【Q&A in MRI】5.1.4 Spatial Frequencies 空间频率

2020-10-30 CTMR技术园蒋强盛

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原著：Allen D. Elster, MD

译注：蒋强盛

What do you mean by spatial frequency? 你所说的空间频率是什么意思？

The concept of spatial frequency is fundamental to all aspects of imaging, including photography, television, general radiology, as well as MRI. You may have seen test patterns like the one pictured below used to evaluate the fidelity of your printer or computer monitor.

Image resolution is often described in terms of " line pairs per mm," the closest spacing of black and white bars that can be resolved. Since a pair of lines are like crests and peaks of adjacent waves, line pairs per mm is a measure of spatial frequency.

空间频率的概念对于所有成像技术都很重要，包括摄影，电视和普通 X 平片，磁共振成像也是如此。你可能见过如下图所示的测试图，用来评估打印机或者计算机显示器的保真度。

图像的分辨率经常用“每毫米线对”来表示，即所能分辨的最小的黑白条纹间隔。由于一组线对就好比波的相邻波峰，每毫米线对就是空间频率的一种测量方式。

Just as a 1-dimensional MR signal can be decomposed into a sum of simple sine and cosine waves, a 2-dimensional image can be decomposed into a set of planar waves varying in phase, frequency, amplitude, and direction.

正如一维磁共振信号可以分解成简单的正弦与余弦波之和；二维图像能够分解成一组不同相位、不同频率、不同幅度与不同方向的平面波。

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【译者注1】

由傅立叶变换可知，任何时域的一维时间变量信号都可以拆解成三角函数基之和。

同样，任何二维图像可以拆解成若干（二维的）平面波之和：

也就是图像由所有方向的不同幅度、不同相位、不同频率的平面波叠加而成，可以用求和公式示意如下：

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The planar waves must be considered from all possible directions: along the image x-axis (at 0º), along the image y-axis (at 90º), and from every other perspective (at 145º, −126º, etc.)

平面波的方向必须是所有方向都要考虑到：沿着图像 x-轴（0°），沿着图像的 y-轴（90°），和分别沿着其他的方向（145°，−126°，等等）。

"Egg-crate" pattern created by two planar waves offset by 90º.

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【译者注2】

由两个方向相互垂直的平面波（方向相差 90°）可以融合成一个装鸡蛋的条板曲面形式：

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As planar waves from more directions are added, the image begins to emerge.

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【译者注3】

当叠加的不同方向的平面波越来越多就越来越逼近想要的图像。

由于电脑经常重装系统，导致 Matlab 和 Mathematica 都不能运行，要不然可以多模拟几个实例。等下次软件重新装好了再模拟。

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All images can thus be decomposed using Fourier analysis into planar waves with various frequencies, phases, amplitudes, and orientations. This is easy to appreciate in the zebra photo, but holds true for images even without sharp boundaries. In fact, the human visual cortex contains cells that respond to sinusoidal spatial frequencies in different orientations. The k-space representation of an MR image is merely an organized collection of these spatial frequencies reflecting that image's spatial harmonic content.

通过傅立叶分析能够将所有图像都分解成频率、相位、幅度和方向不同的平面波。从斑马的照片当中很容易领会到其中真意，但它对于所有图像都是如此，即使没有分明边界的图像。事实上，人类大脑的视觉皮层含有能够分辨不同方向的正弦空间频率的细胞。 k 空间能够代表磁共振图像，只不过是因为它有组织地收集了能够反映图像空间频率的空间谐波分量。