GAMES101_notes_05
GAMES101: 现代计算机图形学入门
笔记05:
Lecture 17: Materials and Appearances
What is Material in Computer Graphics?
Material == BRDF
Diffuse / Lambertian Material 漫反射材质
Light is equally reflected in each output direction
- Suppose the incident lighting is uniform:
- $L_o(\omega_o) = \int_{H^2}f_rL_i(\omega_i)\cos\theta_i\mathrm{d}\omega_i = f_rL_i\int_{H^2}\cos\theta_i\mathrm{d}\omega_i = \pi f_r L_i$
- $f_r = \frac{\rho}{\pi}$, 反射率$\rho$
Glossy Material 类似金属的光滑材质
Ideal reflective / refractive material(BSDF)
Perfect Specular Reflection
- $\omega_o + \omega_i = 2\cos\theta\vec{n} = 2 (\omega_i\cdot \vec{n})\vec{n}\quad \Rightarrow \quad \omega_o = -\omega_i + 2(\omega_i \cdot \vec{n})\vec{n}$
Specular Refraction
- In addition to reflecting off surface, light may be transmitted through surface.
- Light refracts when it enters a new medium
Snell’s Law
- Transmitted angle depends on
- index of refraction(IOR) for incident ray
- IOR for exiting ray
- $\eta_i\sin\theta_i = \eta_t\sin\theta_t$
- $\cos\theta_t = \sqrt{1 - (\frac{\eta_i}{\eta_t})^2(1 - \cos^2\theta_i)}$ 当入射介质折射率大于折射介质时，可能会发生全反射现象（Snell’s Window/Circle）
Fresnel Reflection / Term 菲涅耳项
- Reflectance depends on incident angle(and polarization of light)
- 对于绝缘体，当入射角角度非常小的时候，反射的能量就非常小，当入射角接近90度时，几乎没有折射发生（大部分能量被反射）
- 对于导体，其菲涅耳项与绝缘体不同，无论入射角方向如何，反射的能量都占大多数
- Approximate: Schlick’s approximation 对两种介质都有较好的近似
- $R(\theta) = R_0 + (1 - R_0)(1 - \cos\theta)^5$
- $R_0 = \left(\frac{n_1 - n_2}{n_1 + n_2}\right)^2$ 基准反射率
Microfacet Material
Microfacet Theory
- Rough surface
- Macroscale: flat & rough 从远处看是材质
- Microscale: bumpy & specular 从近处看是几何
- Individual elements of surface act like mirrors
- Known as Microfacets
- Each microfacet has its own normal
Microfacet BRDF
- 如果微表面各处法线在宏观上差别不大，那么就得到了glossy材质
- 如果微表面各处法线在宏观上区别很大，那么就得到了漫反射材质
Isotropic/Anisotropic Materials(BRDFs) 各向同性/各向异性
Key: directionality of underlying surface
Properties of BRDFs
- Non-negativity
- $f_r(\omega_i\rightarrow\omega_r)\ge 0$
- Linearity
- $L_r(p, \omega_r) = \int_{H^2}f_r(p, \omega_i\rightarrow \omega_r)L_i(p, \omega_i) \cos \theta_i\mathrm{d}\omega_i$
- Reciprocity principle
- $f_r(\omega_r\rightarrow\omega_i) = f_r(\omega_i\rightarrow \omega_r)$
- Energy conservation
- $\forall \omega_r,\quad \int_{H^2}f_r(p, \omega_i\rightarrow \omega_r)L_i(p, \omega_i) \cos \theta_i\mathrm{d}\omega_i \le 1$
- Isotropic vs. anisotropic
- If isotropic, $f_r(\theta_i, \phi_i; \theta_r, \phi_r) = f_r(\theta_i, \theta_r, \phi_r - \phi_i)$
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Then, from reciprocity, $f_r(\theta_i, \theta_r, \phi_r - \phi_i) = f_r(\theta_i, \theta_r, \phi_i - \phi_r) = f_r(\theta_i, \theta_r, \phi_r - \phi_i )$