Okay. Let's talk about the last step, the phase encoding. So far, we talked about the slice selection and also frequency encoding which generate a signal called echo, and that can be used to fill a one k-space line along x direction called the kx direction on the frequency domain called K-space. The one last direction is along y direction, and the goal is to fill the K-space for the whole K-space region. To do that, we have to fill one. We can fill one K-space line by using frequency encoding. The goal is to repeat this frequency encoding at different location along y direction. That is the goal. How can we do that? As I mentioned in the previous video lectures, the K-space location or frequency domain location is determined by the integral area of the gradient pulsing. To change the location of this frequency encoding lines along y direction, that can be performed by applying for a gradient pulsing before data sampling. That is about the phase encoding. This can be performed, as I mentioned, just adjusting the height of the horizontal line to be filled in K-space. That can be done in the phase encoding, by called the phase encoding. Again, the gradient strength and timing, so integral area determines K-space location. That is the same thing for the y direction, too. To fill all K-space as shown here, we have to repeat this procedure. This means, this whole procedure from here to here, so that is continuously repeated, we have to keep repeating. The same process as you keep repeated, but with different gradient area as shown here, gradient strengths but with a fixed timing and that changes integral area of gradient along y-direction before data sampling. Before data sampling, this gradient changed, area changed from plus to minus, and then that will change the original location. This pre-phase gradient as I mentioned, pre-phasing gradient will move the frequency domain location all the way to the left. And this gradient, vision coordinating gradient will change the location along Y-direction. That determines the start point of this data sampling on the frequency domain and that can be changed here, here, and here. By changing this gradient and repeating this data location, and this repetition of this data location will complete the cumulative fill this K-space frequency domain data. This encoding scheme is called phase encoding. Now, we've completed pulse sequence diagram of gradient echo sequence. Again, this part is a slice selection and slightly focusing gradient, and frequency encoding gradient. This is pre-phasing gradient, and this redox gradients and this portion is combined with the data sampling called analog-to-digital converting to sampling. And before this sampling, we are flying for another gradient along the other direction, so in this case Y-direction. And that will change the location of frequency domain along Y-direction as shown here. By repeating this procedure with a different gradient strength, this represent repeating this procedure with a different intensity or a different integral area. And then we can completely fill the frequency domain signal. And then what would be the next and then offline for Fourier transform to the acquired data will provide MR imaging. Let's consider phase encoding in another perspective. This is pulse sequence diagram as I showed in the previous slide. Light episode of slightly focusing gradient, all the spins are in phase. And then gradient makes the precession slower or faster on the spatial domain and resulting in phase difference along Y-direction. If applying for phase encoding gradient along Y-direction means some of the spins. Right at the point of number one, all the proton spins are in-phase along Y-direction, and applying for gradient along Y-direction will make some of the proton spins rotate slower or faster, and in the middle it remained the same. That's how it would rotate faster or slower. And then, we start applying for gradient, and then acquiring data at the non-point point three. At this point, all the spins have different phase along Y- direction, and that is used for data Kaist. Because of that phase along the Y-directions are different on the acquired data, and that's the reason why it is called the phase encoding. And that is used to get a spatial information along the Y-direction. This is another perspective of phase encoding in the viewpoint of spatial domain. 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.