Lecture 11 : Monte Carlo Integration
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Tianye

I find the slide a little bit confusing, especially with the different occurrence of f(x). It was helpful for me to think of it this way:

our goal is to calculate $$I=\int f(x)dx$$ We can rewrite this as $$I=\int {f(x)\over p(x)}p(x)dx$$ which is $E[{f(x)\over p(x)}]$.

By the derivation of the next slide, we can estimate this with ${1\over N}\sum {f(x_i)\over p(x_i)}$, where $x_i$~$p(x)$

mmp

That's good feedback. In retrospect I think that what you're suggesting--basically starting with the derivation we used later in the lecture, would make the ideas clearer. (The students next year can benefit from this!)

Tianye

Sorry I forgot what V(p,p') was... Is it BRDF?

mmp

V is just a function that is 1 if the two points are mutually visible and 0 if there is another object occluding the line segment between them.

The BRDF comes in starting in next lecture.

atrytko

The phrase "decreases linearly with sample size" was confusing to me. After checking with the teaching staff, I thought I'd post here to clarify that this means 'is inversely proportional'.

brianjo

Small technical note: the right hand side of the first line of the proof should read E[f(X_i)/p(X_i)].

mmp

Good catch--thanks. Will fix in the master slides.