STD: Student's t-Distribution of Slopes for Microfacet Based BSDFs

This paper focuses on microfacet reflectance models, and more precisely on the definition of a new and more general distribution function, which includes both Beckmann’s and GGX distributions widely used in the computer graphics community. Therefore, our model makes use of an additional parameter γ, which controls the distribution function slope and tail height. It actually corresponds to a bivariate Student’s t-distribution in slopes space and it is presented with the associated analytical formulation of the geometric attenuation factor derived from Smith representation. We also provide the analytical derivations for importance sampling isotropic and anisotropic materials. As shown in the results, this new representation offers a finer control of a wide range of materials, while extending the capabilities of fitting parameters with captured data. This document illustrates the fitting obtained for all the material data provided in the MERL database [3]. It is performed on the Cook-Torrance model [1]: fCT (i, o, n) = ρ π + ρs F (i, h) D(h) G(i, o, h) 4|ih||oh| , (1) where h = i+o ‖i+o‖ is the half angle vector between i and o, F (i, h) corresponds to Fresnel’s reflectance (with the refractive index denoted as ni), ρ is the diffuse reflectance, ρs is the specular albedo, and G is the Smith’s GAF. The distribution D is set to Beckmann, GGX or STD. We also provide fitting results with the model of Löw et al. [2]: fLow(i, o, n) = ρ π + F (i, h) S( √ 1− |hn|) G(i, o, h) |ih||oh| (2) where G is the Torrance-Sparrow GAF [4] and S is the ABC distribution adapted by Löw et al. in order to be included in a microfacet BRDF model S(f) = A (1 +Bf2)C . (3) 1 Parameter A allows to control the height of the specular lobe (in the same manner of ρs in Cook-Torrance model), while B and C control the shape. The remainder of this document contains one page for each material, with: 1. The curves corresponding to 3 incidence directions, for each RGB channel, and CookTorrance BRDF model with STD (first row), GGX (second row), Beckmann (third row), and Löw’s BRDF model (fourth row); 2. The parameter values obtained by the fitting process for each model; 3. Rendered images with each model and the corresponding color-coded image relative differences. Details on the genetic algorithm The fitting is done with a genetic algorithm performing a minimization of the normalized absolute difference ε (called normalized distance in the remainder) between raw BRDF data from MERL database and the model to fit: ε = ∑N k |MERLR − fR|+ ∑N k |MERLG − fG|+ ∑N k |MERLB − fB| 3 N , (4) where N is the number of sampled directions i and o used to evaluate the error andMERLc and fc are the BRDF values for the MERL data and the fitted model respectively. Figure 1 illustrates this error obtained for each of the four models (Cook-Torrance with Beckmann, GGX or STD distribution and Löw’s model). In practice, the number of samples (couples of incident and outgoing directions) is set to N = 49160 . Incident directions are chosen with their azimuthal angle φ = 0 and outgoing directions cover all the hemisphere with an equal solid angle distribution. Figure 2 provides the RMSE between raw BRDF data from MERL database and the model with the obtained fitted parameters: RMSE = √∑n k (MERLR−fR) n + √∑n k (MERLG−fG) n + √∑n k (MERLB−fB) n 3 (5) Each channel (red, green and blue) is fitted separately, starting from the red channel where ni, σ and γ. In the case of Löw’s model, B and C are also fitted. For the green and blue channels, only the specular albedo is fitted. In the case of Cook-Torrance based models, the red component of the specular albedo is fixed to 1, which is clearly a drawback compared to the Löw’s model. The diffuse component is fitted separately: It corresponds to the mean of the BRDF values for a normal light incident direction (explaining why all models have the same Lambertian component ρ). Our genetic procedure is summarized in 2 Algorithm 1 (nbLoopMax is set to 100 in our implementation). For each channel, a new population is generated (5000 individuals in our implementation). For the red channel, the model parameters are generated uniformly randomly. For the two other channels (green and blue), parameters ni, σ, γ,B, and C are fixed with the best individual parameters obtained in the previous step. The error ε is computed for one channel at a time. The final set of parameters is chosen as the individual with the minimum error for all 3 channels. The selection function performs a guided selection depending on the error of each individual. The individuals are randomly removed from the population, according to their distance to the best one. They are replaced during the crossover process given by Algorithm 2. Finally, the mutation process applies to an individual with a chance of 50%, except for the best individual, which is never changed because it corresponds to the potentially final set of parameters. If an individual is selected for mutation, only one parameter can be changed. This parameter is uniformly randomly chosen in the set of parameters. It has a chance of 80% to be randomly regenerated, and 20% to be slightly mutated.