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What is k-space? what it is how it works ? vs k-space 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. 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 we’ll get into the harder stuff in the second k-space tutorial: spatial encoding. © D M Higgins

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what it is … 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. The most essential data is found in the centre of k- space. © D M Higgins NB. we create the MR image from this data It has a mathematical relationship to the image (Fourier transform) We need to fill a lot of k-space, line by line, before we have enough data to create a good MR image It is where we store our MR signals

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“ space” storing data in k-space many radiofrequency signals raw data 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. 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 array of numbers in k-space is sometimes referred to as raw data space. That’s just what it is! © D M Higgins

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k-space is an array of numbers whose Fourier transform is the MR image. 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. 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… © D M Higgins 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.

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1m 2 wavelengths per metre 3.25 wavelengths per metre k = 2 m -1 k = 3.25 m -1 out of phase k-space is the plot of all possible wave numbers (it’s a k-plot!) The k of k-space 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. Almost completely (180°) out of phase Almost in phase 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! © D M Higgins

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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. 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. 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. 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. The movie shows the knee image beginning to form. © D M Higgins

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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. 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. Now we are using only higher spatial frequencies, and the lower spatial frequencies have been omitted. 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. © D M Higgins

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

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