Transcription
Hello and welcome back. So we've spent two talks looking at exactly how we select a specific field of view, how we select a specific bandwidth, the range of frequencies along the x-axis of a particular slice, and how the combination of both the field of view and the bandwidth will also determine the gradient field strength required in order to generate that specific bandwidth over that specific field of view.
And we've seen that it's the bandwidth that determines the frequencies, or the range of frequencies, that we can accurately sample when we take an analog signal and convert that into a digital signal. And once we store those digital data values within k-space, can we then use those digital data values to accurately predict the range of frequencies that are within our slice? And I touched on briefly that if we have frequencies that are outside of that bandwidth, we get what's called aliasing, where we misrepresent an analog signal with a falsely calculated digital signal. And that leads to what's known as a wraparound or an aliasing artifact within our MRI imaging.
Now, today we're going to look specifically at what aliasing is, why aliasing occurs, and then we're going to touch on a few steps that we can take in order to reduce the amount of aliasing that happens within an image. So, say we have this image, for example, a specified field of view, and we specify the matrix that we want, the resolution that we want within our image. Now, the matrix size, the term, the number of phase encoding steps that we need to have within our pulse sequence, and it determines the number of data samples that we need to take during the data acquisition period whilst that frequency encoding gradient is on.
The number of samples we take determines the matrix size along the x-axis, and the number of phase encoding steps that occur within our pulse sequence determine the resolution in the y-axis, the number of pixels that we can accurately put within the y-axis of our image. Now, say we want to look at a specific region within this image and we want to get better resolution within this image here. What we want to do now is be able to determine two pieces of anatomy that lie within this specific pixel here.
Now, in order to do that, what we can do is keep the matrix size the same, keep the number of data points along the x-axis the same, and the number of phase encoding steps that we have in this particular matrix, but reduce our field of view. Now, as we reduce the field of view, you can see that the width of the matrix of each pixel within the matrix gets smaller. And because the dimensions of the pixels are smaller, we're getting better spatial resolution. The matrix size has stayed the same, but the field of view has reduced.
Now, as we reduce that field of view, we can see we are cutting out some regions that are going to be giving off signal. There are still hydrogen atoms within the flanks of this brain here that will be giving off signal that our receiver coil will be able to detect. Now, when we determine a specific bandwidth here and get a range of frequencies along the x-axis, as well as get a range of frequencies or complex spatial frequencies in the phase encoding direction, we are going to get signal that we are going to misrepresent that's outside of our field of view. That's going to show up in the image in the form of aliasing. So, see how the signal that was coming from the patient's right-hand side here has wrapped around and is being displayed on the image as a misrepresented piece of anatomy on the left-hand side of the patient in this part of the field of view. This is what's known as aliasing.
Now, why exactly does aliasing occur? Well, in the frequency encoding direction, there's a slightly different mechanism than that occurs in the phase encoding direction. So, we're going to look at both of those separately. The underlying principle, though, remains the same, and we're going to then look at how aliasing occurs via the conversion of an analog signal into a digital signal, regardless of whether that's coming from the phase encoding direction or the frequency encoding directions.
So, let's start by looking at the frequency encoding direction. As we apply a gradient field across the field of view that we've predetermined, and we select a specific bandwidth that determines the range of frequencies across our specified field of view, we are going to get different frequencies coming from the slice based on the x-axis location of the protons within that slice. And we see that as the gradient's field strength in the frequency encoding direction increases here from the null point, we get an increase in frequency of the signal that's coming from those x-axis locations. And then, as we take the null point and move to a decrease in relative frequencies, a relative frame of reference here, we get an increase in the rate of change of frequencies within this part of the slice.
And we can see that we can then use those frequencies to accurately represent where those frequencies are along the x-axis. And we've seen that it's the bandwidth that determines the maximum frequency that we want to be able to accurately detect and convert into a digital data set. And this frequency here is what's known as our Nyquist limit. And we've seen that the Nyquist limit determines the sampling rate, how quickly we actually have to sample this specific frequency in order to accurately convert that analog frequency into a digital frequency. And that sampling rate can then be used to calculate what's known as our sampling interval.
Now, if these are unfamiliar to you, go back to the previous talks and see how we calculate sampling rate, which just so happens to be the same as our bandwidth, and how we can use that value, in this case, 50,000 Hertz, to determine the sampling interval, how long we have to take each sample within that frequency encoding gradient. And this is also determined by the number of pixels or the matrix size along the x-axis of our slice.
Now, using these calculations, we're going to accurately be able to represent the signal that's coming from this part of the field of view. But we've seen that there are actually signals coming from outside of our field of view that have higher frequencies than the frequencies represented within our bandwidth here. We're going to be able to try and sample these frequencies, and as you'll see, we're going to inaccurately sample those frequencies because our sampling rate isn't fast enough, it doesn't reach the Nyquist limit, and we're going to misrepresent these frequencies and falsely put them in a different region based on its x-axis location. And that's going to cause a wraparound of the aliasing.
Now, in the phase encoding direction, the mechanism is slightly different. We've seen this representation of our slice here before, where we've got spins differentiated on both the x-axis location and their y-axis location. And if we separate the signal coming from the different y-axis locations here, we can see that when the signal is in phase, we've got our maximum signal coming from the slice, and all of the individual y-axis components are going to add up together with one another. They are completely in phase with one another.
Now, what we can do is, before applying our frequency encoding gradient, we can apply a phase encoding gradient. Now, the strength of that phase encoding gradient is going to determine how much dephasing occurs of the spins based on their y-axis location. And we can see now that the peaks of these waveforms here are no longer in line with one another. They have been dephased. We then apply the frequency encoding gradients, and the processing spins are going to process at different frequencies based on that frequency encoding gradients, but they've got this phase encoding baked into that signal. And it's the combination of this phase encoding gradient combined with this frequency encoding gradient when we read out the signal that is going to give us our complex signal. And that complex signal has a two-dimensional frequency complex signal within it that can be then Fourier transformed to figure out where exactly these signals are coming from based on this difference in phase.
If we were to look at a different phase encoding strength, but one that was sufficient enough to cause this spin at the distal portion or the far portion of our image along the y-axis to dephase a full 360 degrees, if we apply that phase encoding gradient, a much stronger phase encoding gradient, we can see that these spins have been a full 360-degree dephasing based on this gradient here. The adjacent spins, depending on how many phase encoding gradients we are going to apply, the number of pixels that we want to separate our y-axis on, there's going to be a certain phase encoding gradient that is going to mean adjacent spins based on y-axis location are completely out of phase with one another. You can see that these frequencies now, depending on y-axis location, are completely 180 degrees out of phase with one another.
If we were to apply any more phase encoding gradients, say we were to apply a phase encoding gradient that caused the spin not to dephase 360 degrees, but 370 degrees, effectively what has happened then is that spin has only dephased 10 degrees out of phase because we've got a full 360 degrees round and then we are gaining 10 more degrees. If you were to think of a spin lying outside of this slice and we were to look at this phase encoding gradient that we're applying here, the spins outside of the slice would have dephased more than 360 degrees, and they will then be measured as only having whatever over 360 degrees dephasing has occurred, and we're going to misrepresent that dephasing as somewhere else in our image. And that's what's causing the aliasing artifact within the phase encoding direction of our spins.
Now, in this example, we've only separated our y-axis into five separate planes. Obviously, in an image, that's going to be much greater, and there's going to be a phase encoding gradient where adjacent spins are completely 180 degrees out of phase with one another. That is the maximum phase encoding gradient that we can apply before getting aliasing in our image.
Now, how is that phase encoding gradient determined? It's determined by the field of view that we set in the phase encoding direction of our image, and it's determined by the difference in each sequential phase encoding gradients that we apply here. If we apply a specific strength of phase encoding gradients, we then reach TR, those spins start to relax, we flip them again in our second cycle at this 90-degree RF pulse, and we apply the next phase encoding gradient, a slightly stronger phase encoding gradient than the one before. The difference in that phase encoding gradient is what is known as delta k.
Now, the smaller the difference in that phase encoding gradients, the greater we can make the field of view in the phase encoding gradient direction. The larger that difference, the smaller the field of view that we can have in this y-axis because we are going to reach this limit quicker with larger differences between phase encoding gradients. We're going to reach this maximum phase encoding gradient here. Spins that experience a phase encoding gradient that is more than this will then experience aliasing.
Now, when we are looking at phase encoding gradient, it becomes quite difficult because we are dealing with a complex signal. We're dealing with a signal that is both a combination of the dephasing in phase encoding that we've applied with our phase encoding gradient, as well as the frequency encoding gradient that we're applying once we're reading out that analog signal with our set sampling interval and set number of data acquisition points during the frequency encoding direction. The combination of those two, depending on the phase encoding gradient strength, is going to give us a complex frequency 2D signal that can then be baked into k-space. And it's comparing separate lines of k-space that is going to then allow us to Fourier transform and determine where the signal is coming from along the y-axis.
This can be a difficult concept to grasp, and sometimes it's better to think of it in this way. If we were to apply a phase encoding gradient to the phase encoding direction here of our slice, the y-axis of our slice, spins in this red region here are going to dephase much less with each phase encoding gradient than spins right out at the edge of our image. With this phase encoding gradient here, the spins here are only going to experience a small amount of dephasing compared to those spins further along the y-axis. The rate of change of dephasing with each sequential phase encoding gradient is going to be much higher here, the rate of change between phase encoding gradients, than it is going to be here. And we can kind of represent that rate of change or phase as a frequency value.
Now, this is looking at it just in the phase encoding direction. We're not baking in that frequency encoding direction that allows us to get those two-dimensional frequency patterns that occur within the slice itself while we're reading out the analog signal for that specific slice. You can think of the rate of change as representing a frequency, and as we apply greater and greater phase encoding gradients, the rate of change in the phase is going to be higher than the rate of change here. And you can see that these signals are going to alias much before these signals here. If we think of each signal as being a slightly darker sine wave here, we can see we've gone from light to darker to darker to darker to darker. The next phase encoding gradient is going to equal our initial phase encoding gradient. We have reached the 360-degree dephasing, and any further phase encoding gradient is going to cause an aliased signal.
Now, when we look at aliasing, we can see it happens in the frequency encoding direction, like we've seen here, and this complex 2D frequency signal that comes out in our phase encoding gradient can also be aliased if it is outside of our field of view. And it's these higher frequencies that we are not able to measure and not able to accurately represent on our image. And we're going to have to use certain mechanisms to try and tease out or reduce that aliasing within the image, which is what we're going to look at today.
Now, let's look at the specific analog signal that is going to be coming out, the signal that's actually coming from our slice that's going to be detected by the receiver coils, and then we take that signal and we digitize it. We can't store an analog signal within the computer. We need to store that as discrete numerical values. So, we digitize that analog signal by sampling the analog signal very quickly. And as we sample it, we get a specific numerical value, and then we can look at the trends of those numerical values and estimate what is the frequency that we are measuring here.
Now, remember, we measure a complex signal from the entire slice, and we use a Fourier transformation to tease out these specific frequencies. And we can only determine those frequencies accurately enough if we're sampling that complex signal fast enough. So, let's take a look at this signal here that's coming from this location within our image. This could be either in the frequency encoding direction, like we're looking at here, or in the phase encoding direction. In both directions, we are looking at a frequency change. We see the analog signal that is coming out from this region here, and these dotted lines represent the times that we are digitizing that signal. And we can take those digitized signals and see the trends of the digitized signals and then make logical conclusions to connect those digital signals and estimate what the frequency is. And if we're sampling quick enough, we can see that we are going to get a very accurate representation of that frequency.
As that frequency gets faster, we are going to sample that frequency at the same rate, but we can see now that each wavelength has fewer samples on it because that frequency is getting quicker. We can then digitize those signals again, join those digital signals, and represent the frequency. In this case, accuracy, our sampling rate is high enough, then we can reach our Nyquist limit that's been set by our bandwidth in the frequency encoding direction, or it's been set by the field of view and the phase encoding strength along the phase encoding direction. We know this is the maximum frequency that we are able to detect with this specific sampling rate. We digitize that again, connect that signal, and we can accurately represent where that signal is coming from, be it on the x or the y-axis.
Once we have frequencies that are outside of our field of view, outside of the predetermined signal that we are trying to accurately measure that we've set with a combination of our field of view and either the bandwidth or the delta k, the amount of phase encoding that we are having in our y-axis direction, anything outside of those frequencies with our sampling rate, we are no longer going to be able to digitize that signal and accurately represent this frequency. We are going to misregister that frequency because the computer, the data set that we have entered into the computer, no longer accurately represents the frequency of the analog signal that we're trying to detect. We have now calculated a slower frequency, and it turns out that this slower frequency now actually matches up with frequencies at this part of our image. You can see how these signals now are going to be misrepresented and placed on this side of our image.
As that frequency gets even higher, we get further and further away from the null point. We can see that we get less and less accurate with our data acquisition. We digitize the samples coming from this signal, and we can see that the frequency that we've determined here is much lower than the actual frequency of that signal, and we are now representing this part of our image as here in our image. You can see how wraparound or aliasing is occurring. These signals we are falsely representing in our image as lying in different parts of our field of view. This is what's known as aliasing.
Now, obviously, we don't want aliasing to occur because the image we look at, we want to accurately represent the anatomy within the patient, and aliasing is going to prevent us from analyzing that image properly. So, how then do we go about reducing aliasing within an image? Now, the reason aliasing occurs is because we've reduced our field of view. Now, why we reduced our field of view? We want to effectively be able to look at a smaller part of our image but keep the same resolution. If we were just wanting to look at that part of the image, we could just zoom in on our previous image. We don't want to zoom in and lose that spatial resolution. We want to be able to look at a smaller part of our image, a smaller field of view, but still keep the resolution within that image.
Now, the first way that we could get rid of aliasing is by preventing tissue from being outside of our field of view. If you're taking an MRI of the thorax and the patient's got their arms next to their thorax, but you've set the field of view to just be their thorax, the arms are going to provide signal and alias into the image. So, you want your patient to lift their arms, to not have tissue outside of the field of view. In this case here, where we've got aliasing, it's because we've narrowed down that field of view. And if we just wanted to get rid of that aliasing, we could then just go and increase our field of view.
Now, what has happened now that we've increased our field of view? We've now created an image without aliasing, but we have stretched out those pixels. If we keep the number of data acquisition points during our frequency encoding direction the same, and if we keep the number of phase encoding steps the same, we've reduced our resolution because we've kept the same matrix size, but we've just increased our field of view. Now, that may be sufficient for what we want. Maybe we've taken an image and we've seen aliasing and say, "Oh, we've set the wrong field of view. Let's make the field of view bigger." But what happens if you want to look at a smaller portion of the brain here, but you don't want to lose this resolution?
Well, the first way that we can do it is by what's known as oversampling. Here, in this example, we've got a set number of data acquisition points within our frequency encoding direction. What happened if we were to increase the number of data acquisition points? That Fourier transformation now has more data points and allows us to plot more x-axis values. It allows us to then get an image whilst keeping our resolution, and this is what's known as oversampling.
Now, oversampling, taking more samples than we need for our specific field of view, will keep our resolution within that field of view, but it will prevent aliasing from happening because we are actually measuring these parts accurately because we've oversampled. And then when we've displayed our image, we've just taken out that data set and we've only displayed the field of view that we want. We still have accurately sampled this region, but we just don't include it on our image. We've got good resolution within this field of view that we want.
Now, it depends whether we are oversampling within our frequency encoding direction or whether we're oversampling within the phase encoding direction. Now, when we look at the frequency encoding direction, the number of samples here is determined by the number of samples that we take during our frequency encoding gradient, the number of data acquisition points. Remember that sample interval that we've calculated? If we want to increase the matrix size in the frequency encoding direction, we just need to take more samples during this frequency encoding gradient. It's not adding extra time to our pulse sequence. And in practice, what you'll actually see is that in the frequency encoding direction, we often oversample. Say we want an image that has a pixel value along the x-axis of only 256. In theory, the number of samples that we take during this frequency encoding gradient will only have to be 256 samples. But in many MRI machines, we'll be taking double or quadruple the number of samples. We're actually oversampling the frequency encoding direction because there's no real downside. We're not adding time to our pulse sequence. We're going to prevent aliasing because we're getting that signal, we're able to differentiate a much wider field of view, but then we only display the specific field of view that we have pre-selected. And that allows us to keep our matrix size within our field of view, but still have this signal not being aliased into our image.
Now, if this was the other way around, say we were getting aliasing within the phase encoding direction. Here, we've set the number of phase encoding steps that we want to take for this specific field of view that we've predetermined. We want this resolution within our y-axis here, but there is signal that is going to be coming from outside of that field of view, and that 2D complex frequency signal is going to then alias into our image. If we wanted to oversample in the phase encoding direction, what we would need to do is take this pulse sequence that has given us this specific field of view, and we would need to add extra phase encoding steps. Each phase encoding step allows us to add more data points or allows us to increase our matrix in the y-axis direction.
The problem with phase encoding is when we run our pulse sequence, we apply one phase encoding gradient, read out our sample, wait till TR, and then we go back and repeat the process with a different phase encoding gradient. In each phase encoding gradient is going to fill a different line of k-space, and we compare those lines in k-space. We compare the changes in phase based on the amount of phase encoding that has happened between the different lines in k-space to then tease out these y-axis values to where exactly that signal is coming from the y-axis.
Now, if we were to add phase encoding gradients to phase oversample, we are going to significantly increase the amount of time it takes to actually create this image. Now, we're going to look at ways to speed up generating images in a later talk, but for the moment, you can think of the total scan time as the length of time it takes from our first 90-degree RF pulse to the TR, the time of repetition, that period of time times by the number of phase encoding steps because each phase encoding step, we need to repeat another TR. And there's a value that we still need to multiply in called the number of excitations or the number of signal averages. It turns out for each slice, we don't only create that image once, we do that multiple different times, combine those images, average out the signal to give us better signal and to reduce noise. That's something we're going to look at later on. But phase oversampling comes at a cost because it significantly increases the total scan time, and it often increases those times to times that are not sustainable in a busy radiology department. So, this often isn't a viable step for us.
Now, we've seen that we can first increase our field of view size to reduce aliasing. We can oversample either in the frequency encoding direction, which is often done, or in the phase encoding direction, but that has a drawback of adding extra time to our scan. The third mechanism for reducing aliasing is we can change our phase and frequency encoding directions around. Because the time-dependent portion of our scan is the phase encoding direction, we want the shortest axis of our image to be the phase encoding direction. So, we can still get good resolution along that short axis without taking too much time and keep our frequency encoding gradient along the long axis of our scan because the frequency encoding gradient, the number of samples that we take there, doesn't increase the scan time. We can do that during the frequency encoding gradients.
Now, there are reasons why we can't do it this way around. We're going to look in our next talk at something called chemical shift artifact that only occurs in the frequency encoding direction, and maybe we can't have the frequency encoding direction along the long axis of our scan. And there are certain motion artifacts or flow void artifacts that depend on the orientation of our scan. So, not always can we make the phase encoding direction the short axis of our scan, but when we can, we should make that short axis the phase encoding direction because that may reduce aliasing. You can see how when we turn this image here, we no longer had aliasing in the phase encoding direction because the short axis fit within our field of view.
Now, the last step that I'm going to talk about today for reducing aliasing is what's known as parallel Imaging. You can see that if we were to take this image here that we've looked at, we are going to alias these regions of the scan if we don't phase oversample here. Now, we're going to look at a technique that allows us to not phase oversample but still allows us to create an image without aliasing, and that's what's known as parallel Imaging.
If we were to determine this specific field of view, but have signal coming from this larger field of view here, and then set the number of phase encoding steps that were required at specific delta k values, at specific phase encoding gradient values, to accurately represent this path, this field of view, and then created an image from it, we would get an image that looked like this: an aliased image. We would get a wraparound of the anterior part of the brain, this part here, on the posterior part of our image, and the posterior part of the brain on the anterior part of our image. This is aliasing happening in the phase encoding direction.
Now, when we are creating these images, we're thinking of the signal coming and being received by one receiver coil, and that receiver coil is measuring the entire signal from the entire slice, and it's that entire signal that we eventually convert into data points along our k-space. Now, when we look at parallel Imaging, we have two receiver coils, separate receiver coils, and these receiver coils are going to receive different signal intensities from different regions within our scan here. If I held a compass in my hand and a magnet in my other hand, and I was moving the magnet here, there would be changes on the compass. The changes on the compass would be much more drastic if the magnet was right next to the compass here. The same happens in this image. Signal coming from this part of the scan is going to induce much greater changes in coil 1 than signal coming from this part of the scan. There's going to be an intensity differential based on how far away the signal is coming from from the coil, and those intensity differentials are going to be different for coil 1 and coil 2.
So, if we were to look at this scan and take two separate points, we have two points that have the same x-axis location in our image but different y-axis locations. And it turns out that when this image aliases, those two locations are going to appear at exactly the same location here. Now, when we look at the signal coming from this part of the image, it is a combination of both the yellow and the pink signal coming from our main image. Now, if we were to get this aliased image from coil 1, the signal coming from the yellow region is going to be much more intense than the signal coming from the pink region because the pink is further away from coil 1. And the combination of these two signals from our aliased signal in coil 1 is going to give us a data value.
If we were to look at the same aliased image but from coil 2, the signal coming from the pink region on our image will be higher than the signal coming from the yellow region on our image. And using those two separate images, we can use linear algebra and use a simultaneous equation to figure out where exactly those signals are coming from based on the differentials between those two coils. So, let's look at coil 1 and coil 2 here. We've got coil 1 with the large field of view containing all of the anatomy that we are imaging, and coil 2 with the same large field of view. See how signal is higher closer to coil 1 and peaks out as it gets further away from coil 1. These are known changes from the specific coil. We know how signal changes as it heads away from this coil. The same happens from coil 2, but in reverse. They're parallel to one another, they're opposite to one another, and we get signal intensity changing based on y-axis location here.
Now, because we are still only taking this number of phase encoding gradient directions, half the number than we needed if we were to create the entire image here, what we're going to get is aliased images, one being represented from coil 1 and one being made up from coil 2. Now, you can see that the signal coming from this aliased part in coil 1 is actually more intense on this part of the image than the same image that has been made from coil 2. That's because this aliased signal is actually coming from the anterior part of the scan. It's just been misrepresented because of that miscalculation of the frequency. So, the signal is high here because it was closer to coil 1. In coil 2, that signal that is here was coming from further away from coil 2, and you can see now that these signals are different here.
Now, for each and every point within our image, we have two variables. We've got two known variables: the way that signal changes as it moves away from each of the coils. We know those two variables, those are known signal changes. And we've got two unknown variables: the signal coming from the aliased part of our image, and the signal coming from the actual part of the image that truly lies within that field of view. We've got two separate equations here with two unknown variables. We can use linear algebra, use a simultaneous equation, then to tease out, to mathematically tease out, which part of the signal is actually aliased signal and which part is actually true signal coming from within our field of view. And that's the crux of parallel Imaging.
Now, why would we do this? Well, we've now been able to take these two images and combine them and create an image here that truly represents the anatomy within the large field of view. We have only had to take half the number of phase encoding steps, though. We created an image with a set number of phase encoding steps that would only phase encode for half the field of view. Now, to create that half the field of view was twice as quick as it would have been if we were to phase oversample here. We didn't have to do those extra phase encoding steps. We created an aliased image, but because we had two receiver coils and known signal intensity changes, we were able then to unfold those aliased images. We have still maintained the resolution, though, that was in our initial field of view. We haven't stretched out the y-axis. We've calculated those aliased parts with good resolution. So, we phase sampled only half of the field of view, but we've effectively gotten double the number of phase encoding steps, and we've kept the resolution in the y-axis.
Now, this does come at a slight cost because we have actually only phase encoded half of that field of view. We are getting a slightly reduced signal, and it turns out we will get a reduction in the signal-to-noise ratio by a square root of that speeding up. We've in fact only sampled that specific field of view for half the amount of time. So, obviously, there's going to be a reduction in the amount of signal that we actually detect.
Now, there are other mechanisms to reduce aliasing, but I've covered the main four here. We can increase the field of view or move tissues that are outside of the field of view away from that slice. We can oversample in the frequency encoding direction, which is commonly done. And we can oversample in the phase encoding direction if we have the time. We can change the frequency and phase encoding directions to allow for the phase encoding direction to be on the short axis of our slice. And we can use parallel Imaging to reduce the number of phase encoding steps that we require and then combine the two separate aliased images to form a single image that has better resolution in the y-axis direction.
Now, you may hear of terms like SENSE or ASSET when it comes to parallel Imaging, and that's taking two separate images like we've looked at now and unfolding them using this parallel Imaging. You may hear of a term called GRAPPA, and that's where you take the actual k-space data without creating the images themselves and then unfold it, reducing the aliasing and creating an image like this. Those are slight technicalities. They all work on the same basis of using two receiver coils to then go about creating two aliased images, whether it be within k-space or actual images that we've calculated, and then we can compare those two data sets, or those two images, to unfold those images and get rid of aliasing.
Now, aliasing comes up over and over again in exams. I haven't seen it asked in a huge amount of detail, and I've tried to keep this at a good conceptual understanding level, not to get too far into the mathematics behind how all of this occurs. What's really important is to understand the concepts of why aliasing occurs, that misrepresentation of those higher frequencies, and then what we can do to reduce aliasing within our image. So, that's all for this talk. In the next talk, we're going to be looking at a different type of artifact known as chemical shift artifact. So, until that talk, I'll see you there. Goodbye, everybody.