[GAMES202 笔记] Lecture 10 – 融雪时分的博客

[GAMES 202] Lecture Note 10

Overview

Microfacet models (its usage in RTR (without multi-bounce) is not phisically accurate) and Disney Pricipled BRDFs (the model itself is not phisically accurate) are among the mostly used PBR surface materials in RTR.

For volumes: fast and approximate single/multiple scattering in volumes.

Microfacet BRDF

f(i,o) = \frac{\mathbf{D}(\theta_h)\mathbf{F}(\theta_d)\mathbf{G}(\theta_i, \theta_o)}{4 \,\cos\theta_i\cos\theta_o}

Fresnel Term $\mathbf{F}(\theta_d)$

Given the index of refraction and the angle which the incident ray makes with the surface normal, the Fresnel equations specify the material’s corresponding reflectance (i.e. the amount of light reflected from a surface) for two different polarization states of the incident illumination. (PBRT)

Physically Accurate: The Fresnel Equation

R_{\mathrm{s}}=\left|\frac{n_{1} \cos \theta_{\mathrm{i}}-n_{2} \cos \theta_{\mathrm{t}}}{n_{1} \cos \theta_{\mathrm{i}}+n_{2} \cos \theta_{\mathrm{t}}}\right|^{2}=\left|\frac{n_{1} \cos \theta_{\mathrm{i}}-n_{2} \sqrt{1-\left(\frac{n_{1}}{n_{2}} \sin \theta_{\mathrm{i}}\right)^{2}}}{n_{1} \cos \theta_{\mathrm{i}}+n_{2} \sqrt{1-\left(\frac{n_{1}}{n_{2}} \sin \theta_{\mathrm{i}}\right)^{2}}}\right|^{2}

R_{\mathrm{p}}=\left|\frac{n_{1} \cos \theta_{\mathrm{t}}-n_{2} \cos \theta_{\mathrm{i}}}{n_{1} \cos \theta_{\mathrm{t}}+n_{2} \cos \theta_{\mathrm{i}}}\right|^{2}=\left|\frac{n_{1} \sqrt{1-\left(\frac{n_{1}}{n_{2}} \sin \theta_{\mathrm{i}}\right)^{2}}-n_{2} \cos \theta_{\mathrm{i}}}{n_{1} \sqrt{1-\left(\frac{n_{1}}{n_{2}} \sin \theta_{\mathrm{i}}\right)^{2}}+n_{2} \cos \theta_{\mathrm{i}}}\right|

$R$ 指Reflection，即反射的部分比例。相应地根据能量守恒，透射（折射）的比例为 $T=1-R$ ，这部分光在物体内部发生吸收和散射（散射表现为漫反射，次表面散射以及透射）。注意 $R$ 随着波长变化，即金属颜色的反映。
根据金属和电介质的 $F_0$ 特性，金属（ $F_0 \ge 0.5$ ）的外观更多地由反射决定，而电介质（ $F_0 \approx 0.06$ ）的外观则更多地由吸收和散射特性决定；不过在出射角接近掠射角时，菲涅尔项均近似为 $1$ ，因此反射起到了决定性作用。

In which $n_1$ and $n_2$ is the index of refractive (ior) of the incident and transmitted media, $n=\frac{c}{v}$ . Dielectrics have real valued ior usually in $[1,3]$ . Conductors have complex-valued ior $\bar{\eta}=\eta + \mathrm{i}k$ , where $k$ is referred to as the absorption coefficient. Ior varies as the light’s wave length changes.

For unpolarized light (which is often assumed), the Fresnel reflectance is:

R_{\mathrm{eff}}=\frac{1}{2}(R_{\mathrm{s}} + R_{\mathrm{p}})

Due to conservation of energy, the energy transmitted by a dielectric is $1-R_{\mathrm{eff}}$ .

The Approximation by Schlick:

\mathbf{F}_{\text{Schlick}}(\theta_d) = \mathbf{F}_0 + (1-\mathbf{F}_0)(1-\cos\theta_d)^5\\ \mathbf{F}_0 = \left(\frac{n_1-n_2}{n_1+n_2}\right)^2

$F_0$ is the specular reflectance at normal incidence.

Normal Distribution Function $\mathbf{D}(\theta_h)$

There are many different models to discribe NDF!

(1) Beckmann NDF

D(\theta_h)=\frac{e^{-\frac{\tan ^{2} \theta_{h}}{\alpha^{2}}}}{\pi \alpha^{2} \cos ^{4} \theta_{h}}

Insights: (1) $\alpha$ is roughness, shape similar to Gaussian distribution with $\alpha$ acts like v ariance~ (2) slope space: the $\tan \theta_{h}$ acts as variant. (If we use $\theta_d$ then it doesn’t attenuate to $0$ when achieving $90^\circ$ ) (3) The result should be $1$ when integrated over the projected solid angle on the surface of upper hemisphere. (为什么积分为1？因为这就是一个概率密度函数？)

(2) GGX (or Trowbridge-Reitz, TR)

Property: Long tail (more gentle specular, “光晕”)

Extending GGX: Generalized TR, GTR [Burly.] Adjustable tail length with $\gamma$ !

Shadowing Masking Term $\mathbf{G}(\theta_i, \theta_o)$

Idea: self-occlusion in microfacet. generally: more occlusions when increasing $\theta_i$ or $\theta_o$ , drops to $0$ when arriving grazing angle.

Another perspective: the $\frac{1}{4\cos\theta_l\cos\theta_v}$ goes to infinity at grazing angles, amplifying Fresnel effect， but the infinity is OK because the geometry shadowing $G$ term.

Smith

Insights: Decoupling shadowing (from light direction) and masking (from view direction): $G(\text{l,v,h})\approx G_1(\text{l,h})G_1(\text{v,h})$ .

Kulla-Conty Approximation

The Issue: Multiple bounces in Microfacet ?

Missing energy! (observed under the white furnace test) Since multiple bounces in microfacet should be considered, especially when high roughness.

To get the overall energy when uniform incident radiance:

\begin{align} E\left(\mu_{o}\right) & = \int_{0}^{2 \pi} \int_{0}^{\pi} f\left(\theta_o, \theta_i, \phi\right) \cos\theta_i \sin\theta_i \,\mathrm{d} \theta_i \mathrm{d} \phi\\ &=\int_{0}^{2 \pi} \int_{0}^{1} f\left(\mu_{o}, \mu_{i}, \phi\right) \mu_{i} \,\mathrm{d} \mu_{i} \mathrm{d} \phi \, ,\, \mu_i=\sin\theta \in [0,1] \end{align}

Idea: (1) Compensate the lost energy with another BRDF lobe, which can be integrated to $1-E(\mu_o)$ in uniform incident radiance. (注意此处没有考虑菲涅尔项，即表面没有吸收或透射光线，因此认为所有能量均被反射) (2) The new BRDF should be reciprocal, i.e. in the form: $c(1-E(\mu_i))(1-E(\mu_o))$ . (since all brdf should be reciprocal w.r.t. $\text{i and o}$ !) So……

f_{\mathrm{ms}}\left(\mu_{o}, \mu_{i}\right)=\frac{\left(1-E\left(\mu_{o}\right)\right)\left(1-E\left(\mu_{i}\right)\right)}{\pi\left(1-E_{\mathrm{avg}}\right)}, E_{\mathrm{avg}}=2 \int_{0}^{1} E(\mu) \mu \mathrm{d} \mu

Validation:

\begin{aligned} E_{\mathrm{ms}}\left(\mu_{o}\right) &=\int_{o}^{2 \pi} \int_{o}^{1} f_{\mathrm{ms}}\left(\mu_{o}, \mu_{i}, \phi\right) \mu_{i} d \mu_{i} d \phi \\ &=2 \pi \int_{o}^{1} \frac{\left(1-E\left(\mu_{o}\right)\right)\left(1-E\left(\mu_{i}\right)\right)}{\pi\left(1-E_{\mathrm{avg}}\right)} \mu_{i} d \mu_{i} \\ &=2 \frac{1-E\left(\mu_{o}\right)}{1-E_{\mathrm{avg}}} \int_{0}^{1}\left(1-E\left(\mu_{i}\right)\right) \mu_{i} d \mu_{i} \\ &=1-E\left(\mu_{o}\right) \end{aligned}

For each material (BRDF), get $E_{\mathrm{avg}}$ by precomputation: $E_{\mathrm{avg}}\left[\mu_o, \alpha \right]$ (Numerical fitting is not quite precise)

If the surface transmits/absorbs energy…

这里使用cosine-weighted average Fresnel近似总共被反射的能量。这种近似代表假设光线的多次散射都是diffuse的。

F_{a v g}=\frac{\int_{0}^{1} F(\mu) \mu \mathrm{d} \mu}{\int_{0}^{1} \mu \mathrm{d} \mu}=2 \int_{0}^{1} F(\mu) \mu \mathrm{d} \mu

这是我们考虑Fresnel项后，多次散射中除去被吸收部分后的总能量（近似）。假设每次有 $F_{\mathrm{avg}}(1-E_{\mathrm{avg}})$ 比例的能量是参与下一次散射的，而 $F_{\mathrm{avg}}E_{\mathrm{avg}}$ 的能量直接可以观测到。 *这个逻辑我还是不明白…

\frac{1}{1-E_{\mathrm{avg}}} F_{\mathrm{avg}} E_{\mathrm{avg}} \sum_{k=1}^{\infty} F_{\mathrm{avg}}^{k}\left(1-E_{\mathrm{avg}}\right)^{k}=\frac{F_{\mathrm{avg}}^{2} E_{\mathrm{avg}}}{1-F_{\mathrm{avg}}\left(1-E_{\mathrm{avg}}\right)}

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功能

[GAMES 202] Lecture Note 10

Overview

Microfacet BRDF

Fresnel Term $\mathbf{F}(\theta_d)$

Normal Distribution Function $\mathbf{D}(\theta_h)$

(1) Beckmann NDF

(2) GGX (or Trowbridge-Reitz, TR)

Shadowing Masking Term $\mathbf{G}(\theta_i, \theta_o)$

Smith

Kulla-Conty Approximation

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[GAMES 202] Lecture Note 10

Overview

Microfacet BRDF

Fresnel Term \mathbf{F}(\theta_d)

Normal Distribution Function \mathbf{D}(\theta_h)

(1) Beckmann NDF

(2) GGX (or Trowbridge-Reitz, TR)

Shadowing Masking Term \mathbf{G}(\theta_i, \theta_o)

Smith

Kulla-Conty Approximation

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Fresnel Term $\mathbf{F}(\theta_d)$

Normal Distribution Function $\mathbf{D}(\theta_h)$

Shadowing Masking Term $\mathbf{G}(\theta_i, \theta_o)$

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