Advanced Computer Graphics: Difference between revisions
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==Monte Carlo Integration== | ==Monte Carlo Integration== | ||
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> | |||
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> | |||
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Revision as of 21:06, 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 \(\displaystyle E[\frac{b-a}{N}\sum f(X_i)] = \frac{b-a}{N}\sum E[f(X_i)] = \frac{1}{N}\sum \int_{a}^{b}(b-a)f(x)(1/(b-a))dx = \int_{a}^{b}f(x)dx\)