This paper presents a new mathematical model of microsurface normal distribution. The proposed model retains a long highlight tail even with roughness coefficient extremely close to zero which matches the behavior of measured material samples. The proposed model obeys the energy conservation law and can be used in Physically-Based Rendering (PBR) shaders to create realistic and visually pleasing specular highlights. The results of using the proposed model are demonstrated and discussed. The highlight shape created by this model is compared against several existing models as well as empirical data.
Nowadays there are several models of microsurface normal
distribution available for specular Bidirectional Reflectance
Distribution Function (BRDF) ranging from minimalistic
ones like Blinn-Phong to complex models like
CookTorrance [
The research presented in this paper is aimed at creating a new model of normal distribution for specular BRDF that would retain realistically long highlight tail with low roughness values.
There is a relatively large number of research papers devoted to solving similar tasks.
Thus, Dupuy et al. proposed in [
Gartley et al. presented in [
In [
Heitz in [
Hanika et al. in [
Walteret et al. in [
In [
Despite of availability of these and other promising results a need of new models development still exists.
The proposed model is based on an assumption that the surface contains cavities of different depths with deeper cavities being less probable. In order to calculate the brightness of a given surface the following steps are performed: • Calculation of the depth of a cavity that would reflect given light direction vector to the direction of given view vector; • Calculation of the probability of required cavity.
The reflection of light vector which hits a cavity is demonstrated in Fig. 1.
The height of a cavity which reflects a given light vector into view vector is calculated using following the formula: (1) where h is the height of the cavity and b is the Blinn term.
The plot of cavity height function in relation to Blinn term is presented in Fig. 2.
After calculation of the cavity height, the probability distribution must be applied. Because Gaussian Distribution Function does not provide the desired result, the proposed model uses a custom distribution function:
The integral of this function over ϕ ∈ (0; π) does converge and is equal to: (2) where f is the distribution coefficient, h is the height of the cavity, and r is the surface roughness coefficient. This function is not normalized (does not integrate to 1) and, thus, should be divided by its integral to be used as a probability distribution. a b However, the integral of this function over h ∈ (0; ∞) does not converge. But instead it is possible to integrate the total amount of reflected light over ϕ ∈ (0; π) where ϕ is the angle between the cavity normal and the surface normal, thus, preserving the energy conservation law.
The dividing of the proposed distribution function by this integral yields the normalized NDF ready to be used for modelling specular reflections: where P (ϕ, r) is the probability of the required cavity, r is the surface roughness coefficient, and cos(ϕ) is the Blinn term.
In order to maintain consistency, it is proposed to use the described NDF with a custom geometric masking function based on the same microsurface model.
The proposed geometric masking function describes the decrease of visible highlight due to microsurface cavity being obscured from light source and from the viewer by adjacent microsurface apex as demonstrated in Fig. 3. (4) (5) The masking term is derived by dividing the length of visible (for view masking) or lit (for light masking) part of microsurface cavity slope by its total length (Fig. 4). where f is the distribution coefficient, r is surface roughness, and ϕ is the angle between the surface normal and the cavity normal as well as cos(ϕ) is the Blinn term. (3) The total microsurface slope length can be described as: where ltotal is the total microsurface slope length and ϕ is the angle between the macrosurface normal and the light vector.
The visible part of the slope is equal to: where lvisible is the visible part of the microsurface slope, ltotal is the total microsurface slope length, ϕ is the angle between the macrosurface normal and the light vector, and β is the angle between the microsurface slope and the macrosurface.
Thus, the masking coefficient is equal to: where m is the masking coefficient, ϕ is the angle between the macrosurface normal and the light vector, and β is the angle between the microsurface slope and the macrosurface.
The masking function can be also presented as follows: Thus, the final masking coefficient formula is: (6) (7) (8) (9) (10) where m is the masking coefficient, ϕ is the angle between the macrosurface normal and the light vector, and β is the angle between the microsurface slope and the macrosurface.
The total masking coefficient should be calculated as the lowest value between light masking and view masking: (11) where m is the total masking coefficient, γ is the angle between the macrosurface normal and the view vector, ϕ is the angle between the macrosurface normal and the light vector, and β is the angle between the microsurface slope and the macrosurface.
The proposed model creates a long highlight tail which looks more similar to empiric measurements as demonstrated in Fig. 5 and Fig. 6.
In fact, the highlight tail of the proposed model is brighter than empirical data and, thus, may require further adjustment to achieve better realism in renders.
Using proposed model for rendering specular highlights creates a smoother and more realistic result as shown in
Fig. 7.
a b c d Fig. 5. Demonstration of highlights: a – measured data from chrome sample, b – proposed model, c – Trowbridge-Reitz model, d –
Beckmann model [
The proposed model retains a long highlight tail even with roughness coefficient extremely close to zero yielding more appealing renders. However, it is also computationally intensive. Besides, it must be noted that the highlight tail of the proposed model in fact seems brighter than measured data and, thus, it may need to be adjusted in order to achieve better realism.