MRI and k-space

MRI images are created from raw data contained in a raw data space, called k-space. This is a matrix where MR signals are stored throughout the scan. K-space is considered a bit of a tricky topic, so I will only outline a brief explanation of what k-space is and how it relates to the MR image.

Kspace.pngThis is a visualisation of k-space.

K-space is what? The first thing to recognise is k-space is not a real space. It is a mathematical construct with a fancy name – a matrix used to store data. The data points are not simple numbers, they are spatial frequencies. Unlike ‘normal’ frequencies, which are repetitions per time, spatial frequencies are repetitions per distance. They are, in other words, waves in real space, like sound waves. We usually measure them as cycles (or line pairs) per mm. The number of cycles per mm is called the wavenumber, and the symbol used for wavenumber is ‘k’ – hence k-space.

It may be easiest to envision these spatial frequencies as variation in the brightness of the image (or variation in signal spatial distribution, if we are to get more technical). So we have a large matrix of brightness variations, which together makes up the complex image. The matrix is full when the image is completed (i.e. the scan is done).

kspace.jpg

K-space relates to the image how? The second thing to recognise is that k-space corresponds (although not directly) to the image. If the image size is 256 by 256, then the k-space will have 256 columns and 256 rows. This doesn’t mean, however, that the bottom right hand k-space spatial frequency holds the information for the bottom right hand image pixel. Each spatial frequency in k-space contains information about the entire final image. In short, the brightness of any given ‘spot’ in k-space indicates the amount that that particular spatial frequency contributes to the image. K-space is typically filled line by line (but we can also fill it in other ways).

Frequencies across k-space. Typically, the edges of k-space has high spatial frequency information compared to the centre. Higher spatial frequency gives us better resolution, and lower spatial frequencies better contrast information. Think of it like this: when you have abrupt changes in the image, you also get abrupt variations in brightness, which means high spatial frequencies. No abrupt changes means lower spatial frequencies. This in effect means that the middle of k-space contain one type of information about the image (contrast), and the edges contain another type of information (resolution/detail). If you reconstruct only the middle of k-space, you get all the contrast/signal (but no edges) – like a watercolour – and if you reconstruct only the edges of k-space, you get all edges (but no contrast) – like a line drawing. Put them together, however, we get all the information needed to create an image.

watercolour-1578069_960_720.jpg

Transformation: Notes and chords. The route from k-space to image is via a Fourier transform. The idea behind a Fourier transform is that any waveform signal can be split up in a series of components with different frequencies. A common analogy is the splitting up of a musical chord into the frequencies of its notes. Like notes, every value in k-space represents a wave of some frequency, amplitude and phase. All the separate bits of raw signal held in k-space (our notes) together can be transformed into the final MRI image (our chord, or perhaps better: the full tune). One important aspect of the Fourier transform is that it is not dependent on a ‘full’ k-space. We may leave out a few notes and still get the gist of the music, so to speak.

Cutting the k-space. We have types of scan that only collect parts of k-space. These are fast imaging sequences, but they have less signal to noise. Less signal to noise means that the amount of signal from the object being measured is reduced compared to the noise that we encounter when we image the object. We always get a certain amount of noise when imaging: from the scanner, the environment and sometimes also the object being imaged. A low signal to noise ratio is typically bad for image quality. Nevertheless, we can get away with collecting fewer lines in k-space when we image something relatively symmetrical, like a human brain. This allows us to mathematically fill in the missing lines in k-spaces, from an assumption of symmetry. Some fast scan sequences sample only every second line in k-space, and we can still reconstruct the image from the information gathered.

The problem of motion. If there was any motion during the scan, there will be inconsistencies in k-space. The scanner does not ‘know’ that the object moved, so it tries to reconstruct the image from data that is not consistent. This means that the reconstruction via the Fourier transform will be flawed, and we can get signal allocated to the wrong place, signal replicated where it should not be or signal missing altogether. We see this as distortions or ‘artefacts’ in the final image. As the data is collected over time, the longer the collection time, the more vulnerable the scan is to motion. Data collected in the phase encoding direction (which takes a relatively long time, see my previous post on gradients) is therefore more vulnerable than data collected in the frequency encoding direction. In a later blog post, I will discuss how we can use motion measurements to adjust the spatial frequencies k-space in a way that removes the effect of the movement. This is partly possible due to us not needing the full k-space to do a Fourier transform and get the image, as described above.

Summary. The take home message on k-space is as follows:

  1. K-space is a matrix for data storage with a fancy name.
  2. It holds the spatial frequencies of the entire scanned object (i.e. the variations in brightness for the object).
  3. We can reconstruct the image from this signal using a Fourier transform.
  4. The reconstruction can still work if we sample only parts of k-space, meaning that we can get images in much shorter time.
  5. Problems during data collection (i.e. motion) may cause errors in k-space that appears as ‘artefacts’ in the image after the Fourier transform.

 

Caveat. There is more to k-space than what I have described above, as a medical physicist would tell you. While it is good to have an appreciation of what k-space is and is not, a detailed understanding is generally not needed for those of us simply using MRI in our work rather than working on MRI. This post is meant as a brief(ish) introduction, not a detailed explanation, and contains some simplifications.

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2 thoughts on “MRI and k-space

  1. Really interesting post once again, thanks for taking your time writing it. I had to read this couple of times to wholly digest it. But it’s really interesting to get glimpse how complex the whole sequence actually is (I sort of thought that it must be complex, but couldn’t really understand how bunch of magnets and computer make an MRI image or three dimensional volume even).

    Liked by 1 person

    • Glad you liked it, and I hope it seems less complex now. While the basic principle underlying MRI is lovely and straight-forward, K-space is trickier. The things I covered in the post is actually pretty close to the level of detail given in the graduate MR course I attended for my first post doc. Most of us never went beyond that.

      Medical physics and data processing aside, MRI is also fascinating from an engineering point of view. I get to help with MR maintenance at work, which allows me glimpses of the inner workings of the machine (the mechanics, not the code). It’s pretty exciting stuff!

      Like

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