TU Wien Rendering #16 - Monte Carlo Integration: Hit or Miss

let's go to monte carlo integration i promise you something if you learn what monte carlo integration is you will never ever in your life will have to evaluate any more integrals never i promise to you i give you my word and this is a simple method to approximate integrals and basically what we are looking for is we would like to integrate the function and we can take samples of this function what does it mean we will check it out in a second so we will take some samples of this function and we would like to reconstruct the integral of it if we do this is what's called

monte carlo integration and this was founded during the second world war bar by stanislav and his co-workers during the manhattan project so this was the atomic bomb project they had unbelievably difficult integrals to solve and they had to came up with a numerical solution in order to at least approximate so this is what they came up with there's two different kinds of monte carlo integration the keys i have this function f of x and i would like to integrate this from a to b so this is a definite integral what i can do is hit or miss monte carlo or sample mean monte carlo 99. 9 of the case we use the sample mean but just for the intuition and to visualize what is going on i will show you the intermission as well and at all times we can choose how to take samples from this function so let's take a look at this is the recipe like for wonderful vinash midsole for

vinash midsole this is the recipe for monte carlo integration you draw this function that you have on a paper you close it in a box that you know the size of and let the size of the box be a you throw lots of random points on this paper and for every single point you have you determine if it is above or below this function and then you have a magical formula you use this formula and you will get the integral the more points you have on the paper the better so i compute the ratio of hits the points below the curve the function compared to all the samples that i have how does it look like this looks more or less like this so this immediately gives you the intuition that the reds are above the function the blues are below the function so i would like to know the ratio of blues to all samples because this gives you exactly what intervals mean integrals mean the area below this curve so get this if i would be on a summer holiday i could have some beers and get a crazy idea that i would go on top of my house and imagine that i have a pool of water and i will start throwing beach balls in this pool and after doing this for long enough i could approximate the value of pi it sounds like black magic right provided that the walls are small enough and i am patient enough that this can happen okay what is the recipe let's go through it let's draw a unit square somewhere so the area of this square is going to be one let's draw a quarter of a unit circle inside this box and this is also of unit radius now we start throwing these points we compute the ratio what is inside and how much is outside we multiply the result by 4 and then we get pi so let me say this again we multiply the results by 4 and we get pi now how is this i mean it doesn't sound like that it makes any sense but this is black magic and this works let's take a closer look at why this works so i would like to compute the integral would be below this function this is the one quarter of a sphere of a circle sorry and what is the area of the sphere r square of times pi well r is one so it's pi over four so what we are essentially approximating here is pi over four so i have this when i solve this integral what i get as a result is pi over four but what do i need to do with this in order to get pi multiplied by or louder four yes exactly so shadow cooper will be proud for all of us physicists do a lot of this excellent yes what if you have a surface this also works for multi-dimensional functions and you will actually compute such a thing in the next assignment and it's going to be absolutely trivial so this trivially generalizes for arbitrary dimensions well it better do because the rendering equation is infinite dimensional so it has to take care of high dimensional functions somehow