New answers tagged importance-sampling
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Think of this way: when integrating uniformly over the hemisphere, it's like you are importance-sampling with a constant pdf of $1/2\pi$.
The multiplication by $2\pi$ at the end, then, can be seen as the division by the pdf, factored out of the sum because it's constant.
The constant pdf is $1/2\pi$ (as opposed to some other number) because it needs to be ...
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