Reflection Models

• Reflection Models
  • reflection models
    • categories
    • geometric settings
  • specular reflection and transmission
    • Fresnel Reflectance --- amount of reflection
      • Index of refraction 定义
      • two dielectric media
      • a conductor and a dielectric medium
      • semiconductors (not included)
    • specular reflection
    • specular transmission
  • Lambertian Reflection
  • Microfacet Models
    • Oren–Nayar Diffuse Reflection
      • main thoughts
      • model
    • Normal Distribution Function
      • Beckmann and Spizzichino Model
      • Trowbridge and Reitz Model
      • comparison
    • Masking and Shadowing
      • Smith masking-shadowing function G_1(w, w_h)
        • Beckmann–Spizzichino distribution
        • Trowbridge–Reitz distribution
      • both direction geometry function G(w_o, w_i)
        • independently for w_o, w_i
        • dependently
    • The Torrance–Sparrow Model
      • basic thoughts
      • Torrance–Sparrow BRDF
      • Torrance-Sparrow BRTF
  • Fresnel Incidence Effects
    • basic principle
    • Ashikhmin and Shirley model
      • glossy specular term
      • diffuse term
  • FourierBSDF
    • objective
    • represents isotropic BSDFs
    • specified BSDF value
      • Catmull–Rom spline interpolation

reflection models

categories

• diffuse
• glossy specular
• perfect specular
• retro-reflective eg. velvet 天鹅绒

  geometric settings

  

specular reflection and transmission

• 镜面反射
• 遵从 Snell’s law的透射
  • 理论上，the index of refraction varies with the wavelength of light (eg. 色散), 但是为了简化，忽略波长的因素

Fresnel Reflectance --- amount of reflection

Index of refraction 定义

the ratio of the speed of light in a vacuum to the speed of light in the medium.

two dielectric media

• index of refraction
• complex index of refraction --- absorption

a conductor and a dielectric medium

semiconductors (not included)

specular reflection

specular transmission

• Fresnel现象：transmission is stronger at near-perpendicular angles

Lambertian Reflection

models a perfect diffuse surface that scatters incident illumination equally in all directions

• const Spectrum R gives the fraction of incident light that is scattered
• BRDF f = R / pi (pi 是 cos(theta)在半球上的积分)

Microfacet Models

rough surfaces can be modeled as a collection of small microfacets

• a representation of the distribution of facets
• a BRDF that describes how light scatters from individual microfacets

Oren–Nayar Diffuse Reflection

main thoughts

• real-world objects do not exhibit perfect Lambertian reflection
• describes rough surfaces by V-shaped microfacets described by a spherical Gaussian distribution with a single parameter theta, the standard deviation of the microfacet orientation angle.

model

Normal Distribution Function

• NDF必须归一化

  Beckmann and Spizzichino Model

  • isotropic
  • anisotropic

    Trowbridge and Reitz Model

    

    comparison

    Trowbridge–Reitz has higher tails—it falls off to zero more slowly for directions far from the surface normal. This characteristic matches the properties of many real-world surfaces well.

Masking and Shadowing

Smith masking-shadowing function G_1(w, w_h)

• 必须满足归一化条件
• basic thoughts forward-facing facets 的投影面积 减去 backward-facing facets的投影面积
  • auxiliary function

Beckmann–Spizzichino distribution

• assume that there is no correlation between the heights of nearby points on the microsurface
• function
• a rational polynomial approximation

Trowbridge–Reitz distribution

both direction geometry function G(w_o, w_i)

independently for w_o, w_i

dependently

microfacet visibility is more likely the higher up a given point on a microfacet is

The Torrance–Sparrow Model

basic thoughts

modeled surfaces as collections of perfectly smooth mirrored microfacets only consider the half vector.

Torrance–Sparrow BRDF

perfect specular reflection

Torrance-Sparrow BRTF

perfect specular transmission

Fresnel Incidence Effects

basic principle

这个关心的是上一层材质的Fresnel reflection是如何影响下一层材质的入射光照的。 Fresnel reflection reduces the amount of light that reaches the bottom level of layered objects.

Ashikhmin and Shirley model

models a diffuse underlying surface with a glossy specular surface above it.

glossy specular term

diffuse term

R_d diffuse surface reflectance, R_s specular surface reflectance

FourierBSDF

objective

一些复杂的材质很难用一个通用BXDF公式去描述。 一种思路是存a large 3D or 4D lookup table，但是这样数据量会太大。 想要寻求的是a more compact representation that still represents the BSDF accurately， FourierBSDF就是这样的一个解决方案。 核心思想是sums of scaled cosine terms using the Fourier basis.

represents isotropic BSDFs

因为isotropic，所以有对称性，所以没有sin项。

specified BSDF value

找到临近的离散值，做一定的插值得到Fourier系数，然后计算BSDF值。

Catmull–Rom spline interpolation

• objective 对于一个function f, 对每一段区间[x_i, x_i+1]希望用a cubic polynomial来拟合这个函数： 使得这个多项式满足
• calculation 因为原函数未知，导数不好求所以用近似的方法