We will talk about the concept of K-space here.

We will talk about mathematical requirement to generate an image,

and then that will give intuition of

why frequency encoding and phase encoding are necessary to form an MR image.

Then, we change back after explanation of frequency encoding and phase encoding,

and then we will change back how they can be used to form an image.

The first viewpoint is why do we need

frequency encoding and phase encoding in the mathematical viewpoint of MR image.

So that is about the concept of K-space,

and we'll talk about this mathematically and also with the physical meaning.

We talked about the slice selection in the previous video lecture.

Now, we have selected a slice position of interest and the thickness to be excited.

So what slice selection do is that it's only magnetization in the slice is tipped,

excited or resonance by RF pulse and has

the transverse magnetization component which is now in

phase after applying for the slice refocusing gradient.

So as shown here, RF pulse and slice selection gradient, they are combined.

Then right after this,

we will have a lower signal intensity because all the proton spins excited.

Within that excited slice,

we will have different precession frequency along the G direction.

Then applying for a slice refocusing gradient will

make the excited speeds within the slice will have in-phase.

Then we will start to see some Free Induction Decay as shown here.

And then theoretically, we can acquire data during this period or

another analog-to-digital converter or sampling or readout all these terms

will be used in interchangeable way.

So generally, Free Induction Decay is not directly used for imaging and the reason is

FID at this point does not have a spatial information along within the excited play.

So it had a spacial information only the G direction or the excited slice direction

but it does not have spatial information in the in-plane directions.

So we need time for spatial encoding by using gradient course.

But there are some exceptions some of

the imaging techniques called Spiral imaging or Radial

imaging case we directly samples data right after the slice selection.

But that is a spatial imaging techniques and other than that the general

or the most popular imaging techniques that something we are going to

talk now and in this case we generally need time for

spatial encoding to get the in-plane information which is converted to image.

Okay. So let's consider right after the slice selection period.

So the FID signal can be mathematically modulated are represented as shown here.

So Free Induction Decay signal S(t) can be mathematically represented as a summation of

all the signals and h(x,y ) represent the spatial location information.

So that is the image that we want to get so h(x,y) so that is multiplied by e to

the minus j two pi f_0 t. So which means this

represent the spins precess in the Larmor frequency.

So the spins will be precess and then that is

induced in the RF coil with the frequency of Larmor frequency two pi

f_0 t. And that is going to be applied to all the protons pinned within

the space with no differences along x and y if there is no gradient.

So all the proton spins along special direction is going to have

sinusoidal signal and that is all summed together to form on Free Induction Decay signal.