(2021-10-10)3D-Reconstruction Review(4)

Microfacet Models for Refraction through Rough Surfaces

这篇文章中提到了一个很重要的模型GGX,前面的几篇文章中都用了该模型.

1.introduction

2.Related Work

参见论文.

3.Microfacet Theory

• If restricted to only reflection or transmission,it is often called the BRDF or BTDF, respectively, and our BSDF will be the sum of a BRDF, $f_r$, and a BTDF, $f_t$ ,term. Since we want to include both reflection and transmission as $f_s(i,o,n)$
• In microfacet models, a detailed microsurface is replaced by a simplified macrosurface (see Figure 4) with a modified scattering function (BSDF) that matches the aggregate directional scattering of the microsurface (i.e. both should appear the same from a distance).即microsurface的细节太小了,只有far-field directional scattering pattern是重要的,同时忽略wave effect,只考虑几何光学,only single scattering is modeled
  - 它假设了microsurface可以被两个量描述:distribution function D and a shadowing-masking function G, together with a microsurface BSDF $f_s$

  3.1 Microfacet Distribution Function, D

• D($\bold{m}$), describes the statistical distribution of surface normals $\bold{m}$ over the microsurface.
  $D(\bold{m})dω_m dA$ is the total area of the portion of the corresponding microsurface whose normals lie within that specified solid angle.D需要满足下面的性质
• Microfacet density is positive valued
  $$
  0 \leq D(\mathbf{m}) \leq \infty
  $$
• Total microsurface area is at least as large as the corresponding macrosurface’s area:
  $$
  1 \leq \int D(\mathbf{m}) d \omega_{m}
  $$
• The (signed) projected area of the microsurface is the same as the projected area of the macrosurface for any direction $\mathbf{v}$ :
  $$
  (\mathbf{v} \cdot \mathbf{n})=\int D(\mathbf{m})(\mathbf{v} \cdot \mathbf{m}) d \omega_{m}
  $$
  and in the special case, $\mathbf{v}=\mathbf{n}$ :
  $$
  1=\int D(\mathbf{m})(\mathbf{n} \cdot \mathbf{m}) d \omega_{m}
  $$

  3.2 Shadowing-Masking Function, G

• The bidirectional shadowing-masking function G(i,o,m) describes what fraction of the microsurface with normal m is visible in both directions i and o
  尽管它对于BSDF的形状只有relatively little effect(除了near grazing angles or for very rough surfaces),但是他需要用来保持能量守恒.他需要保持下面的性质
• Shadowing-masking is a fraction between zero and one:
  $$
  0 \leq G(\mathbf{i}, \mathbf{o}, \mathbf{m}) \leq 1
  $$
• It is symmetric in the two visibility directions:
  $$
  G(\mathbf{i}, \mathbf{o}, \mathbf{m})=G(\mathbf{o}, \mathbf{i}, \mathbf{m})
  $$
• The back surface of the microsurface is never visible from directions on the front side of the macrosurface and viceversa (sidedness agreement):
  $$
  \begin{aligned}
  G(\mathbf{i}, \mathbf{o}, \mathbf{m})=0 & & \text { if } &(\mathbf{i} \cdot \mathbf{m})(\mathbf{i} \cdot \mathbf{n}) \leq 0 \
  & & \text { or } &(\mathbf{o} \cdot \mathbf{m})(\mathbf{o} \cdot \mathbf{n}) \leq 0
  \end{aligned}
  $$
• 确切的G函数需要微表面的细节,一般是rarely available的,但是,可以通过各种统计模型以及假设来对其进行简化.

3.3 Macrosurface BSDF Integral