【Q&A in MRI】5.1.7 Signal and Spatial Frequency 信号与空间频率（1）

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原著：Allen D. Elster, MD

译注：蒋强盛

I don't understand how you can simply plug the digitized MR signal data directly into k-space. How are these points the same as spatial frequency?

我不太理解为什么你能够简单地将数字化的磁共振信号直接填充在 k 空间？这些数据点是如何跟空间频率一致的？

When a frequency-encoding gradient is applied during evolution of the MR signal, successive data points in the echo (or FID) reflect progressively increasing spatial frequencies. This allows the sampled raw data from the MR signal to be plugged directly into the k-space matrix.

In other words, each data point of the MR signal is already in "Fourier format." I call this surprising fact the "Miracle" of k-space! Why this miracle occurs is not intuitively obvious and will require some explanation.

换句话说，磁共振信号的每一个数据点已经是“傅立叶格式”了。我称此令人惊讶的事实为 k 空间的“奇迹”！为什么这一奇迹会出现却并不怎么直观明了，还需要做一些解释。

The "Miracle" of k-space

由于信号读出时在横轴方向施加了一个散相梯度 G(x) ，那么 FID 信号就经历了一个加速散相的过程。G(x) 可以是频率编码梯度也可以是相位编码梯度。为了简化讨论，我们假设 G(x) 随着位置 (x) 呈线性变化，遵循公式 G(x) = G • x, 其中 G 的单位是毫特斯拉每米 mT/m 或其他等效单位。

While G(x) is being applied, the total magnetic field at a given location is B(x) = Bo + G(x). The corresponding resonance frequency at location x is therefore 

当施加 G(x) 的时候，某点处的总的场强为 B(x) = Bo + G(x). 那么在点 x 处的的共振频率为：

f(x) = γB(x) = γBo + γG(x) = fo + γGx.

where γ is the gyromagnetic ratio and fo is the Larmor frequency of the main magnetic field (Bo). Under action of the gradient, the resonant frequencies increase from left-to-right across the image as long as the gradient is being applied. Once the gradient is turned off, the resonant frequencies all revert to fo.

其中 γ 为磁旋比， fo 为主磁场 (Bo) 所对应的拉莫尔频率。只要梯度打开，那么在梯度的作用下，共振频率从左到右逐渐升高。一旦梯度关闭，所有位置的共振频率又将会回到原来的相同的共振频率 fo.

While the gradient is applied, protons in the higher parts of the field will precess more rapidly and gain phase compared to those in lower parts of the field. This phase shift persists even after the gradient has been turned off. Because phase = frequency x time, the phase gain is directly proportional to the length of time (t) the gradient is applied. As a function of position (x), the phase gain is given by

当梯度施加的时候，处于共振频率高处的质子进动频率将更快，并且相较于低频处的质子将获得相位。这种相位偏移当梯度关闭后仍然保留。由于相位 = 频率 × 时间，获得的相位正比于梯度所持续的时间 (t) 。获得的相位的大小是位置 (x) 的函数：

ϕ(x,t) = (γGx) • t = (γGt) • x = kx(t) • x

where kx(t) = γGt. This is the same "kx" of k-space fame, expressing phase cycles per unit distance along the x-direction. 

其中， kx(t) = γGt. 这跟 k 空间中的 "kx" 是一样的，表示 x-方向的单位距离内相位周期数。

Another way to think of k(t) is as the gyromagnetic ratio (γ) times the area under the gradient (G) curve at time (t). Even though we have assumed a linear, rectangular gradient waveform varying along the x-axis in our simplified example above, the same definition holds for any arbitrary gradient shape and direction (r).  Specifically, 

另一种表示 k(t) 的方式是磁旋比 (γ) 乘以 (t) 时间内的梯度 (G) 曲线下的面积。即使在上面的例子中我们假设了梯度是沿着 x 轴的线性矩形梯度波形，这个公式同样适用，它适用于任何形状与任何方向 (r) 的梯度。用公式表示如下：

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【译者注】

When t = tearly, kx is small and the spread of phases across the image is very small. 

当 t = tearly（信号采集早期）, kx 的值很小，整个图像不同位置的相位差很小。

When  t = tlate, kx is large and many phase cycles encompass the image. 

当 t = tlate（信号采集后期）, kx 的值很大，整个图像不同位置的相位差许多相位周期。

The resultant MR signal at each point in time (t) thus reflects increasing spatial frequencies and the summed phase angles from all locations in the image. Hence each successive point along the MR signal reflects progressively higher spatial frequencies whose values can be used to "miraculously" fill k-space cells directly.

最终，不同时刻 (t) 采集的磁共振信号数据点反映了逐渐增加的空间频率，和图像中所有位置的相位角总和。因此，相继采集的磁共振信号数据点反映了逐渐增高的空间频率，它们“神奇地”可以被用来直接填充 k 空间单元。

If the above discussion was too mathematical, perhaps the example below will provide some additional insight. Here two objects, one of uniform density (blue) and the other of periodic density (red, like a fence), have been subjected to the same gradient field. We assume this gradient has induced a set of phase shifts (green) across both objects, whose spatial frequency happens to match that of the red object.

Because the uniform (blue) object extends over multiple phase cycles, no signal would be observed as magnetization vectors would all cancel. In other words, for each pixel in the uniform object with phase shift φ, it would be easy to find another pixel with phase shift −φ. Conversely, the red object would generate a strong signal at this phase encoding, as the spatial frequency inherent in the object would exactly match that produced by the gradient. 

由于多个相位周期内蓝色物体密度都是均匀的，那么由于磁化矢量全部抵消将探测不到信号。换句话说，密度均匀的蓝色物体中的每一个像素如果它的相移为 φ，那么很容易找到另一个像素，它的相移为 −φ. 相反，由于红色物体内在的空间频率与梯度所产生的空间频率十分匹配，那么在这个相位编码梯度作用下，红色物体将产生一个十分强的信号。

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