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

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

Refractive objects

For mirrors

In practice, refractive objects are handled as a distinct case.
You do not need to sample a direction.

For glass

• Randomly sample either reflected or refracted ray with given probability.
• Typically produces a lot of noice

Emitting surfaces

• If a ray accidentally hits emitting surface, don't add emission
  • Exception: If eye ray hits emitting surface
  • Exception: If ray is generated from a refractive surface

Advanced Sampling Techniques

Multiple Importance Sampling

Weighted sampling between \(\displaystyle F_a\) \(\displaystyle F_d\)

• Take \(\displaystyle N\) samples from each technique (j=1,...,N)
• \(\displaystyle F=\frac{1}{N} \sum_{j=1}^{N} \sum_{i=1}^{n} w_i(X_{i,j}) \frac{f(X_{i,j})}{p_i(X_{i,j})}\)
  • Make sure \(\displaystyle \sum_{i=1}^{n}w_i(x) = 1\)

• Weights for provable variance reduction
  • Balance heuristics: \(\displaystyle w_i(x)=\frac{p_i(x)}{\sum_{k=1}^{n}p_k(x)}\)
  • Power heuristics: \(\displaystyle w_i(x)=\frac{p_i^2(x)}{\sum_{k=1}^{n}p_k^2(x)}\)

Stratified Sampling

• Intuition: clumping of samples is bad
• Instead of canonic uniform random variables, generate variables in strata
• Also known as "Jittered sampling"

Other stratified sampling patterns

• N-rooks (Latin hypercube)
• Quasi Monte Carlo

Bidirectional Path Tracing

Trace path from eye and light
Example:

• from eye we get path \(\displaystyle z_0, z_1, z_2\)
  
• from light we get \(\displaystyle y_0, y_1\)
  
• Then make shadow rays from every pair of z, y
  

Path of length k with k+1 vertices

• s vertices from light, t from eye
• Path denoted \(\displaystyle \bar{X}^{s,t}\) (e.g. \(\displaystyle \bar{X}^{2,3}\))
• We also get probability density for this path \(\displaystyle p_{s,t}\)

Participating Media

Transmittance

• Multiplicative property
  • \(\displaystyle T(s)=T(s_0) * T(s_1)\)
• Beer's law \(\displaystyle T(s)=e^{-sigma_t s}\)
  • For homogenous media where \(\displaystyle \sigma(x) = \sigma\) is constant

Phase Functions

Henyey-Greenstein phase function

• \(\displaystyle p(\cos \theta) = \frac{1-g^2}{4\pi(1+g^2-2g\cos \theta)^{1.5}}\)

Properties

• Unitless
• Reciprocity
  • \(\displaystyle p(\omega' \rightarrow \omega) = p(\omega \rightarrow \omega')\)
• Energy conservation
  • Integrates to 1
• Average phase angle determined by g

Volume Rendering Equation

Integro-integral form

• \(\displaystyle L(\mathbf{x}, \omega) = \int_{0}^{\infty}\exp(-\int_{0}^{s'}\sigma_t(\mathbf{x}-s''\omega)ds'')S(\mathbf{x}-s'\omega, \omega)ds\)
  • \(\displaystyle \exp(-\int_{0}^{s'}\sigma_t(\mathbf{x}-s''\omega)ds'')\) is Transmittance \(\displaystyle T(s')\) due to extinction
  • \(\displaystyle S(\mathbf{x}-s'\omega, \omega)\) is source (emission, in-scattering)

Subsurface Scattering

BSSRDF

bidirectional surface scattering reflectance distribution function

• \(\displaystyle S(\mathbf{x}_i, \omega_i, \mathbf{x}_o, \omega_o)\)