[MUSIC] Let's talk about the concept of sampling here. So what is sampling? And then we have two different signals, one is continuous signal and the other is discrete signals. So continuous signal, many signals that we want to measure is just a continuous signal, in many cases. But we have to, to measure the signal, should be stored in the computer. So most cases, they should be saved as a discrete signal, so as shown here. So we have to consider converting the continuous signal into discrete signal requires the concept oversampling. So which is digitization, as shown here. So if you see any signal on the computer that looks like continuous, in fact, they have some. If any signal is displayed in the computer, then that is already digitized because the sampling is so high, it looks like continuous. But all the signals that we can see and analyzed on the computer should be digitized in their discretion. So I'm trying to remind you the concept of point impulse, unit impulse function or Delta function with the same slide. I'm not going to go through details again, but this data function is typically used to describe the concept of sampling. And it's typically denoted as the arrow as shown here and we put the number, so that represent to the area of that delta function, okay. So let's consider a shifted impulse and the concept of sampling. So if we shift delta function in the time domain, then, let's consider delta t minus tau. So this function is shifted delta function, and this function is shifted along, in the amount of tau along time domain as shown here, and the area is still one. And we represent that as an arrow as shown here, and put the number which represents the area of this function. And then, we can multiply this shifted impulse with any arbitrary function, like x(t). If we multiply these two functions, and then what will happen? So, x(t) multiplied by delta t minus tau is going to be the same as x tau, delta t minus tau. So now the function x(t) is replaced by x tau and the reason is delta function has all the variables are if a t is different from tau, as shown here. But that is the definition of delta function. So, x(t) multiplied delta t minus tau is going to be the same as x(tau), delta t minus tau. So now the area is nought 1, and that is scaled by x(tau) as shown here. So this multiplication is just picking up a certain value of the signal from the signal x(t). So this property is called sampling property, so multiply shifted impulse to the original function that we want to analyze. And that is the concept with sampling property, picking up certain value. If we take integration to this sampled function, and then we can get the signal x tau, okay. So they're just picking up a value at a specific point, t equals tau. So this is concept of sampling. And then what is sampling? Let's talk about that conceptually. A sampling is to electronically store and process continuous signals using computers, and we must transform them into collections of numbers. The continuous signal, and sampling, and then discrete signal. Okay, this is one dimensional sampling function. So delta s and function is X and spacing is delta X. And then this sampling function is represented as a linear summation of shifted delta functions as shown here, okay? And these shifted delta functions, shifted delta functions which we called the sampling function. So that is multiplied to the original signal f(x) and then f(x) multiplied by this sampling function can be considered as a discrete signal, discrete version of the original signal f(x). So we can represent this discrete signal, but now we are trying to represent this discrete signal in the continuous domain, with variable X. That can be mathematically represented as a multiplication of the sampling function with the original function, so that represent discrete the signal in the continuous with a continuous variable X. Okay, so this sampling function also can be extended to the two dimensional case, which is applicable to MR imaging. So sampling in 2D is the picking off values on a grid or matrix of points, and typical sampling points of MRI are 64 by 64 to 512 by 512. So two dimensional sampling function can be represented as delta S x, y, delta x, delta y. So here are delta x and delta y is spacing between the sampled points. So it can be mathematically represented as a linear summation of these shifted impulses. So delta( x- m delta x, y- n delta y). So this sampling function can be applied to the original signal. So original signal is actually our brain, as shown here, which is definitely continuous. But the detected MR images, to measure the MR images, is not going to be continuous. It cannot be continuous, but we have to sample the data at a certain point. So we can sample densely, as shown here, or as partially, as shown here. So here, f (x,y) is 2D continuous unit. That's the desired images that we want to get. These are the structure of our brain, okay? Fs (x, y) delta x, delta y. This is the sample of two dimensional signal that we measure using MRI. And delta x and delta y are sampling periods and 1 over delta x and 1 over delta y is going to be sampling frequencies. So fewer samples, if delta x and delta y becomes bigger. So spacing between the sampling points are bigger and then the resolution is going to be lower. And sampling frequency becomes lower and this calls, there is advantage and disadvantage. So the disadvantage is the partial volume of effects gets bigger. So the spatial resolution gets lower, but because each pixel has a bigger effect, so the SNR gets higher. If we change this opposite direction, then this advantage and disadvantage is going to move in the opposite direction. So let's consider the sampling property in the continuous signal, so continuous domain. So fs(x, y) is going to be multiplication of original f(x, y), so our brain, okay, that is multiplied by a sampling function. Mathematically, that can be represented as shown here. So this is definition of this sampling function. And then, this portion can be, x and y can be replaced by m delta x, n delta y, based on the sampling property, okay? Because this property of this shifted impulse is that all the remaining portion. If x = n delta x, y = n delta y, then there is called the meaningful values. Otherwise, all of them are zeros, so just this portion can be replaced by this constant, x and y can be replaced by these constant variables. So this is sampling property like we just mentioned, okay. That is multiplied by the shifted impulses, and what this means, okay? This means picking up some barriers at that specific point, m, n. And this is this tight portion of the sigma, and this tight m, n. So given discrete signal ft(m,n), so we can calculate the continuous signal f then for the version fs(x, y). So again, this fs(x, y) is presenting the discretized the signals in the continuous domain x, y, okay? And then, so this conversion is obvious, as shown here. And then, the goal is to represent original brain structure, f(x,y), mathematically. And the mathematical question is, can we reconstruct the original signal f(x,y), okay, from its sampled version, fs(x,y)? Okay, fs(x, y) is going to be the one we get from MRI, so the one you get from MRI is this portion, okay sampled version. And then that can be mathematically represented in the continuous domain, f(x, y) based on this mathematical formula. And then, the ultimate goal is to reconstruct original brain images. And then, can we reconstruct original f(x,y) for its sampled version, okay? So that is important concept for us to achieve. Okay, we will talk about that in the next video lecture.