(For the answer to the homework k-space diagram given at the end of Part Nine, see Note 1.)
K-space in two dimensions
As anyone knows who has encountered MRI professionally, whether in research or medicine, there seems to be an endless array of pulse sequences to choose between. The variety can be overwhelming at first. Nor is the situation helped by different vendors using different acronyms - we always use acronyms in MRI! - for what are essentially the same sequence.
It's little wonder, then, that most neophytes' eyes glaze over when it comes to comparing and contrasting any two pulse sequences if the taxonomy appears to be ad hoc. Where on earth to start? But it turns out that most pulse sequences can be categorized fairly easily, and their heritage traced, by separating the part(s) of the sequence that is responsible for spatial encoding, from the part(s) of the sequence that will provide the tissue or functional contrast. Occasionally there is overlap within the sequence of these two missions, but even then it's usually straightforward to understand the spatial encoding and interpret its genesis.
A useful pictorial representation of imaging pulse sequences
It turns out that there are only a handful of spatial encoding methods in common use these days, almost all with roots in the late 1970s or early 1980s. While new pulse sequences appear in the literature all the time, when you look at their k-space representations you'll be able to see how each new method has developed from a small number of key ideas from those early years. It's possible to categorize the encoding methods without k-space, but the k-space formalism makes comparisons trivial (in MR terms).
Spatial encoding methods can be separated into families derived from a central idea. For instance, following Lauterbur's original imaging paper in 1973 (which led to the family of projection reconstruction methods), in 1975 Richard Ernst's group came up with a sequence that utilized a 2D Fourier transform to yield the final image. (See Note 2.) It was a remarkable breakthrough and is the grandparent of nearly all medical/biological sequences still in common use today.
Still, even geniuses miss opportunities every now and then. And in 1980 a group at Aberdeen came up with a far more practical implementation of Fourier imaging, using amplitude-modulated gradients in a "constant time" pulse sequence, rather than the fixed amplitude, variable time scheme of Kumar, Welti and Ernst. It is this constant time scheme, which the Aberdeen group termed "spin warp" phase encoding, that provides the basis for most clinical (anatomical) scanning used today. It's also a good scheme to look at when first encountering 2D k-space, so we'll consider it in detail in this post.
The goal revisited
In the first part of the last post (see Part Nine) I used two examples of digital images to illustrate how the information content in a 2D plane of image pixels can be equivalently represented in reciprocal 2D space, or k-space. I mentioned that both the images and the k-space comprised 512x512 points, but later on when I started to draw (one-dimensional) k-space trajectories I did so on a k-space plane that was represented by just a set of axes, not discrete points. In case you think that image space and k-space in MRI are continuous, I'm going to spend a moment considering the digital k-space plane explicitly. (Like real space, k-space can also be continuous rather than digital, but that's not how MRI works.)
Here is a 16x16 plane of k-space points (see Note 3) overlaid on some actual signals to reinforce the point that we're digitizing a continuous process:
|Courtesy: Karla Miller, FMRIB, University of Oxford.|
The goal is to traverse the entire k-space plane, i.e. to use our gradients to follow a trajectory that crosses every single point (as defined by the white grid itself), acquiring data (with our receiver coil), one point for each grid coordinate, as we go. Once we have traversed the entire 2D plane (and assuming a suitable data acquisition scheme) we will have 16x16 k-space data points and will then be in a position to apply a 2D FT and get a 16x16 image out. (See Note 4.)