[GAMES202 笔记] Lecture 5 – 融雪时分的博客

预先知识

更好理解地本节课程的环境光照IBL部分，可能需要对于Microfacet BRDF模型的形式以及其各种近似有一定了解。

[GAMES202] Lecture Note 5

Distance Field Soft Shadows

Distance Functions, SDF

Defination: The minimun distance to the closet location on an object Application Example: Interpolation SDF of a moving object… Calculating SDF: at each point, take the minimum of every object’s SDF value

Applications:

Ray Marching, finding ray-SDF intersection: At each point p, travel $\text{SDF}(p)$ as a safe distance.

Soft Shadows, at an point $p$ , to get the approximate percentage of the occlusion of $p$ towards an area light: From $p$ , each ray-marching step $p_i$ use $\text{SDF}(p_i)$ to calculate the safe angle.

min\{\frac{k\cdot \text{SDF}(p)}{p-o}, 1.0\}, \text{where k is the hardness of the shadow}

Pros: Fast rendering, high quality Cons: precomputation, storage (a 3D texture) and artifacts

Environment Lighting, IBL

An image light source from infinite distance, i.e. the incident radiance is only dependent with the direction $w_i$ , or the spatial $(\theta, \rho)$ .

L(p, w_o) = \int _{\Omega^+} L(p, w_i)f(p, w_i, w_o)\cos\theta_i dw_i\text{ , where L defined by IBL}

General solution: Monte Carlo Integration

A BRDF Observation:

\int_\Omega f(x)g(x)dx \approx \frac {\int_\Omega f(x)dx}{\int_\Omega dx} \cdot \int_\Omega g(x)dx

Separate the radiance term by approximation

L(p, w_o) \approx \frac{\int_{\Omega^+}L(p, w_i)dw_i} {\int_{\Omega_+}dw_i} \cdot \int _{\Omega^+} f(p, w_i, w_o)\cos\theta_i dw_i

Consider the diffuse constant-intergration and the specular lobe. (Recall: the accurate condition of the approximation: small support or smooth integrand)

Precompute Solution: The Split Sum Approximation

The radiance term

Prefiltered environment map, Query: given the incident direction of the lobe and filter size (lobe area) , return the intergrated radiance.

In detail: the diffuse reflection queries alongside the macronormal, while the specular reflection queries along the reflected direction

The BRDF term

Idea: Precompute the integration for different view direction ( $v, \text{ or } w_o$ , or $\cos{\theta_v}$ for isotropic materials) and surface parameter combinations. (roughness in NDF and G, base color in Fresnel, etc., different BRDF approximations need different set of parameters. resulting a 5+dim texture!)

An optimization: Extract $F_0$ ( $F_0$ is the specular reflectance at normal incidence) using the Schlick Fresnel approximation $F_\text{Schlick}$ . (小疑问：ppt上将Fresnel项化简为下方第二行的式子，将原BRDF分为两部分积分从而提取出 $F_0$ 。但不化简好像也能达到同样的效果呀？)

\begin{align} F_\text{Schlick} & = F_0 + (1-F_0)(1-\cos\theta_d)^5 \\ & = F_0(1-(1-\cos\theta_d)^5) + (1-\cos\theta_d)^5 \\ \end{align}\\ \theta_d \text{ is the halfangle between viewdir and lightdir}

Finally, we got the two integration parts that only depends on $\text{roughness}$ and $\cos\theta_v$ (i.e. $\text{viewdir}\cdot\text{normal}$ ). We can precompute this integration, and store it into a 2D texture with 2 channels (called the BRDF LUT or BRDF Integration Map).

\int _{\Omega^+} f(p, w_i, w_o)\cos\theta_i dw_i \\ = F_0\int _{\Omega^+} \frac{f(p, w_i, w_o)}{F_\text{Schlick}}(1-(1-\cos\theta_d)^5)\cos\theta_i dw_i + \\ \int _{\Omega^+}\frac{f(p, w_i, w_o)}{F_\text{Schlick}}(1-\cos\theta_d)^5\cos\theta_i dw_i

The 2D BRDF LUT. — 2 channel radiance + brdf term

Remaining problem: the shadows (visibility) from environment lighting?

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[GAMES202] Lecture Note 5

Distance Field Soft Shadows

Distance Functions, SDF

Applications:

Environment Lighting, IBL

A BRDF Observation:

Precompute Solution: The Split Sum Approximation

The radiance term

The BRDF term

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[GAMES202] Lecture Note 5

Distance Field Soft Shadows

Distance Functions, SDF

Applications:

Environment Lighting, IBL

A BRDF Observation:

Precompute Solution: The Split Sum Approximation

The radiance term

The BRDF term

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