1. What is k-space? ? what it is vs how it works
If you were asked to explain how an automobile works, you could divide your answer into two parts. 1: what it is and what it does, and 2: how it works. An automobile is a box on wheels that gets you from one place to another under its own power, and under your control. The practical information is in your answer, but do you know how a car works? Not at all! K-space is a bit like this: the devil is in the detail. The challenge is in understanding phase encoding, and why we acquire MR data in many steps, and it’s not very intuitive. ?
what it is vs
how it works
If you were asked to explain how an automobile works, you could divide your answer into two parts. 1: what it is and what it does, and 2: how it works. An automobile is a box on wheels that gets you from one place to another under its own power, and under your control. The practical information is in your answer, but do you know how a car works? Not at all! K-space is a bit like this: the devil is in the detail. The challenge is in understanding phase encoding, and why we acquire MR data in many steps, and its not very intuitive.
However, it is worth persevering, because knowledge of the properties of k-space will be very rewarding in your understanding of MR techniques. The link between the data acquisition method and the effect on the image is an understanding of what happened in k-space.
We will start by considering what k-space is, and some of its properties. Then well get into the harder stuff: spatial encoding.
This presentation is occasionally highly animated in order to better represent concepts under discussion. If you are reading this on paper, know that you can find the tutorial somewhere on this website:
k-space
We will start by considering what k-space is, and some of its properties. Then we’ll get into the harder stuff in the second k-space tutorial: spatial encoding.
However, it is worth persevering, because knowledge of the properties of k-space will be very rewarding in your understanding of MR techniques. The link between the data acquisition method and the effect on the image is an understanding of what happened in k-space.
© D M Higgins

2. The most essential data is found in the centre of k-space.…
It is where we store our MR signals
It has a mathematical relationship to the image (Fourier transform)
K-space is simply an array of numbers. Those numbers have been translated into grey-scale values when you see k-space like this. Because those numbers represent a certain type of data, that array has interesting properties. An array with those properties is called k-space.
NB. we create the MR image from this data
what it is
We need to fill a lot of k-space, line by line, before we have enough data to create a good MR image
K-space is where we store our MR signals. K-space hasn’t been invented because of MRI, rather, the data we collect in MRI is the same data you find in what is referred to as k-space. It is simply an array of numbers, but because those numbers represent a certain type of data, that array has a number of interesting properties. An array with those properties is called k-space. Why k? Why space? Well get to that later in this tutorial!
The most essential data is found in the centre of k-space (find out why later).
The most essential data is found in the centre of k-space.
© D M Higgins

3. many radiofrequency signals
storing data in k-space
many radiofrequency signals
We have illustrated the digitisation of just one line in k-space. In fact many MR signals need to be acquired under different imaging conditions. This is why the acquisition of MR data can take some time.
The different samples (numbers) are illustrated here by using different grey-scale values.
This is a schematic representation of one of the many MR signals we need in order to create a single image. We sample the amplitude of the signal and put these numbers into a list. This is digitisation.
The radiofrequency signals that we measure are digitised before being stored in the raw data space (k-space). It’s just a list of numbers!
K-space is just an array of complex numbers, which is why we can assign grey-scale values to those numbers and plot k-space like an image, as on this slide.
[Note that this is a magnitude image of k-space, such that each data point plotted is equal to √(Real2+Imaginary2). The phase angle of each point in k-space may be calculated using φ = tan-1(Imaginary/Real).]
“ space”
raw data
This array of numbers in k-space is sometimes referred to as raw data space. That’s just what it is!
© D M Higgins

4. k-space is an array of numbers
whose Fourier transform is the MR image.
It’s impossible to guess what the image is going to be of, but there are some hints about the quality of the image one may expect, from looking at the raw data. Read on…
Here are some examples of k-space. Notice how it can differ quite considerably from one to the next. Remember that there is no relationship between a value at a point in k-space and the same point in the MR image. In fact, every point in k-space contributes information to the whole MR image. This is a difficult concept, and you will need to understand spatial frequencies and the Fourier transform to understand why. That comes later in this tutorial.
The images are:
(1)knee image, (2) test tubes with gels, (3) a single image from a dynamic perfusion study, (4) some fruit.
The movie shows the knee image k-space in 3D. It animates a magnitude plot of k-space. (That is to say, the points plotted are equal to √(Real2+Imaginary2). Remember k-space is complex data. More on why that is later.)
Advanced:
Can you see any reduced scan percentage in the k-space images? Which k-space array contains more noise? Why?
Here are some different k-spaces. You can see that the information they contain can vary quite a lot. Some are noisy data sets or have clear areas of higher signal (e.g. the third). This is partly to do with how we acquire the data, partly because of the nature of what were scanning, and partly to do with the nature of k-space.
Click the image for 1MB movie of k-space (magnitude images).
NB!! If the movie plays in a new window, press Alt-F4 to return to the tutorial.
© D M Higgins

5. Almost completely (180°) out of phase
The k of k-space
2 wavelengths per metre
k = 2 m-1 1m
wave number
Stationary plane waves may be characterised by an amplitude and a wave number. The term wave number refers to the number of complete wave cycles that exist in one metre of linear space (units: cycles m-1). The wave number is traditionally denoted by the letter k.
3.25 wavelengths per metre
k = 3.25 m-1
The spectrum of frequencies is described by a function on k-"space". K-space is the space (or plot) of all possible wave numbers. Plane waves with identical wave numbers (same frequency) may still differ with regard to their phase. For this reason, the data on k-space is complex data, so that the phase information is included. This is why k-space is two-dimensional and not just a line of possible wave numbers!
out of phase
Stationary plane waves may be characterised by an amplitude and a wave number. The term wave number refers to the number of complete wave cycles that exist in one metre of linear space (units: cycles m-1). The wave number is traditionally denoted by the letter k. [I realise the next question might be: why is wave number denoted by "k"? !! I can't answer that one except to say "convention"!] Wave number is related to the frequency of a wave by the speed at which it travels. The spectrum of frequencies is described by a function on k-"space". K-space is the space (or plot) of all possible wave numbers. Plane waves with identical wave numbers (same frequency) may still differ with regard to their phase, and the data on k-space is complex data, so that the phase information is included. This is why k-space is two-dimensional for a 2D image, and not just a 1D line of possible wave numbers! We measure MR echoes which are functions made up of a combination of plane waves of certain frequencies (or wave numbers), and phases. Conceptually, k-space is infinite, but we concern ourselves only with the frequencies which we have used in the spatial encoding of the data. You might see k-space referred to as spatial-frequency space, or Fourier-space in some contexts.
Almost completely (180°) out of phase
k-space is the plot of all possible wave numbers (it’s a k-plot!)
Almost in phase
© D M Higgins

6. The movie shows the knee image beginning to form.
Don’t believe it? Click on the image below to play a movie [0.1MB, MPEG4 codec required]. Only 81 of the spatial frequencies are added in. (They are the lowest spatial frequencies, the 9*9 most central dots in k-space.)
NB!! If the movie plays in a new window, press Alt-F4 to return to the tutorial.
Just as the three dots illustrate here, every point in k-space represents a wave in the image: a different spatial frequency at a different angle. The brightness of the wave in the image depends on the brightness of the dot / the value of the number at that point in k-space!
In fact, every value in k-space represents a wave like the ones seen here! If we superimpose all these waves at their various frequencies, angles and brightness, we get the image of the knee.
I have increased the value of the number at this point in k-space. It is now represented by a much brighter dot. Look at the effect on the image! Large undulating bright and dark areas. [An RF-noise spike might cause such an artefact in real life.]
This time the bright dot is a little further out in k-space. Notice how the frequency of the wave in space (in the image) is increased.
If we introduce extra data into the k-space of the knee image we can note the effect on the image. This helps us understand exactly what the k-space data really is. (I’m sure you’ve heard the term spatial frequencies before, but what are they really?!) In the first three pairs of images a light dot has been inserted into k-space and the knee image has been re-created. A light dot is just a high number in the array of numbers. You can see that depending on the location of the dot, a different set of lines appear on the image. Do you see the difference between each set of lines? The difference is in the frequency of their light and dark areas in space, isn’t it? Shall we call these frequencies spatial frequencies?
You’re getting closer to what k-space really is! All the individual numbers (or dots in the k-space image) represent a different wave at different angles. The value of those numbers (brightness of the dots) dictate how bright the wave that that point represents should be plotted on the MR image. Now were getting down to the point: an MR image is simply (!) the superposition of thousands of these waves, all plotted on the same image, over top of one another. When you add them all together, you get the image (like the knee in this example). It sounds improbable, but its true.
The last two pairs of images on this slide show what happens if you miss out certain spatial frequencies. The first misses out the wide spatial frequencies, which are in the middle of k-space. (This is a low-pass filter, allowing only low spatial frequencies. The rest of the array is filled with zeroes.) You’ll notice that the re-created knee image lacks the fine detail of the original. Most of the contrast of the image remains however. This is because it is not possible to create small, detailed structures without using high (narrow) spatial frequencies. Similarly, with the second pair of images the low spatial frequencies have been omitted and the high spatial frequencies only remain. In this case it is only possible to re-create the fine detail (edges, etc) in the knee image. The contrast information has been lost because it is not possible to create the big contrast information (i.e. large general areas of light and dark) without low (wide) spatial frequencies.
Now we need to connect up your understanding of spatial frequencies with spatial encoding. Lets see how we get the data.
[All images and k-space have been prepared for this tutorial using IDL. See for more info.]
Moving the dot further out increases the spatial frequency again. Notice that the angle of the wave in the image is at right angles to a line joining the dot and the centre of k-space.
The movie shows the knee image beginning to form.
© D M Higgins

7. Large areas of bright and dark (i. e
Large areas of bright and dark (i.e. contrast information) cannot now be drawn on the image because this cannot be done without large bright/dark waves: low spatial frequencies. The image has only the edges and detail information.
The resultant image contains mainly contrast and little detail / edge information. This makes sense, because we need to add high frequency waves to be able to draw fine-detail.
Here we have used only the low spatial frequencies to create the image. Remember, these data points in k-space produced only large waves on the image.
Now we are using only higher spatial frequencies, and the lower spatial frequencies have been omitted.
© D M Higgins

8. …this is the point of no return!
To learn why k-space has these properties we have to discuss
Keep going!
Spatial Encoding
<< see the How K-Space Works tutorial >>
© D M Higgins