We will talk about the concept of a field of view in this video lecture. So, let me try to remind you of the concept of sampling again. So, we have a signal, one dimensional signal, and this is impulse stream for the sampling. Then sample the signal, we'll have spectrum slightly different from the previous one. So, this is the spectrum and this is the original signal intensity on the spatial domain. So let's say this is frequency distribution of this signal, original signal, and this is sampling for impulse stream. So, its Fourier transform is another impulse stream with longer distance than the previous impulse stream. Then these two sampling means, there's convolution between these two signals, which means, the original spectrum is going to be repeated on the frequency domain. If there is aliasing between these spectrums, and we cannot recover the signal completely. But if this sampling frequency is high enough, so that sampling frequency determines the distance between these two events. So, that means distance between these two repeated spectrum, this repeated spectrum distance is longer, bigger than the maximum bandwidth or twice the maximum bandwidth of the signal or the full range of the spectrum. One side represents maximum frequency or bandwidth of the signal. So, the full range is twice the bandwidth of the signal. So, if twice the bandwidth of the signal is smaller than the sampling frequency which determines the distance between these two spectrum, then there will be no aliasing. So, that is the concept of micro Sampling Theorem. The sampling frequency, the distance between the repeated spectrum should be bigger than the twice the maximum bandwidth of the signal or the full range of the spectrum. So, we can apply for low pass filter to the sampled version of the signal to recover the original signal completely. So, this low pass filter bandwidth or antialiasing filter bandwidth or placebo bandwidth, and the sampling frequency are different, but often in MR imaging they represent almost the same thing, considered the same. Sometime they are used interchangeable way. They present definitely different things, but they sometimes are used to represent the same thing. Let's consider the sampling in MRI. The MR imaging is a procedure that samples for the K-space and analog to digital converter, so ADC, performs the digitization of the signal. The acquired data is stored as a form of K-space which is frequency domain of those images. For instance, this is an echo signal, we have sampled about sampling and this interval is called sampling frequency, involves all the intervals in the sampling frequency. This is going to be measured signal. So, here the T is the sampling time for N points. So, total point from here to here is denoted as T. Sampling time for N points, N is number of sampling points and delta T is sampling time per point. So, the relationship between these three variables that you want to be T equals N delta T. Here the sampling rate is going to be one over delta t. The sampling frequency is defined as the rate at which the signal is sampled and digitized, and antialising filter in ADC is receiver bandwidth is typically one over sampling rate per one over two sampling frequency, and four minus one over F_s port two to F_s port two. So, F_s represent sampling rate or sampling frequency. Total number of samples taken is determined by the number of voxels desired in the frequency encoding direction, and that determines the number of points or also that determines resolution of the images for the given field or view. So, these typical range is 64 to 512 in MR imaging. Let's consider the concept of a field of view here. So, we have readout gradient which assigns different frequency along spatial direction. So, this particular direction represent frequency which is proportional to magnetic field or precession frequency, they can be interchangeable, used can be interchangeable way. So, this readout frequency range is related to readout frequency range, receiver bandwidth, this frequency range is related to the field of view of the imaging. So, receiver frequencies is on one side of readout direction can be considered a positive, as shown here, and those on the other side can be considered negative. Then high sampling rate means high receiver bandwidth. If we increase this receiver bandwidth, then what will happen? If we increase this, then field of view will get bigger, if its gradient strength is maintained. Then if this readout bandwidth becomes higher, and then field of view will increase or if the field of view is fixed, and the frequency gets increased, then what should change? The gradient should be steeper, it should be steeper to maintain the same field of view. These three parameters affect each other. But again, this MRI operator determines field of view. So, based on the field of view given by the operator, the sequence program it automatically calculate the necessary gradient strength. So, when gradient is fixed, the sampling frequency determines the size of the imageview along the readout direction FOV_x, we just mentioned. So, that is it can be derived based on the Larmor equation. Omega equals gamma b or f equals gamma-hat b. Same thing. Then based on this equation, this B field is the gradient multiplied by spatial domain length which is a field of view X. So, field of view X is multiplied by the gradient strengths and that determines frequency range. And that is multiplied by gamma-hat and then that determined frequency range, which is obvious based on the Larmor equation. Or it can be represented in a slightly different manner. So, FOV_x in the viewpoint FOV_x that exactly F_s over gamma-hat G_x, and then that is the same as one over gamma-hat G_x multiplied by delta t. The sampling frequency is inverse of delta T. So, it can be moved to the denominator as shown here. So, FOV_x equals one over gamma-hat G_x delta T. Here G_x is a readout gradient strength as shown in the viewpoint of sequence diagram. G_x is readout gradient strength, the height of the gradient, and delta T is sampling period. So, this portion represent that the area of one sampling determines the whole range of the image view which is field of view. Let me try to describe that in some sentences. So, sampling along the frequency encoding direction is accompanied by the readout gradient, G_x and the sampling frequency of the interval determines the size of the image view along the readout direction. So, again, it can be represented in these equations as mentioned in the previous slide. The denominator represent a step on the K-space in the horizontal axis of the K-space. So, one step, one delta K_x is inverse of the field of view. So, frequency domain, K-space domain and the image domain, they have inverse relationship. Maximum view on the image domain, which is a field of view that is proportional to the one step size on the K-space. So, FOV in the frequency encoding direction is inverse of the step size in the horizontal direction or the K-space. So, let's try and compare between K-space and imaging. So, one step size is delta K_x, so that determines immediate field of view, whole range for the image. Actually opposite is also true, the whole K-space range determines one pixel size. So, that is the concept of a major loser. Opposite is also true and that is going to be the concept we talk in the next video lecture. So, let me try to summarize again. Sampling along the y direction is accompanied by phase encoding gradient, G_y. The same thing for the phase encoding direction is the same thing as the frequency encoding duration, and delta A_y is step size in phase encoding gradient area, which is one step size on the K-space. Delta K_y that is inversely proportional to field view along the y direction. So, FOV in the phase coding direction is also the inverse of the step size in the vertical direction of the K-space. Same thing is also true for the field of view along the phase encoding direction.
