Advanced Computer Graphics: Difference between revisions
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Suppose we want to estimate <math>I_1 = \int_{a}^{b}f(x)dx</math>.<br> | Suppose we want to estimate <math>I_1 = \int_{a}^{b}f(x)dx</math>.<br> | ||
Then we can use <math>\hat{I_1} = \frac{b-a}{N}\sum f(X_i)</math> where <math>X_1,...,X_n \sim Uniform(a,b)</math>.<br> | Then we can use <math>\hat{I_1} = \frac{b-a}{N}\sum f(X_i)</math> where <math>X_1,...,X_n \sim Uniform(a,b)</math>.<br> | ||
This is because <math>E[\frac{b-a}{N}\sum f(X_i)] = \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</math><br> | This is because <math>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</math><br> | ||
Note that in general, if we can sample from some distribution with pdf <math>p(x)</math> then we use the estimator: | |||
* <math>E\left[ \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)} \right]</math> | |||
===Importance Sampling=== | |||
Suppose we can only sample from pdf <math>g(x)</math> but we want to sample from pdf <math>p(x)</math> to yield a more reliable (less variance) estimate.<br> | |||
Then we can sample from <math>p(x)</math> using <math>Y = F_{p}^{-1}(F_{g}(X))</math>.<br> | |||
Then apply the above equation. | |||
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Revision as of 21:15, 12 February 2020
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
Geometrical Optics
- Light are rays which reflect, refract, and scatter
Solid Angle
- Solid angle = area / radius^2 on a sphere
BRDF, Reflection Integral
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:
- \(\displaystyle E\left[ \frac{1}{N} \sum \frac{f(X_i)}{p(X_i)} \right]\)
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.