Diffuse / Lambertian Material: Light is equally reflected in each output direction. Suppose the incident lighting is uniform

Suppose BRDF ($f_r$) is constant, incidence radiance ($L_i(\omega_i)$) is constant, incident radiance ($L_i$) is equal to the exit radiance ($L_o$).

Define a reflectance $\rho$ (-albedo( color)) ($0<\rho<1$), so $\Large f_r=\frac{\rho}{\pi}$

Glossy Material (BRDF)

Ideal Reflective / Refractive Material (BSDF)

BSDF (Scatter) = BRDF (Reflection) + BTDF (Transmitted)

Perfect Specular Reflection

Specular Reflection: In addition to reflecting off surface, light may be transmitted through surface. Light refracts when it enters a new medium.

Snell's Law (Law of Refraction): Transmitted angle depends on

• index of refraction (IOR) for incident ray.
• index of refraction (IOR) for exiting ray.

$\eta$: index of refraction is wavelength dependent (these are averages)

Total internal reflection: $\large1-(\frac{\eta_i}{\eta_t})^2(1-\cos^2\theta_i)<0$

When light is moving from a more optically dense medium to a less optically dense medium: $\large\frac{\eta_i}{\eta_t}>1$

Light incident on boundary from large enough angle will not exit medium.

Snell's Window / Circle

Fresnel Reflection / Term: Reflectance depends on incident angle (and polarization of light)

The left image is Dielectric ($\eta=1.5$), and the right image is Conductor.

Fresnel Term - Formulae:

• Accurate: need to consider polarization

• Approximate: Schlick's approximation

$n_1,n_2$ is $\eta_1,\eta_2$

Microfacet Material

Rough surface

• Macroscale: flat & rough
• Microscale: bumpy & specular

Individual elements of surface act like mirrors

• Known as Microfacets
• Each microfacet has its own normal

Microfacet BRDF

• Key: the distribution of microfacets' normals

  • Concentrated <==> glossy

    

  • Spread <==> diffuse

Isotropic / Anisotropic Materials (BRDFs)

• Key: directionality of underlying surface

• Anisotropic BRDFs: Reflection depends on azimuthal angle $\phi$, $f_r(\theta_i,\phi_i;\theta_r,\phi_r)\ne f_r(\theta_i,\theta_r,\phi_r-\phi_i)$

  Results from oriented microstructure of surface, e.g., brushed metal, Nylon, Velvet

Properties of BRDFs

• Non-negativity $\large f_r(\omega_i\to\omega_r)\ge 0$

• Linearity $\Large L_r(p,\omega_r)=\int_{H^2}f_r(p,\omega_i\to\omega_r)L_i(p,\omega_i)\cos\theta_id\omega_i$

  

• Reciprocity principle $\large f_r(\omega_r\to\omega_i)=f_r(\omega_i\to\omega_r)$

  

• Energy conservation $\Large\forall\omega_r\int_{H^2}f_r(\omega_i\to\omega_r)\cos\theta_id\omega_i\le1$

• Isotropic vs. anisotropic

  • If isotropic, $f_r(\theta_i,\phi_i;\theta_r,\phi_r)=f_r(\theta_i,\theta_r,\phi_r-\phi_i)$
  • Then, from reciprocity, $f_r(\theta_i,\theta_r,\phi_r-\phi_i)=f_r(\theta_r,\theta_i,\phi_i-\phi_r)=f_r(\theta_i,\theta_r,|\phi_r-\phi_i|)$

Measuring BRDFs

• Image-Based BRDF Measurement: gonioreflectometer

  General approach:

  for each outgoing direction wo
      move light to illuminate surface with a thin beam from wo
      for each incoming direction wi
          move sensor to be at direction wi from surface
          measure incident radiance
  

  Improving efficiency:

  • Isotropic surfaces reduce dimensionality from 4D to 3D
  • Reciprocity reduces # of measurements by half
  • Clever optical systems...