[MUSIC] Hi, we will talk about the signal processing theory that is helpful to understand MRI fundamentals in this week. The purpose of this vid is to better understand how MR images are generated and processed, and you need to be familiar with by the concept of Fourier transform and sampling. So we will discuss concept of Fourier transform and the sampling in the viewpoint of MRI in this week. Let's talk about the Fourier transform and MR images. So what's Fourier transform? The Fourier transform decomposes a signal. It's typically a function of time into the frequency components that make it up, similarly to how a musical chord is expressed as the amplitude for loudness of its constituent notes. The Fourier transform of a signal is a complex valued function of frequency. Which absolute value represent the amount of that frequency component present in the original function. And whose complex arguments, phase angle is the phase offset of the basic sinusoid in that frequency. Okay, let's talk about that in detail. So this is mathematical definition of Fourier transform, so this f(t) is always known as time-domain number. And F(u) is frequency-domain, it's Fourier transform or it's frequency component. And then the definition of F(u) is integration of f(t), original function f(t) multiplied by e to minus t 2pi ut dt. So this is the definition of Fourier transform. So you can understand this equation as a multiplication of complex, sinusoidal signal, e2 minus t 2pi ut, which has now, with opposite polarity. So e2 minus t 2pi ut. So that is multiplied to the original function f(t). So if any sinusoidal component in f(t), that has the same frequency as e to the plus t 2pi ut, then they will cancel out, and then this integration will have a meaningful value, right, that will be converted to F(u). But if f(t) has different sinusoidal component with a different frequency then, it would j 2pi ut then, this multiplication we will have a component of oscillating component. And then, this integration will make that oscillating component to be 0 because oscillating signal will have positive and negative value, they cancel each other. So that is a very brief explanation of this Fourier transform. So just multiply the complex sinusoidal signal with opposite polarity to the original function and take integration, that gives the sinusoidal component of that function F(u). So this is the advantage of the Fourier transform, and the statistic region represent inverse Fourier transform. So, f(t) equals integration of F(u) e to the j 2pi ut du, now the integration is performed as a function of u. So in this case, this equation, inverse Fourier transform is very similar to the Fourier transform. The difference is this polarity here so e to j 2pi ut. So this equation, this is transforming the frequency function F(u) into the original time domain function f(t). So this function, inverse Fourier transform can be understood as the original signal f(t) is now represented as a summation, a linear summation of complex sinusoidal signal e to j 2pi ut, that is scaled by F(u). And that is linear result. So this inverse Fourier transform represent what is original signal f(t) as a linear combination of a complex sinusoidal signal. Okay, this is the concept of Fourier transform and the inverse Fourier transform. So, let's do some example. This is time domain signal where sinusoidal signal in the time domain is cosine t, then each Fourier transform will have only one frequency component. So that is omega component equals just one, so we will have on the frequency spectrum, that we will have only one component on the frequency of 1 or -1. And cosine function is now cosine 3t, then the frequency increased three times compared to the case A. And then, this Fourier transform of this function B will be very similar to the previous one but the frequency is now increased by three times compared to the previous one. And if we see another example with 2 cosine 3t. So now, the amplitude increases twice compared to the case B. And then, the frequency component remain the same but the amplitude increase twice, okay? So this example shows the signal itself is a sinusoidal signal, but many time domain signal may be represented as just summation of sinusoidal component. So decomposition of a signal into complex exponential signal, so complex sinusoidal signal is a Fourier can be considered as a Fourier transform. So let's talk about the Fourier transform in the two dimensional case because all that MR images we talked will two dimensional or three dimensional. So we have to talk about two dimensional Fourier transform. So two dimensional Fourier transform is very similar to the one dimensional case but it's just extended to another dimension. So now we will have signal in the image domain as f(x, y). And then this image domain sooner, can be transformed to the frequency domain by applying for two dimensional Fourier transform. And that is denoted as F(u, v) here. And then F(u, v) can be represented as integration of x, y original immediate domain signal. That is multiplied by complex sinusoidal signal, so equal to -j, 2pi ux plus vy, that is integrated as a problem dxdy. And that is the definition of two dimensional Fourier transform. And this is definition inverse Fourier transform. So f(x, y) equals integration of F(u, v), e to the j 2pi ux plus vy dudv. So the basic concept is very similar to the one dimensional Fourier transform case. So F(u,v) represents a complex sinusoidal signal in the original image domain signal. And then f(x,y) so inverse Fourier transform represent the original image domain signal as a linear combination of complex sinusoidal signals. So an image is generally formed by two independent variables, typically denoted by x and y. And two-dimensional Fourier transform is the same as a sequential application of a 1D Fourier transform for each of the two variables. So if you see this equation, you can consider this two-dimensional Fourier transform as a sequential one-dimensional Fourier transform as a function of x. And then applying as functional y sequential one dimensional Fourier transform can be considered as a two dimensional Fourier transform. So this is an example of a Fourier transform in the two dimensional case. So this is the case where the constant variables for all the images, there is no change. And then, we will have only very small dot in the middle, on the frequency domain. Because there's no frequency component, so that way, we'll see only DC component here in the middle. And then the case B is the case with sinusoidal signal along x direction but there's no change along y direction. And then each of two dimensional Fourier transform will keep two dots on the frequency domain. So in case of one dimensional, we represent it the particle the x and y represents the height, but in this case, we cannot represent that so we can represent the intensity as a brightness of the images. So please keep in mind that, so now this complex sinusoidal signal along x direction is changed through two dot okay, here and here, so which is similar to the one dimensional case that we just talked in the previous slide. And this is the case with sinusoidal signal applied along y direction but x direction is constant for both these y direction has higher frequency component than the case B. So in this case, we will observe two dots along y direction, but its frequency component is a little bit bigger than this frequency component, okay? So it's a little bit more away from the original signal. And then D, the case with sinusoidal signals are applied along both x and y direction and then, we can observe four dots as shown here. So because we have sinusoidal component along x and y. So the case E represent the case with only one sinusoidal component, but it's angle is oblique. So y direction has a little bit higher frequency component and x direction has a little bit lower frequency component. And then, so we can observe two dots, but x direction is slightly off the center and y direction is more off the center. So they're located along the oblique angle, and two dots with oblique angle, as shown here. And F is the function. This function is not sinusoidal, its sinusoidal signal divided by x, so sine x divided by x. So the function is called the sync function and we will talk about that in this week later. And this function is equal to sync function, it's sine x divided by x. And then that is two dimensional Fourier transform then its signal is rectangular, shape is rectangular along x direction. So we can see the brightness in the middle as a bar as shown here and we'll about that, this relationship later. So anyway, so you're going to change only along x direction so we can see the signal along this direction only. So this is the example of two dimensional Fourier transform and two dimensional Fourier transform describes how much of each frequency component is needed to produce the imaging, okay. That is about the Fourier transform. Okay, let's talk about the Fourier transform concept and MR images relationship. So let's say, the frequency domain as shown here, and images, as shown here. So one thing you may have to remember is that MRI is rather different from other medical imaging devices in that acquired data from the MRI scanner is not the images. But its frequency domain of the images which is called k space and we'll talk about that this concept, not in this week but much later. But all other medical imaging devices acquire the imaging domain as it is. But the MRI acquires data in the spatial frequency component, which is called case based. And applying for the two dimensional Fourier transform to the acquired MR images generate images. Okay, there is a huge difference of MRI from other imaging modalities. So anyway, so this is the acquired MRI data which is in the frequency domain compared to the images and then, this is original frequency domain data, okay? And then if we apply for two dimensional Fourier transform, then we can get full images as shown here. If we caught out the middle and then replace the part with just zeros. And then applying for two dimensional Fourier transform, and then what will happen? And then, the resulting imaging looks like this, for almost there is no contrast. Although we can see only some as shown here. But all right, what happen if we just take only the middle portion and then replace all other peripheral regions with zeros, and then what will happen? And then the resulting images, as shown here. So it's kind of certain contrast. But all the detailed edge information are just gone. So these figures explains the slow signal variation in space, in the imaging, so that corresponds to low frequency components in the center. And the fast signal change variation in space, image domain, corresponds to high frequency component, which is outer region, okay? So the center of frequency domain contains low frequency component information, so which determines image contrast as shown here, okay. And then while the peripheral parts or k-space, this one here contained a high-spatial-frequency components, in which affect spatial resolution, so as shown here. So like you find details and edges and ripples, okay. They are determined by the edge component. So you may see, this is original case image data so majority of the energy is located in the middle. So the middle part determines image contrast which is very important part.
