0:10

So, let me try to remind you of the concept of sampling again.

So, we have a signal, one dimensional signal,

and this is impulse stream for the sampling.

Then sample the signal,

we'll have spectrum slightly different from the previous one.

So, this is the spectrum and this is the original signal intensity on the spatial domain.

So let's say this is frequency distribution of this signal,

original signal, and this is sampling for impulse stream.

So, its Fourier transform is another impulse stream

with longer distance than the previous impulse stream.

Then these two sampling means,

there's convolution between these two signals,

which means, the original spectrum is going to be repeated on the frequency domain.

If there is aliasing between these spectrums,

and we cannot recover the signal completely.

But if this sampling frequency is high enough,

so that sampling frequency determines the distance between these two events.

So, that means distance between these two repeated spectrum,

this repeated spectrum distance is longer,

bigger than the maximum bandwidth or

twice the maximum bandwidth of the signal or the full range of the spectrum.

One side represents maximum frequency or bandwidth of the signal.

So, the full range is twice the bandwidth of the signal.

So, if twice the bandwidth of the signal

is smaller than the sampling frequency

which determines the distance between these two spectrum,

then there will be no aliasing.

So, that is the concept of micro Sampling Theorem.

The sampling frequency, the distance between the repeated spectrum should

be bigger than the twice the maximum bandwidth

of the signal or the full range of the spectrum.

So, we can apply for low pass filter to the sampled version

of the signal to recover the original signal completely.

So, this low pass filter bandwidth or antialiasing filter bandwidth or placebo bandwidth,

and the sampling frequency are different,

but often in MR imaging they represent almost the same thing, considered the same.

Sometime they are used interchangeable way.

They present definitely different things,

but they sometimes are used to represent the same thing.

3:11

Let's consider the sampling in MRI.

The MR imaging is a procedure that

samples for the K-space and analog to digital converter,

so ADC, performs the digitization of the signal.

The acquired data is stored as

a form of K-space which is frequency domain of those images.

For instance, this is an echo signal,

we have sampled about sampling and this interval is called sampling frequency,

involves all the intervals in the sampling frequency.

This is going to be measured signal.

So, here the T is the sampling time for N points.

So, total point from here to here is denoted as T. Sampling time for N points,

N is number of sampling points and delta T is sampling time per point.

So, the relationship between these three variables that you want to be T

equals N delta T. Here the sampling rate is

going to be one over delta t.

The sampling frequency is defined as

the rate at which the signal is sampled and digitized,

and antialising filter in ADC is receiver bandwidth is

typically one over sampling rate per one over two sampling frequency,

and four minus one over F_s port two to F_s port two.

So, F_s represent sampling rate or sampling frequency.

5:21

Let's consider the concept of a field of view here.

So, we have readout gradient which assigns different frequency along spatial direction.

So, this particular direction represent frequency

which is proportional to magnetic field or precession frequency,

they can be interchangeable,

used can be interchangeable way.

So, this readout frequency range is related to readout frequency range,

receiver bandwidth, this frequency range is related to the field of view of the imaging.

So, receiver frequencies is on one side of

readout direction can be considered a positive,

as shown here, and those on the other side can be considered negative.

Then high sampling rate means high receiver bandwidth.

If we increase this receiver bandwidth, then what will happen?

If we increase this,

then field of view will get bigger,

if its gradient strength is maintained.

Then if this readout bandwidth becomes higher,

and then field of view will increase or if the field of view is fixed,

7:28

So, when gradient is fixed,

the sampling frequency determines the size of the imageview

along the readout direction FOV_x, we just mentioned.

So, that is it can be derived based on the Larmor equation.

Omega equals gamma b or f equals gamma-hat b.

Same thing. Then based on this equation,

this B field is

the gradient multiplied by spatial domain length which is a field of view X.

So, field of view X is

8:16

And that is multiplied by gamma-hat and then that determined frequency range,

which is obvious based on the Larmor equation.

Or it can be represented in a slightly different manner.

So, FOV_x in the viewpoint FOV_x

that exactly F_s over gamma-hat G_x,

and then that is the same as one over gamma-hat G_x multiplied

by delta t. The sampling frequency is inverse of delta T. So,

it can be moved to the denominator as shown here.

So, FOV_x equals one over gamma-hat G_x delta T. Here G_x is

a readout gradient strength as shown in the viewpoint of sequence diagram.

G_x is readout gradient strength,

the height of the gradient,

and delta T is sampling period.

So, this portion represent that the area of

one sampling determines the whole range of the image view which is field of view.

9:37

Let me try to describe that in some sentences.

So, sampling along the frequency encoding direction

is accompanied by the readout gradient,

G_x and the sampling frequency of the interval

determines the size of the image view along the readout direction.

So, again, it can be represented in these equations as mentioned in the previous slide.

The denominator represent a step on the K-space in the horizontal axis of the K-space.

10:09

So, one step, one delta K_x is inverse of the field of view.

So, frequency domain, K-space domain and the image domain,

they have inverse relationship.

Maximum view on the image domain,

which is a field of view that is proportional to the one step size on the K-space.

So, FOV in the frequency encoding direction is

inverse of the step size in the horizontal direction or the K-space.

10:41

So, let's try and compare between K-space and imaging.

So, one step size is delta K_x,

so that determines immediate field of view,

whole range for the image.

Actually opposite is also true,

the whole K-space range determines one pixel size.

So, that is the concept of a major loser.

Opposite is also true and that is going to

be the concept we talk in the next video lecture.

So, let me try to summarize again.

Sampling along the y direction is accompanied by phase encoding gradient, G_y.

The same thing for the phase encoding

direction is the same thing as the frequency encoding duration,

and delta A_y is step size in phase encoding gradient area,

which is one step size on the K-space.

Delta K_y that is inversely proportional to field view along the y direction.

So, FOV in the phase coding direction is also

the inverse of the step size in the vertical direction of the K-space.

Same thing is also true for the field of view along the phase encoding direction.