This is mathematical viewpoint of frequency and the phase encoding.

We talked about on opposite way,

and now we go through the mathematical viewpoint from

the pulse sequence starting from

pulse sequence frequency encoding and phase encoding and

what happens to the final MR image.

Without frequence and the phase encoding, the phase signal,

the phase intensity of the signal is going to be

integration of h(x,y) e to the minus j two pi,

f zero d, dxdy.

So this is not going to be a function of X and Y without gradient.

And with frequency encoding gradient,

now the frequency is going to change as gamma P_0 plus xgx.

And this portion, F is going to be changed as shown here.

And this baseband frequency component can be moved out of this integration and now h(x,

y) is multiplied by e two minus j two pi, gamma x_gx tx.

So this x_gx tx.

Delta x is now frequency or precession frequency is a function of x.

And phase encoding is going to be changed.

The phase component, so phi y,

is going to be induced to acquire the data as I just mentioned in the previous slide.

Induced to phase is going to be proportional to

the integral area of the gradient applied.

That is proportional to gamma head g_y,

y_t y and that is multiplied by two pi.

That is going to be induced phase on the acquired data which is a function of y.

And gy ty that determines integral area of the gradient.

The acquired signal is going to have a different phase to the acquired data.

So, h_e two minus j two pi gamma,

gx gx dx that is same,

and now additional phace is induced to the acquired data as shown here.

And this form corresponds and this portion can

be removed after demodulation, and after demodulation,

the acquired data corresponds to fuel transform of the signal h(x, y),

so now we can applying for two dimensional Fourier transform in

the viewpoint by replacing u and v. So gamma gx,

tx and gamma gy,

ty with the u and v. And then this is going to be exactly

corresponds to Fourier Transform equation for the original image h(x, y).

The acquired MR imaging is to 2D Fourier transform

or acquired data on the frequency domain called k-space.

Another viewpoint or frequency and phase encoding.

K-space has a sinc shaped data along

both the frequency and the phase encoding direction as shown here.

In the middle, the sonar is much brighter than the peripheral region,

and it looks like close to the sinc shaped,

so most of the energy are located in the middle.

This portion determines imaging contrast.

And also, the outside region,

the peripheral region determines

the high-frequency information which is edges on the spatial domain.

The frequency encoding gradient is applied with a fixed grade,

fixed strengths but with bearing time points to be sampled,

so that gradient, frequency encoding gradient and data samplings are combined together.

At each sampled point,

they have different time points on the frequency domain or frequency encoding gradient.

Gradient strength is fixed but the time points to be sampled vary as a function of time,

which modulates case-based position.

In contrast, phase encoding gradient is applied with

a fixed time duration but with varying strengths,

which also determines case-based location as shown here.

This explains why they are called frequency encoding and phase encoding.

Eventually, they do almost the same thing.

They just change the location on the frequency domain to be sampled,

which is determined as integral area of gradient

applied along x or y direction at the time point of that data sampling.

And the desired MR signal can be acquired by applying for a

two-dimensional Fourier transform to

the acquired data on the frequency domain called the K-space.

This is the summary of the forming MR image.

See you next week.