Advanced Computer Graphics

Classnotes for CMSC740 taught by Matthias Zwicker

Acceleration

How to speed up intersection calculations during ray tracing.

Object Subdivision

Bounding volume hierarchy

• Create a tree where objects are placed together.
• Each node corresponds to a region covering all the objects it has
• You want to insert in some greedy way:
  • Minimize the size of each region
• Subtrees may overlap and are unorder

Spacial Subdivision

Octtree, kd tree

Radiometry

Main article: Wikipedia: Radiometry

Geometrical Optics

Main article: Wikipedia: Geometrical Optics

• Light are rays which reflect, refract, and scatter

Solid Angle

• Solid angle = area / radius^2 on a sphere

Spectral Radiance

Main article: Wikipedia: Radiance

• Spectral radiance is energy per time per wavelength per solid angle per area
• \(\displaystyle L(t, \lambda, \omega, \mathbf{x})=\frac{d^4 Q(t, \lambda, \omega, \mathbf{x})}{dt d\lambda d\omega dA^\perp}\)

where energy is \(\displaystyle Q(t, \lambda, \omega, \mathbf{x})\)

Radiance

Main article: Wikipedia: Radiance

• power per solid angle per area
• \(\displaystyle L(\omega, \mathbf{x}) = \frac{d^2 \Phi(\omega, \mathbf{x})}{d\omega dA^\perp}\)

Monte Carlo Integration

Suppose we want to estimate \(\displaystyle I_1 = \int_{a}^{b}f(x)dx\).
Then we can use \(\displaystyle \hat{I_1} = \frac{b-a}{N}\sum f(X_i)\) where \(\displaystyle X_1,...,X_n \sim Uniform(a,b)\).
This is because \(\displaystyle E\left[\frac{b-a}{N}\sum f(X_i)\right] = \frac{b-a}{N}\sum E[f(X_i)] = \frac{1}{N}\sum \int_{a}^{b}(b-a)f(x)(1/(b-a))dx = \int_{a}^{b}f(x)dx\)
Note that in general, if we can sample from some distribution with pdf \(\displaystyle p(x)\) then we use the estimator:

• \(\displaystyle \hat{I} = \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)}\)

Importance Sampling

Suppose we can only sample from pdf \(\displaystyle g(x)\) but we want to sample from pdf \(\displaystyle p(x)\) to yield a more reliable (less variance) estimate.
Then we can sample from \(\displaystyle p(x)\) using \(\displaystyle Y = F_{p}^{-1}(F_{g}(X))\).
Then apply the above equation.

Integration

Some notes about integration

• Hemisphere: \(\displaystyle d\omega = \sin \theta d\theta d\phi\) where \(\displaystyle \theta \in [0, \pi/2)\) and \(\displaystyle \phi \in [0, 2\pi)\)

Path Tracing

PDF for sampling light sources

\(\displaystyle p_{\omega}(x) = \frac{1}{\# lights} * \frac{1}{area of light} * conversion\)