So the measured signal that we have, so

what we want is we want a measurement over each of these voxels.

But alas, the measured signal that we have combines information from the whole brain.

So basically the signal that we're measuring is basically the combination

of all the hydrogen atoms over the whole slice.

So we sort of get a measure of the total number of hydrogen atoms.

Unfortunately, this doesn't give us the information we need to figure out

how many hydrogen atoms are in each of the individual voxels.

So we need to make more measurements to get this information.

And also we need to make different types of measurements.

And so here is one of the clever things about MR scanners,

is here we use a second magnetic field which is called magnetic field gradient.

And using these magnetic field gradients we can sort of sequentially control

the spatial inhomogeneities of the magnetic field, and so

we can change the magnetic field across the brain.

And so this allows us to make a new measurement which is now a weighted

integral of the hydrogen concentration across the brain.

So here you see that now we have for two constants, kx and

ky, we can measure S(kx, ky), which is this row of x,y

weighted by this exponential term which depends on x and y, and kx and ky.

Where, again, kx and ky is controlled by S.

So basically what we can do is we can alter values of kx and ky.

And we can get new measurements of row xy until we have enough for

which we can solve this inverse problem and get a reconstruction of the image,

get the individual rho x y's back.