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The K of K-Space

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Why is k-space called k-space?

The Fourier series describes a given function as an infinite linear combination of stationary plane waves, each of which 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. (It has also been suggested that wave number was denoted by "k" because of early NMR research in crystalline lattices: "kristall-raum" meaning "crystal space" in German.) 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.

Further reading on this topic:
Books: Mezrich R. A perspective on k-space. Radiology 1995;195:297-315.

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