Overview

In this section we are going to give an overview of the basic theory. At first we want to point out, that in the scope of the seminar we focused on the preprocessing part of the model, which includes representing of layered materials as scattering tensors. For the final rendering we used PBRT V3 framework [3].

Because every layer represents either a medium or a boundary and we want to be able to store the result, we need to discretize the phase and BSD functions. In the following part we are only going to look at BSDFs, because phase functions can be handled analogous. A BSDF is a function of 4 parameters , which models the illumination behavior of a material given in and outgoing direction. $$ \mathbb{R}^4 \to \mathbb{R} : f(\mu_i, \phi_i, \mu_o, \phi_o) $$ The directions are encoded as the cosine of the elevation angle \( \mu \) and the azimuth angle \( \phi \). Using isotropy assumptions, we are able to reduce the function to a function of 3 parameters. $$ \mathbb{R}^3 \to \mathbb{R} : f(\mu_i, \mu_o, \phi_d) $$ where \( \phi_d = |\phi_i - \phi_o| \). In order to further reduce the dimension of the domain, we sample \( \mu \) from \( [-1, 1] \). Therefore we use the GaussLobatto points. Now the BSDF only depends on one parameter. $$ \mathbb{R} \to \mathbb{R} : f_{\mu_i, \mu_o}(\phi_d) $$ The last step to remove the remaining parameter by projecting the BSDF function into fourier space, by representing it with even fourier basis approximating it with k fourier orders. $$ f_{\mu_i, \mu_o}(\phi_d) = \sum\limits_{l = 0}^{k} c_{\mu_i, \mu_o}(l) \cos(l \phi_d) $$ where c(l) denotes the l-th Fourier coefficient of f as a function of azimuth for fixed \(\mu_i\) and \(\mu_o\).

The main challange is to find a fourier representation of a resonable BSDF model. In the paper they choose the microfacet model by Walter et al. [2]. This model consists of a reflection and a transmission term. $$ f(\mu_i, \phi_i, \mu_o, \phi_o) = f_r(\mu_i, \mu_o, \phi_i - \phi_o) + f_t(\mu_i, \mu_o, \phi_i - \phi_o) $$ The reflection term \(f_r\) is defined as $$ f_r(\mu_i, \mu_o, \phi_i - \phi_o) = \frac{F(\mu_h)D(\mu_h)G(\mu_i, \mu_o)}{4|\mu_i\mu_o|} $$ where F specifies the Fresnel reflectance, D is the Beckmann distribution, G is a shadowing-masking term, and \(\mu_h\) denotes the cosine of the angle between the normal and the half-direction vector of the incident and outgoing directions. However, they were not able to find an analytic transformation for this model due to the significant complexity, for that reason they are forced to turn to numerical integration.

The final discretized function is represented as a tensor, for every fourier level we have a quadratic scattering matrix, which contains the coefficents of this level. The size depends on the amount of samples in \(\mu\). For a level l we get the following scattering matrix: $$ \left[ \begin{array}{c|c} T_l^{tb} & R_l^b \\ \hline R_l^t & T_l^{bt} \end{array} \right] = \pi (1 + \delta_{0l}) \left[ \begin{array}{c|c} F_l^{tb} & F_l^b \\ \hline F_l^t & F_l^{bt} \end{array} \right] W \overline{M} $$ where $$ F_l := (c_{\mu_i, \mu_o}(l))_{ij} \in \mathbb{R}^{n \times n} $$ $$ W := (\delta_{ij}w_i)_{ij} \in \mathbb{R}^{n \times n} $$ $$ \overline{M} := (\delta_{ij}|\mu_i|)_{ij} \in \mathbb{R}^{n \times n} $$ M are the samples from the GaussLobatto method and the W contains the corresponding weights. The result is a block matrix where R represents the reflective part and the T the transmissive of the current BSDF on the given fourier level.

The scattering is now a discrete approximation of the BSDF reflection and transmission behavior. Since the goal was to rrepresent layered materials, we are going to stack two materials by combining the tensor of both materials. For this step they use the adding equation: $$ \overline{R}^t = R^t_1 + T^{bt}_1(I - R_2^t R_1^b)^{-1} R_2^t T_1^{tb} $$ $$ \overline{R}^b = R^b_2 + T^{tb}_2(I - R_1^b R_2^t)^{-1} R_1^b T_2^{bt} $$ $$ \overline{T}^{tb} = T_2^{tb}(I - R_1^b R_2^t)^{-1} T_1^{tb} $$ $$ \overline{T}^{bt} = T_1^{bt}(I - R_2^t R_1^b)^{-1} T_2^{bt} $$ This results in new scattering matrix for each level with represents the combined material.

Implementation

For our implementation we decided to use the script language python 3.4, together with the packages numpy, scipy and matplot lib for plotting the matices. For the data exchange between our implementation and the render we used the format proposed by the paper, which is also supported by PBRT V3. The following code snippet shows how to create layers in our implementation and how to combine them to a new layered material.

            
albedo = 0.5
eta_top = 1.5      
alpha  = 0.02  # Beckmann roughness
eta_bot = get_rgb(gold)

alpha_top = 0.1  # Beckmann roughness of top layer (coating)
alpha_bot = 0.1  # Beckmann roughness of bottom layer (gold)

n = 196                
m = 267        
mu, w = gaussLobatto(n)

# 1. Creating coating layer
coating = layer(mu, w, m)
coating.setMicrofacet(eta_top, alpha_top)

output = []
for channel in range(3):
    # Construct gold bottom layer for each channel
    # 2. Creating metal layer
    l = layer(mu, w, m)
    l.setMicrofacet(eta_bot[channel], alpha_bot)

    # Apply coating
    # 3. Applying coating..
    output.append(layer.addToTop(coating, l))          
            
            

Results

In this section we are going to describe our results. The start with the comparsion of the BSDF/ phase functions to their fourier approximation. Afterwards be compare for each level our scattering matrix with the scattering obtained by using the reference implementation. Finally we visually compare our results of rendering layered materials with reference images from the paper.

Fourier approximation

Scattering matrices

Adding equation

Renderings

References

• [wenzel2014,1]

  Wenzel Jakob, Eugene D'Eon, Otto Jakob, Steve Marschner "A Comprehensive Framework for Rendering Layered Materials"
  ACM Transactions on Graphics (ACM SIGGRAPH 2014)
  Online: http://www.cs.cornell.edu/projects/layered-sg14/
  Last accessed: 18. January 2016

  
  
  
• [walteretal,2]

  Bruce Walter, Stephen R. Marschner, Hongsong Li, and Kenneth E. Torrance. "Microfacet Models for Refraction through Rough Surfaces"
  In Proceedings of EGSR 2007
  Online: https://www.cs.cornell.edu/~srm/publications/EGSR07-btdf.html
  Last accessed: 18. January 2016

  
  
  
• [pbrt,3]

  Matt Pharr, Greg Humphreys "Physically Based Rendering, Second Edition: From Theory To Implementation"