Preface
K-space is the core of MRI image reconstruction. Like the author, many people may wonder why the k-space is the frequency domain space of the image. Looking at the filling process, it is obviously the sampling and filling of the spatial signal. There are very few articles discussing this issue on the Internet, so I write down my understanding here for your reference.
MRI position coding
The current MRIscan is one fault or multiple faults. The so-called fault is a thick surface, which zis selected by the direction gradient field. The purpose of scanning a fault is to obtain the 各位置proton status of the fault . They carry information about the properties of the tissue, and what they reflect is the signal strength. If the entire fault uses the same magnetic field strength, and we can only get the total signal of one fault, it is obviously impossible to distinguish which signal comes from where, and what we get is just a piece of chaos. Only differences can be distinguished. The so-called position coding (frequency coding and phase coding) is used to make this difference. By adding xand ygradient field direction, so that the magnetic field strength of each of the fault location are different (although both are not simultaneously added, but the final effect together so that the magnetic field intensity is different for each position), a w=γBdifferent field The spin frequency is different, so that the total magnetic moment of the proton at each position can accumulate different phases, and finally the phase accumulation becomes the basis for separating the position signals in the total signal. As shown in Figure 1 (modified from the PPT of teacher Jia Guang of Xidian ), when the x、ydirection is independently applied with the gradient field, each position only taccumulates the phase situation under the action time of the gradient field magnetic field (without considering the main magnetic field) .
Due to the nature of the computer, the spatial coordinates are discrete, and the fault has a thickness, so each position is a voxel determined by (x, y). In this way, the above position refers to the voxel here. In the process of signal generation and acquisition, the gradient field in the y direction is added for a period of time, and then the gradient field in the x direction is added for a period of time to realize the position encoding. It is easy to know that each voxel has a certain effect on the ydirection and the xdirection gradient field. The phase accumulation after time is the sum of the phase accumulations acting on the same time in two directions. Now we assume that the signal intensity of a voxel (x, y) is f(x, y), and it is a sinusoidal signal, then the signal s(x, y) generated by the voxel can be used with amplitude and gradient The phase caused by the field is expressed by equation (1):
Since the signal collected each time is the superposition of the voxel signals, the collected signal can be given by equation (2). It is easy to see that γ and G are constants, and x and y are integrated, so the total signal is not a function of x and y, but can only be a function of the duration t of the gradient field in the x and y directions.
k space
K-space is the core of magnetic resonance image reconstruction and is a genius invention. Through it, the intensity distribution f(x,y) of each voxel signal in the slice can be quickly obtained, that is, the nuclear magnetic resonance image. Observing the expression of the collected fault signal s, that is, equation (2), it can be found that it is very similar to the two-dimensional Fourier expression in equation (3), so a genius scientist thought of using Fourier transform to process this problem.
At this time, if formula (2) is used,
then formula (2) can be transformed into formula (4). It is easy to find that this formula is the intensity distribution of each voxel signal in the fault f(x, y), Fourier transform to two-dimensional The expression of the frequency domain signal. If the total signal value s obtained by each sampling, the value S obtained after simple variable substitution, is filled into the storage location in the specified order, then a space will be formed after filling, since the value in it is a function of k, it is called It is k-space. Obviously, it must be the frequency spectrum of f(x,y). (Such variable substitution can be done directly through hardware)
If we get the frequency spectrum (that is, the k-space is filled), after one inverse Fourier transform, f(x,y) is the intensity distribution, that is, the nuclear magnetic resonance image. The expression is as (5)
Note: In the process of k-space filling, kx is changed by sampling time tx. Once sampled, tx increases by Δt, and kx also increases by Δkx. This is why a group of signal values are sampled to fill one row of k-space, and ky is changed by changing Gy. To achieve this, after a TR changes ΔGy, and then changes Δky, that is, ky within a TR is the same, which also explains why a set of signal values are sampled to fill one row of k-space.
reference
https://www.bilibili.com/video/BV1JJ411W7Fv
https://www.dushu.com/book/13293977/