Hi, we would like to understand now a bit better why this fifth effect projection thing

works and before we try to understand this let's first make an excursion. Let's call

it Playground Tomography. And let's call this game Playground Tomography. You consider

a 2x2 square with unknown entries in those squares and your friend says that the sum

over the first column is 1, the sum over the second column is 2, the sum over the first

row is 2 again and the sum over the second row is 6. So you would like to find the correct

values in those squares such that this plus this is equal to 2, this plus this is equal

to 6, this sum is 6 and this sum is 2. So you don't even know whether this is solvable

or not but I can give you the algorithm in order to work with this right away and that

is to do back projection. So you know that in the first row you have to have a sum of

2 so why not evenly distribute this sum of 2 into the entries in this row. So what you

do is distribute missing mass in first row. So what does that give you? You don't do anything

in the second row let's just call this 0, 0 and you just put a 1 here and a 1 here.

So that is now good. Let's make this green. So you're done with this row. Then you want

to do the same for the second row. Again this square you don't touch the first row this

is still 1 and 1. Now in the second row you have a mass combined of 6. Now you distribute

this in this row so you put a 3 here and a 3 here and checking for correctness you see

that this is fine, this is fine but you have a problem now with your columns so that's

wrong and this is wrong as well. So what you do now is you distribute missing or excess

mass in your first column. So the combined mass in your first column is 4 which is 2

less than the number 6 that you want to have so you have to add a mass of 2 in total in

the first column so you put 1 and 1 so you have 2 and 4. You don't change the first column

it's not going to the second column so of course you have a problem with the first row

but that's ignore that for a second and you do the same thing in the second column which

is not yet okay. Second column so 2 and 4. The combined mass in the second column is

4 which is 2 more than the 2 you want so you subtract now 1 from each so you have a 0 here

and 2 here and as you can see this is now correct so you have the correct sums in all

rows and columns so of course you don't know whether this algorithm works all the time

whether it gives a unique answer and things like that so I can tell you that if there

is a solution and this converges to the solution you might need more steps than just looking

at each row and column once but it will converge to a solution. The solution is non-unique

but nevertheless this gives you a solution so the idea here is to take this well this

is kind of tomography right here so you're putting a virtual x-ray horizontally through

here and it encounters two objects they count six objects horizontal here six objects vertically

here and two objects vertically here so it's kind of a type of tomography you're counting

volume so to speak or mass according to or along certain x-rays. The idea here of playground

Tomography is to take the mass in each of these entries and just project back

what is missing. And this is something that you might think could work for

actual computerized tomography as well. So we're going to look at another Python

notebook. So this is similar to something I already showed you but I've changed the

code a bit now we'll give you the link to this notebook. The image we're looking

at will be an L-shaped image because the geometry is much easier than working

with actual tomography data which of course you can do. So this is the image

is a 100 pixel by 100 pixel square image. Light colors mean mass and dark colors

mean no mass. In this case this image is zero everywhere and it's one on this L

shaped contour here. So this is you could think of this as being a rigid object in

the form of an L you're trying to do CT with it. Now let's again talk about the

forward radon transform. So how does that work? You pick an angle this in this case

you pick the angle well it corresponds to the angle theta equal to zero this

means that you're projecting down or upwards it always depends on