for Light Reflection,” Computer Graphics, vol. 25, 1991, pp. 175-186. [5] H. W. Jensen, S. Marschner, M. Levoy and P. Hanrahan, “A Practical Model for Subsurface Light Transport,” Proceedings of SIGGRAPH, 2001, pp. 511-518. [6] R. Lu, J. Koenderink, and A. Kappers, “Optical properties of velvet,” Applied Optics, 37(25), 1998, pp. 5974-5984. [7] R. Lu, J. Koenderink, and A. Kappers, “Optical properties of shot fabric,” Applied Optics, 39, 2000, pp. 5785-5795. [8] E.P.F. Lafortune, S. Foo, K.E. Torrance and D.P. Greenberg, “Non-linear Approximation of Reflectance Functions,” Proceedings of SIGGRAPH, 1997, pp. 117-126. [9] W. Matusik, H. Pfister, M. Brand and L. McMillan, “A DataDriven Reflectance Model,” ACM Transactions on Graphics, 22(3), 2003, pp. 759-769. [10] C.L. Vernold, and J.E. Harvey, “A Modified BeckmannKirchoff Scattering Theory for Non-paraxial Angles,”Proc. SPIE, vol. 3426, 1998, pp. 51-56. [11] LB. Wolff “Diffuse Reflectance Model for Smooth Dielectric Surfaces,” J. Op. Soc. Am. A, 11(11):2956-2968, 1994.
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